Partial differential equations: Theory and numerical solution
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Partial differential equations Theory and numerical solution

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at Stony Brook B. Moodie, University ofAlberta S. Mori, Kyoto University L.E. Payne, Cornell University D.B. Pearson, University of Hull I. Raeburn, University of Newcastle G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University ofAdelaide

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Charles University, Prague

(Editors)

Partial differential

equations

Theory and numerical solution

0

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CONTENTS Preface Arkhipova A. A.: On the Global in Time Solvability of the CauchyDirichlet Problem to Nondiagonal Parabolic Systems with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . Bendali A.: Boundary Element Solution of Scattering Problems Relative to a Generalized Impedance Boundary Condition . . . . . . . . Benes M.: Mathematical Analysis of Phase-Field Equations with Gradient Coupling Term . . . . . . . . . . . . . . . . . . . Berestycki H.: Qualitative Properties of Positive Solutions of Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . Bey R., Loheac J.-P., Moussaoui M.: Nonlinear Boundary Stabilization of the Wave Equation . . . . . . . . . . . . . . . . . . Boisgerault S., Zolesio J.-P.: Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow . . . . . . . . . . . . . . . Cagnol J., Zolesio J.-P.: Second-Order Shape Derivative for Hyperbolic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . Cao F., Caselles V., Morel J.-M., Sbert C.: An Axiomatic Approach to Image Interpolation . . . . . . . . . . . . . . . . . . . . Carmona J.: A Class of Strong Resonant Problems via Lyapunov-Schmidt Reduction Method . . . . . . . . . . . . . . . . . . . . . Cioranescu D., Girault V., Glowinski R., Scott L.R.: Some Theoretical and Numerical Aspects of Grade-Two Fluid Models . . Da Prato G.: Large Asymptotic Behaviour of Kolmogorov Equations in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . Frehse J., Malek J., Steinhauer M.: On Existence Results for Fluids with Shear Dependent Viscosity- Unsteady Flows . . . . . . . Gallay Th., Raugel G.: Stability of Propagating Fronts in Damped Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . Hagen T., Renardy M.: On the Equations of Melt-Spinning in Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Hiinlich R., Glitzky A.: On Energy Estimates for Electro-Diffusion Equations Arising in Semiconductor Technology . . . . . . . . Jager W., Mikelic A.: On the Boundary Conditions at the Contact Interface between Two Porous Media . . . . . . . . . . . . . Jochmann F.: On the Semistatic Limit for Maxwell's Equations . . . . Kacur J.: Application of Relaxation Schemes and Method of Characteristics to Degenerate Convection-Diffusion Problems . . . . . . .

1 10 25 34 45 49 64 74 89 99 111 121 130 147 158 175 187 199

Kawohl B.: Symmetrization or How to Prove Symmetry of Solutions to a PDE. . . . . . . . . . . . . . . . . . . . . . . . . . . Knobloch P., Tobiska L.: A Bubble-Type Stabilization of the Ql/QlElement for Incompressible Flows . . . . . . . . . . . . . . . Koch H.: Instability for Incompressible and Inviscid Fluids . . . . . . Lavrentiev M. Jr., Spigler R.: The Kuramoto-Sakaguchi Nonlinear Parabolic Integrodifferential Equation . . . . . . . . . . . . . Meister E., Passow A.: Scattering of Acoustical and Electromagnetic Waves by Some Canonical Obstacles . . . . . . . . . . . . . . Moisan L.: PDEs, Motion Analysis and 3D Reconstruction from Movies Pokorny M.: Steady Flow of Viscoelastic Fluid Past an Obstacle Asymptotic Behaviour of Solutions . . . . . . . . . . . . . . Raitums U .: Properties of Optimal Control Problems for Elliptic Equations Sanfelici S.: On the Galerkin Method for Semilinear Parabolic-Ordinary Systems . . . . . . . . . . . . . . . . . .

214 230 240 248 254 273 283 290 298

Schweizer B.: Modelling the Dynamic Contact Angle Serre D.: £ 1-decay and the Stability of Shock Profiles Tadie: On Positive Solutions of the Equation !J.U + f(ixi)UP = 0 in Rn, n>2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 312

Velazquez J.J.L.: Singularity Formation for the Stefan Problem Wei J .: On the Construction of Interior Spike Layer Solutions to a Singularly Perturbed Semilinear Neumann Problem . . . . . . .

329

322

336

Preface The present volume constitutes the Proceedings of the conference Partial Differential Equations- Theory and Numerical Solutions. The conference took place in August 10-16, 1998, in Praha. As part of the celebration of the 650th anniversary of the foundation of Charles University, it was organized under the auspices of the Charles University, Praha and the Ruprecht-Karls-Universitaet, Heidelberg. The conference was included among the satellite conferences of the International Mathematical Congress ICM'98 in Berlin. With its rich scientific programme, the conference provided an opportunity for almost 200 participants to gather in a convenient environment and to discuss the present status of several important directions in which partial differential equations have been developing recently. The invited plenary speakers, as well as the invited speakers in sections, were carefully chosen by the members of the Programme Committee: H. Berestycki, M. Feistauer, B. Fiedler, A. Friedman, H. Gajewski, M. Golubitsky, J. Haslinger, G. C. Hsiao, T . J . R. Hughes, J. Kaeur, D. Kroener, J . M. Morel, R. Rannacher, M. Renardy and I. Vrkoe. The care with which they performed their important task cannot be overestimated. The scientific programme of the conference consisted of 15 plenary lectures and about 40 section lectures. Together with this invited part of the programme there were about 110 other presentations (short communications and posters). This book consists mostly of the papers submitted by the invited speakers. They are completed by a limited number of the contributions based on the short communications. (In selecting these complementary papers, the articles containing new results were preferred to the survey papers.) The task of organizing the material for publication has been made easy by the thoroughness with which each author has converted the spoken to the written word. We hope that this volume will help to stimulate further research. We are grateful to CRC PRESS (U.K.) LLC publishing house for its prompt and efficient publication of the proceedings. The organizers take pleasure in expressing their thanks to the participants not only for the interesting talks but also for their contribution in creating the friendly and stimulating atmosphere of the conference. They also want to thank Charles University, Prague, for its moral support and Ruprecht-Karls-Universitaet, Heidelberg for its important contribution. We extend our thanks to our colleagues from the Institute of Mathematics of the Academy of Sciences of the Czech Republic for their substantial help with the organization. We cannot imagine how the conference could have been organized without the collaboration of the Prague School of Economy (conference building, perfect service and first class equipment) and the Institute of Chemical Technology (student residence). Last but not least, we should like to express our gratitude to the brewery in the Moravian town of Litovel for sponsoring the conference party by providing excellent beer.

On the Global in Time Solvability of the Cauchy-Dirichlet Problem to Nondiagonal Parabolic Systems with Quadratic N onlinearities A. A. ARKHIPOVA Abstract. In the case of two spatial variables the global solvability for nonlinear parabolic systems with quadratic nonlinearities in the gradient is proved. We assume that elliptic operator of the system has variational structure. The constructed solution is smooth almost everywhere and has at most a finite number of singular points. Keywords:

boundary value problems, nonlinear parabolic systems, solvability

Classification:

35J65

Let n be a bounded domain in lRn In ;::: 2, Q = n X (0, T), T arbitrarily. Let u: Q H RN, N > 1, be a solution of the problem Ut-

Lu = 0

in Q,

> 0 be fixed (1)

Here L is a nonlinear elliptic operator of the second order with quadratic nonlinearity in the gradient; 8'Q is the parabolic boundary of Q. First, we suppose that L is a quasilinear operator, in such case local in time existence of classical solution of (1) was proved by H. Amann (1] and M. Giaquinta and G. Modica [7]. The existence of a solution on any time interval [0, T] has not been studied yet. For the special case when L is the operator of variational type and n = dim[} = 2, the author has constructed global in time almost everywhere smooth in Q solution of (1). Now we shall discuss this result and some of its generalizations. Further we assume that dim[}= 2. To describe L we consider the functional

t'f[v]

=~

L

(A(x, v)v:a: · v:a:}dx,

where the integrand N

2

1 "" afJ (x,v)ppp I 1c , f(x,v,p) = 2(A(x,v)p · p} = 21 "" L...,L...,Aicl 0 k,l

a ,~

(2)

A. A. Arkhipova

2

Vx E il,

v E lRN,

Vz = {v:., }k~N.

2

p E R N;

We suppose that the matrix A = {

A:f}

a :52

satisfies the following conditions on the

M=ilxlRN:

sup M

functions A:f {til' .. . ' vN),

IIA(x, v)ll ~ 1-'i

are twice differentiable with respect to

sup

L L

x =

(x1, x:~)

(3)

and

I(A:f(x,v))' I= h < +oo,

Mk,I~N a ,ti,'T:52 z., sup L L I(A:f(x,v))' I= a< +oo, M

sup M

k,l,m~N a,tJ:52

L

v=

L I(A:f(x, v) )"

k,l,m,.~N a,tl~2

(4)

vm

vmv•

I

= l2

< +oo.

Let L = {£Ckl}k:5N be the Euler operator for Et, then system (1) has the form

u~- (A:f(x,u)u~~)z., +~(A:f:(x,u))~.u~~u~ =0,

k=

1, ... ,N. (5)

System (5) is the quasilinear parabolic system with quadratic nonlinearity in the gradient:

(we do not suppose any smallness condition on the parameter a). For some fixed oo E (0, 1) and t1, t2 E [0, T] we define the set

.K:{(tt, t21} = { U:

[}X

(tt. t2]

~ JRN

lu, U:a: Ut, Uz:z E 0° ([}X (t1, t2); 6) 1

U:a:t E £ 2 ( [}X (t1,t2)) }• and write v E .K:{[t1, t2) }, if v E .K:{[t1, r]}, We denote by 6 usual parabolic metric;

0

1

(6)

On the Global in Time Solvability of the Cauchy-Dirichlet Problem to Nondiagonal . . .

3

If u is a solution of (1), (5) then £,[u] is called the energy of the problem and

is called the local energy of the problem on the set OR(x 0 ) = [} n BR(x0 ), x 0 E 7i, at the moment t (BR(x 0 ) is a ball of radius R in JR2 centered at x 0 ). In [2] the following result was proved. THEOREM 1 (Th. 0.3 [2]). Let[} be a bounded domain in JR2 with C2+ao_ smooth boundary 80 and L be the operator defined by {5}, ,P = ,P(x). If conditions 0

(.1), (4) hold then for any T > 0 and 1/J E W~(O) there exists a global solution of the problem {1} u : Q H JRN, which is smooth on [} x (0, T]\E and the singular set E consists of at most finite many points (xi, ti), j ~ M. For some number e- 0 > 0 (depending on the parameters of the problem and Cl+l_ characteristic of iJ[}) and every point (xi, tJ) E E the following inequality is valid:

Moreover, (1) u E W(Q) =

{v E

0

L 00 ((0,T);W~(O)), 3vt E L 2 (Q)},

and

sup£,[u(t)] ~ Et['I/J];

(O,T)

(2} u is the unique solution of {1} with the pointed properties; (3) u is a weak solution of {1} from the class W(Q) in the following sense:

k(

UtiJ

+ (A(x, u)u.., · IJ..,} + ~(A~(x, u)u.., · u..,}IJ) dQ = 0, 0

v71 E L 2 ({0, T); W~(O}) n L1 ({0, T), L

00

(0));

(4) u(·, t)-+ 1/J in Wi(O) as t-+ 0. To prove this result we followed M. Struwe's idea of the construction of heat flows for harmonic maps in a two-dimensional case [11]. Our considerations were based on two important facts: 1} the result on the local classical solvability of (1} mentioned earlier and 2) some continuation theorem which is the main analytical result of (2]. Now we formulate the continuation theorem.

4

A. A. Arkhipova

THEOREM 2 (Th. 0.2 [2]). Let conditions (3}, (4) hold, an E C2+ao,,p E c2+ao(n),t/J vanishes on an with its first and second derivatives. Assume that u E K:{[O, T0 )} is a solution of (1}, (5} for some To > 0. There exist co > 0 and Ro(co) > 0 such that if sup sup E,[u(t), nRo(x0 )] 0.

(9)

In the next steps we do not use the variational structure of L. Step 2. With the help of the smallness energy condition (7) we derive that

11Uzzii2,Q0 + lluzii4,Q0 $ kt.

(10)

with the constant k 1 depending on v, J.', l1o l2, a and Cl+ 1 - characteristic of an. To get (10) we straighten a part of an and deduce the estimate like (10) in local coordinates. Here and later on we estimate the strong nonlinear term in (5) by using well-known multiplicative inequality ([8], Ch.1): 0 1

.

'v'w E W 2 (D), d1mD = 2.

(11)

Step 3. Now finiteness of the integral J = JQo luzl 4dQ and inequality (11) allow to get the following result. There exists t1 E (0, To) such that for any 'Y < -y1 = ;',. and Tt ~ t1

[~'1.,)

0

'ldx + 1~

2 2

fniut(x, t)l +

In

2 2 3 2 (1uztl lutl " + lutl + 'l) dxdt $

$Co fniut(x, Tt)l2+

"dx,

2

(12)

On the Global in Time Solvability of the Cauchy-Dirichlet Problem to Nondiagonal . . .

5

with eo, t1 depending on v, J.', a, l2 , in addition, t 1 depends on the integral J. Step 4. There exist t2 E (t1, T0 ) and 'Y2 E (0, 'Yt) such that sup lluz(x, t)I2+ 2'Ydx [.,.,,To) n

~ c[lTo .,.,

1 n

2 2 + fniuz(x, r2)l + -rdx],

(luti 2+'Y + luzl 2+2-r)dxdt+

Vr2 ;::: t2, 'Y

~ 'Y2

(13) 1

where c and t2 depend on v, J.', h, a, Cl+l_ characteristic of an, and in addition, t2 depends on J. First, we derive this inequality in the local coordinates by attracting (11) once more. From (13), (12) and (10) it follows that sup lluz(·, t)ll2+2-y,!l ~ k2 (1 + lluz(·, r2)i12+2-y,!l + llut(·, r2)1i2+2-y,n) [.,.,,To)

= mo,

(14)

k2 depends on the same parameters as k1 • From now on we fix r• E [t2, To), Q. = ilxA., A.= (r*,To). As lluz(·,To)ll,,n ~ illnt/'To llu,(·, t)ll,,n ~ mo, p = 2 + 2'Y > 2, and due to the imbedding theorem we have the estimate: supllu(·, t)llc•ocn) ~ m1, A.

6o= -'Y- . 1+'Y

(15)

(All mi, i ;::: 1, depend on the same values as m 0 .) Step 5. From (15) and the first estimate iii (8) it follows ((10], Lemma 4) that (16)

'Y is fixed in (14). Step 6. Let us look now at the system (5) as at the elliptic one for some fixed t E [r*,To):

-(A;f(x, u)u~II)Zo + bk(x, u, u,) = 0,

X

En,

lb(x,u,p)i ~ ~IPI 2 +4>(x,t),

uklan = 0,

k ~ N,

4>(x,t) = Ut(x,t),

(17)

from (12) it follows that ll4>(·,t)ll 9 ,n ~ const, Vt E (r*,To), for some q > 2. To advance in estimating sup llu,(·,t)llc•{i)") with some (J > 0, we consider (17) A.

in local coordinates and use the freezing coefficients method. First of all we prove that the derivatives Uz(·, t) are uniformly bounded on A. in the norm of Morrey

6

A. A. Arkhipova

> 0. It gives us estimate (15) with any 60 < 1. Then we improve this result deriving the inequality

space L 2 •2 -1!(fJ), 'Ve

sup lluz(·, t)ll~2.2+2'(n) $ const, A.

with some (} E (0, 1). At this step our considerations are close to papers [3,6]. According to the property of Campanato space .C2 •2+28 (fJ) the last inequality gives that sup lluz(·, t)llc•(n) $ m3. (18) A.

Step 7. From (18) and (16) it follows ([9], Ch.2, Lemma 3.1) that for some

f3o E (0, 1)

(19)

Now we can attract the linear theory for parabolic systems to assert that uEK:{[O,Tol}. D Remark 1. The monotonicity formula (8o) guarantees that u(·,tm)-+ u 00 (·) weakly in W:}(n) as some sequence tm -+ oo. There arise two opportunities: 1) all singularities appear in some finite interval (0, T], and 2) the solution has singular points at the infinity. In the first situation u(·, tm) -+ u 00 (-) in Wi(fJ) and u 00 appears to be smooth extremal of £,. Moreover, if Et[t/J] is small enough then Uoo = 0 inn (uoo is the minimizer of£,). In the second situation the solution u has a finite number of singular points x 1 , ... , xM at the infinity. If to fix for any r > 0 the set Dr= n \ u~l nr(x') then u(·,tm)-+ Uoo(·) in Wi(Dr) for some sequence tm -+ +oo, u 00 being the almost everywhere smooth extremal of£1 (with finite number of singular points). Remark 2. As it was pointed in [2] we can construct a global solution of (5) under nonzero boundary condition ult=O

= cp E WJ(n),

3,P E c2+ao(n)

such that

ulanx(O,T)

= "'·

0 1

cp- "'

e w 2(n).

(20)

Further we shall discuss the case of nonlinear operator L in (1). Let

where f(x, u,p) and its derivatives/,, /u, /uu. /pu, /pp, /pz are continuous functions on the set M 1 = n x IRN x R2 N and satisfy the following conditions

On the Global in Time Solvability of the Cauchy-Dirichlet Problem to Nondiagonal . . •

I.

voiPI 2 $ f(x, u,p) $ Jio(1 + IPI:J),

7

(21)

1/pl + 1/p:a:l + l/,.1 $ /lt(1 + lpl), 1/ul + 1/uul $/l2(1 + IPI 2), /p~p~{!{1 ;?: vi{I:J, 't/{ E R2 N, 11/ppll $

(22) (23)

(24)

Jl,

where v0 , JiG, ..• , JL are some positive constants (no smallness condition on J.L2 is assumed). II. For some a 0 E (0, 1) functions /pp, /,., /p:a: and /,. satisfy Holder condition with exponent ao on any compact set in M 1 • III. {} is a bounded domain in IR2 with CHao_ smooth boundary an. ~ 11owmg • variant • vve cons1"d er t he 10 Let L -- {L

Sm(h)

Nm(h)

= kn ~m(h),

Nm(h)

= J;,(s- h)Y,:.(s) -

Y,:.(s- h)J:,.(s).

Wave number k and index n being fixed, an asymptotic expansion in powers of h gives an expansion in powers of r5. Thus, Taylor series expansion yields

= h(Ng> + hNg> + O(h2 )), 6m(h) = ~~) + h6l!l + CJ{h2 ), Nm(h)

with

Ng> = -J:;.(s)}~(s) + l';;:(s)J:,.(s), Ng> = ~ (J::(s)Y,:.(s)- l';;:'(s)J:,.(s)), 6~) = .J~,(s}}';,.(s)- }~(s)Jm(s), 6!!> = - J:;.(s)Ym(s} + Y,::(s)Jm(s).

Using the fact that Jm and } ;n are solutions to the Bessel equation of order m, Ng>, Ng> and ~l!l can be expressed in terms of ~~) only: N(l) m

= (m21)6(0) s2 m '

N(2) =_!_(3m2 m

2s

s2

-l)~(O)

m '

~(I) = ~~(OJ. m

s

m

A. Bendali

14

A simple calculation then gives

Sm(h)

= knh (t;: -

l) + /;~~- l)h

1+;

+ O(h3 ).

Expanding more completely the above expression and taking into account that the Fourier-series expansion interchanges derivative by multiplication by im, we arrive at the following approximate conditions. • Zero-order boundary condition (the thin shell is completely neglected)

a,

sJo>I{J = o. • First-order boundary condition

s~l)!p = -6( ~2;f, + k2n2)1{J. • Second-order boundary condition 2

2 2

(2) ( 1 2 2 2 6 1 2 k n ) 86 cp-- 6(R2CJB+kn)+2(R3us-R cp.

To obtain boundary conditions on arbitrary r, we use the formal approach introduced in (21] consisting in writing the above expressions in an intrinsic way by making the following substitutions: 1

li~,..,

where s is the curvilinear abscissa and following relations: T

:= 8,m;

1t

is the related curvature defined by the

8,n =ItT,

m E f,

where r is the unit tangent to r obtained by rotating n by rr /2 in the direct sense. Note however that it is the improvement [1] mentioned above consisting in writing first the differential operator in a divergence form like, for example,

n-3lf, = R- 1a,(R- 1R- 1a,) which leads to the right conditions which have been already rigorously obtained. • Neumann (or zero impedance) boundary condition 80 u = e- 1SJ01 u := 0.

(1.4}

• First-order boundary condition of Engquist-Nedelec 8 u = e- 1SJ11 u := -e-16(&: + k 2 n 2 )u.

(1.5}

0

• Second-order boundary condition of Engquist-Nedelec (2l

8nu = e- 18 6 u := -e- 16(8,(1 +

6~t

2

)a, + k2 n 2 (1-

!S~t

2

)}u.

(1.6}

Boundary Element Solution of Scattering Problems Relative to...

15

The related boundary-value problem, relative to one of the above boundary conditions {1.4), {1.5) or {1.6), is a particular form of the following general setting: Au + k u = 0 in n- I 8nu = -8,(a8,u) + /Ju on r, limt~l-+oo lx! 112 (8r{u- ulnc)- ik(u- u1nc)) = 0, 2

}

{1.7)

a and /3 being two given functions defined on the boundary r. This type of boundary condition relating the Neumann to the Dirichlet data is called a generalized boundary condition. We limit the exposure here to presenting how a boundary integral formulation with nice properties can be obtained for this problem and how it is solved by a Boundary Element Method (designated by BEM as usual in what follows). We refer to [9, 10, 29, 30, 6] for a complete mathematical as well as numerical analysis of the method and for further extension to the Leontovitch boundary condition for the full Maxwell's system. 2. THE BOUNDARY INTEGRAL EQUATION

A threefold objective is assigned to the reduction of the boundary-value problem ( 1. 7) to a boundary integral system on

r.

1. Its {main) unknowns must be the so-called physical unknowns

.>. := -ulr,

and p := -8nulr·

Cauchy data in the terminology of PDE theory. In this way, existence of a solution to the integral equation will be assured a priori. 2. It must reduce to the formulation in terms of the physical unknowns of the problem relative to the Neumann boundary condition for a = /3 = 0. {Since for the Neumann condition p = 0, the integral equation is obtained by representing u as a double-layer potential.) 3. All the integrals involved in the formulation must be weakly singular, that is, with a singularity integrable in the usual sense. Moreover, the final linear system to be solved must have a symmetric matrix and, for a given mesh of r, must have the same order than the related case of a Neumann boundary condition. The starting point is as usual the following representation of the solution u to problem (1. 7) given by the well-known formulae giving the outgoing solutions to Helmholtz equation {cf. e.g., [11, 12]) u(x)

= u10c(x) + Vp(x) +N.>.(x),



r,

{2.1)

where Vp is the single layer created by the density p

l -l

Vp(x) :=

G(x, y)p(y) dl'{y),



r,

and N.>. is the double layer created by the density.>.

N.>.(x) :=

an,G(x, y).>.(y) dl'{y),



r.

A. Bendali

16

The kernel G(x, y) is that giving the outgoing solutions to the Helmholtz equation

in IR2 , 1

G(x, y) := ~H~ )(k lx- yl),

where H~ 1 l := J0 + iY0 is the Hankel function of the first kind and of order 0. It is of fundamental importance to note that the boundary condition results in the following relation linking densities p and ..X: P = (-a,o:a, + f3)..X :=

z..x.

(2.2)

A formal integral equation is formed by forcing the representation (2.1) to satisfy the boundary condition

(anVP)- +(anN..X)-+ a. (o:a, (Vp +N..X)-)- f3(Vp+N..xr = - (anuinc +a. (o:a,uioc) - {3uioc) .

Traces of single- and double-layers potentials are given by the following formulae (cf. e.g., [11, 12])

(Vp)± (x)

(N..X)± (x)

= Vp(x},

X

E

= ±l..X(x) + N..X(x),

(anVP}± (x)

r, X

= ±lp(x) + Kp(x),

X

E

r,

E

r,

= -a, (V(a,..X)) (x)- k 2V(..Xr)(x). r(x} , are the integral operators on r given by the weakly singular kernels

(anN..X)± (x) where V, Nand K

l = -l =l

Vp(x) N..X(x)

Kp(x)

=

G(x, y)p(y) df(y),

an,G(x,y)..X(y}df(y},

anzG(x, y)p(y) df(y) .

Noting that K is the formal transpose -NT of -N for that

l

Kp..X'df=

-l

N..X'pdf,

and that the terms outside the integral sign cancel since p and ..X are linked by relation (2.2), we are led to the "integral equation" which still remains formal: -a.v(a,..x)- k 2 r·V(..Xr) - NT p+ (a,o:a,- {3) (Vp+ N..X)

=9

{2.3)

where the right-hand side g is related to the incident wave by g

= -anuioc- a, (o:a,uioc) + {3uioc.

Equation (2.3) has been called formal because the derivatives not under the integral sign cannot be moved inside and, hence, applied to the kernel. This prevents the

Boundary Element Solution of Scattering Problems Relative to . . .

17

equation from being directly suitable for designing an effective numerical method to solve problem (1.7}. Following a different path, Nedelec (24], for the Laplace equation, and later Hamdi [17], for the Helmholtz one, have already remarked that it will be sufficient to use a variational formulation to get a formulation involving a weakly singular integral only as follows: (anN>.)±>.' df

l

=l

(V (a.>.) a.>.'- k 2 V (>.r). >.'r) df.

A similar approach is used to reformulate equation (2.3) as follows

I

r (V(a.>.)a.>.' - k 2 V(>.r) · >.'r) df-

+I

I

r pN>.' df

r ('Vp + N>.)(a.aB.>.'- {J>.') df

=I

r g>.' df.

Let rl be a test function, relative to unknown p, linked to >.' exactly as p was linked to >. by {2.2}: p'

= (-a.aa. + {J)>.' := z>.'.

(2.4)

We have thus obtained the following variational equation,

a({p,>.},{p',>.'})

= lg>.'df,

(2.5)

where

a( {p, >.} , {rl, >.'}) :=

I-I

rxr

-I

G(x, y) (a.>.(y)a.>.'(y) - k 2 >.(y)r11 • >.'(x)r,.) df(y)df(x)

rxr rxr

G(x, y)p(y)p'(x) df(y)df(x) an,G(x, y) (>.'(y)p(x)

+ >.(y)rf(x)) df(y)df(x) .

Remark 1. Integral equation {2.5} only involves explicit integrals absolutely convergent in the usual meaning. If p and p' are eliminated from the formulation using relations {2.2} and {2.4}, equation (2.5} lies in the clll.Ss of Fredholm integral equations of the first kind and has a variational formulation expressed by a symmetric bilinear form in >. and >.'. For a= {J = 0, equation (2.5} reduces to

I

rxr

G(x, y) (a.>.(y)a.>.'(y)- k 2 >.(y)r11 • >.'(x)r,.) df(y)df(x) =-

f

(2.6)

r anuinc>.' df

which is the most efficient way to solve the boundary-value problem relative to a Neumann boundary condition.

A. Bendali

18

The above remarks show that the threefold objective which has been initially

assigned to the formulation is almost reached. However, it still remains an essential difference with the formulation (2.6} relative to a Neumann boundary condition. The direct elimination of p and p' would have as a consequence that the second-order derivatives of A as well as those of A' arise in the formulation. Apart from the difficulty of implementing a boundary element method with A and A' being polynomials of degree ~ 3 over each element, this approach would require two degrees of freedom per node, namely twice of unknowns of what is involved in solving problem (2.6} on the same mesh. Thus, proceeding in this way would move us away from our initial program. The following treatment finishes the procedure. We will consider relations (2.2} and (2.4} as constraints and deal with them through a system of Lagrange multipliers l and playing the role of a supplementary unknown and its related test function, respectively. More precisely, we consider the following problem whose solution {p, A, l} yields a solution {p, A} to (2.5}. The problem is to find p, A and l so that

e

a({p,A},{p',A'}}+b(l,{p',A'})= b(t,{p,A}) =0,

lrgA'df,}

(2.7)

e

for all p', A' and now without any explicit link between either p and A or p' and A'. We have denoted by

b(l,{p',A'})

= Ire (p' + a,a8,A

1

=I

r lp'

-

{JA')

df

df - I r (aa,ea,A' + {JlA') df.

Remark 2. Formulation {2. 7) involves differential operators of order at most 1 so that its numerical solution can be obtained by a BEM with shape functions of degree ~ 1 over each element only. The above formulation seems to triple the number of unknowns. We will see that, in fact, p and l can be eliminated at the element level during the assembly process only keeping A as unknown in the final linear system to be solved. Clearly, any solution to problem (2.7} leads to a solution to problem (2.5}. Thus, no spurious solution has been introduced by augmenting the problem. It is also interesting to examine the converse, to assure that problem (2. 7) has a solution. Let A and p be a solution to problem {2.3}. Set

l = Vp+NA. Since A and pare linked by relation (2.2}, equation (2.3} shows that {A,p, l} is solution to the following system:

-a. (V(8,A))- k 2 V(Ar). r- NT P- ze = 9 , } -NA- Vp+l = 0,

(2.8}

-Zl+p=O. Clearly, problem (2.7} is nothing else but a variational formulation of problem (2.8}.

Boundary Element Solution of Scattering Problems Relative to...

19

Remark 3. It is easy to prove that the well-posedness of problem {2.3} is equivalent to the following two properties (cf. [29]}: 1. Exterior problem {1. 7} is well-posed. 2. The homogeneous associated cavity problem ~u + k 2 u = 0 inn+, anu+ = -a.(aa.u+) + (3u+

on

r, }

(2.9}

has no spurious solution.

Finally, note that jump relations of single- and double-layer potentials give IJ

1 -

'=liu.

Since v := -ulncln+ is a solution to the interior problem, this relation is a particular case of the following one:

where vln- is the scattered wave relative to a general g as right-hand side in the boundary condition. In the same way, vln+ is the solution to problem (2.9} with a nonzero right-hand side equal to g (cf. [29]}. This relation makes it possible to recover v± as well as 80 v± by a simple post-processing of the solution. This is important in some decomposition domain methods where Cauchy data have to be transmitted through the interface.

3.

THE BOUNDARY ELEMENT METHOD

r is approximated by a polygonal curve, still denoted by r . The vertices lie on the initial curve and are numbered in the direction defined by the above curvilinear abscissa s. This yields a mesh on r that elements are given by segments As usual

a 1, ••• , aN

K; := [a., Bt+d,

i = 1, . .. , N

with the usual convention aN+I = a 1• Next, all functions, unknown and test, involved in the formulation are approximated by a C0 -finite element method whose shape functions are linear over each element K;. As a result, any of these functions can be characterized by a column-vector of length N whose components are its nodal values; as an example, A is completely recovered from the vector {.X}= {A;}:~~ in the following way:

where s :=

lm- a. I and IK;I is the length of K; .

A. Bendali

20

The usual assembly process and evaluation of weakly singular integrals give the

following matrices defined {-\'} T

D {-\} =

I

G(x, y) (o.A(y)o.A'(y)- k 2 A(y)r11 • A'(x)r.,) dl'(y)dl'(x),

r> 0 in fl.

H. Berestycki

36

To achieve this goal, the key property which is used is that w~ satisfies a linear elliptic equation. Indeed, u~ satisfies the same equation as u: ~u~

+ l(u~) =

0

in

n~.

Therefore, w~ is a solution of a linear elliptic equation (2.1)

where

c"(x) := l(u~(x))- l(u(x)). u~(x)- u(x)

(2.2)

Since I is Lipschitz continuous, c" is bounded in sup-norm independently of~ E (~·, 0). Furthermore, since u > 0 and by construction, it is easily seen that w~ ~ 0

on

oo~.

(2.3)

The claim is that (2.1)-(2.3) implies that w~ remains positive for all values of This does not readily follow from the maximum principle applied to (2.1)-(2.3) because there is no assumption on the sign of c". (The maximum principle would indeed apply directly if one were to assume that I is nonincreasing - a rather trivial case.) Instead, it involves a sophisticated continuation argument. I will now give the outline of this argument. It hinges on a version of the maximum principle for small domains (see also [14]). In the present context, it can be stated as follows. Let L be a given constant; there exists a positive number 6, which depends on L and on the domain n for which the following holds. Let D be any subdomain of n such that its measure, IDI, satisfies IDI :56. Then if w satisfies ~ E (~·, 0).

~w

+ -y(x)w :5 0 w

with

1-r(x)l :5 L for all x

~0

in on

D, 8D,

(2.4) (2.5)

E D it must be the case that w ~0

in

D.

(2.6)

For more general statements and the proofs, we refer the reader to [11] and [14]. Applying this maximum principle to the above situation, we first see that w~ ~ 0 when 0 < ~- ~· is small enough. Next, suppose that w~ > 0 for all values of~ such that ~ • < ~ < J.L and that J.L is maximal for this property. The claim is that J.L = 0. We argue by contradiction and suppose that J.L < 0. Take a compact subset K c n,. with 10,. \ Kl < 6/2. From the definition of J.L it follows that w" ~ 0 in and, in fact, by the strong maximum principle, w" > 0 in n,.. Therefore, for some 17 > 0, we have w" ~ 17 on K. By continuity, for 0 :5 ~ - J.L small enough - say ~ :5 J.L + e with e > 0- we see that w~ ~ 17/2 > 0 on K. This implies in particular that w~ > 0 on 8K. Since, by construction, w~ ~ 0 on 00~, it follows that w~ ~ 0 on 8(!1~ \ K). If e is sufficiently small, the set D := (!l~ \ K) has measure smaller than fJ. Hence, we may apply the previous maximum principle in the set D. We infer that w~ > 0 first

n,.

Qualitative Properties of Positive Solutions of Elliptic Equations

37

in D and then in all of n~ since we already know it in K . This holds for all A < p. + c, contradicting the maximality of p.. Hence, p. = 0 and the proof is complete. In [11), we also consider much more general equations of the type

F(x , u, Du, D 2u)

=0

in

n

(2.7)

that is completely nonlinear equations and I refer to that paper for precise statements. For unbounded domains, the argument just outlined does not suffice, essentially because of the lack of maximum principle and for an underlying compactness argument in the continuation part. Three main approaches have been used to overcome this difficulty. One is to use some transformation such as an inversion or a Kelvin transform to get back t.o a bounded domain setting (an example is [18]). Another one is to apply the maximum principle in the unbounded part of the underlying domain, when for instance one makes assumptions on I which imply that ~ (see 2.2 above) has a negative sign outside some large ball. Here an example is [29, 30). Another one is to have special forms of the maximum principle exploiting the geometry of the domain. Some of the results presented here use the latter in their proofs. 3.

SPHERICAL SYMMETRY IN ALL OF SPACE

The case of positive solutions of 1.1 in all of space /R" was the first one to be studied and is indeed motivated by the study of ground states in some nonlinear field equations. One considers solutions of ~u

+ l(u) = 0

in u > 0 in /Rn, u(x) -+ 0 as lxl -+ oo.

(3.1)

(3.2) (3.3}

The limit in (3.3) is understood to be uniform. The first result for this type of problem is due to Gidas, Ni and Nirenberg [25). A more general version of this result was then obtained by Congming Li [29), who also extended it to the case of fully nonlinear equations in all of space. In these papers, it is essentially assumed that 1'(0) < 0. This result has been further extended by Yi Li and W.M. Ni [30) (who also consider the case of fully nonlinear equations). Here is a result from this paper:

Theorem 3.1. Suppose that I is of class C 1 and that on some interval (0, s 0 ) (with s 0 > 0}, the function I is nonincreasing with 1(0) = 0. Then all positive solutions of {9.1} {9.3} must be mdially symmetric about the origin (up to tmnslation) and Ur < 0 for r > 0 .

Congming Li uses the moving plane method handling the unbounded part with the sign of~ (as was explained above) . Indeed, in view of the assumption on f in theorem 3.1, one sees that ~(x) :50 outside some large enough sphere. Congming Li also considered in [29) fully nonlinear elliptic equations in all of space. In a very recent paper, Serrin and Zou [39) have further extended these results to more general classes of quasilinear elliptic equations involving singular operators, like

38

H. Berestycki

the p-Laplace operator, and are able to handle solutions which may have compact

support. 4. DOMAINS BOUNDED BY A LIPSCHITZ GRAPH Here, one considers bounded positive solutions of (1.1)-(1.2) in an unbounded open set n defined by

n = {x E JRn;

Xn

> cp(x., . ..

(4.1)

,Xn-tH.

where cp: JRn- 1 ~ lR is a globally Lipschitz continuous function. For x E JRR, we use coordinates x = (x',xn),x' = (xt. . . . ,Xn-t)· In this case, the specific issue we are concerned with is that of monotonicity properties for positive solutions of such problems. We make the following assumptions : 0 0 and, in addition, for some 0 < s 0 < s 1 < f(s) ~ 0 in fl. This result is proved in [7) by the sliding method (see [11) and the references therein) where various further properties are derived. In particular, we show that under the conditions of the theorem, the solution of {1.1}-(1.2) is unique in this case. Actually, theorem 4.1 yields monotonicity with respect to a whole cone of directions about the direction Xn· This problem was motivated by an earlier work on regularity in some free boundary problems ([9]). The first result of the nature of theorem 4.1 is due to M. Esteban and P. L. Lions [22]. They consider the case of a "coercive" Lipschitz graph, that is, they assume that lim cp(x') = +oo.

lrl-+oo

(4.6)

Here is a more general version of their result (with no smoothness assumption on the graph): Theorem 4.2. Suppose cp: ./Rn- 1 ---. lR is a locally Lipschitz gmph satisfying (,1.6) above. Let u be a positive solution of (1.1}-(1.2){u is not assumed to be bounded) with f locally Lipschitz. Then 8':. > 0 in Q.

Qualitative Properties of Positive Solutions of Elliptic Equations

39

This more general version includes for instance the case of a cone, which was not considered in [22]. It just suffices to observe that the method of moving planes as described above (like in (11]) applies with no changes since, in this case, the domain n~ cut out by planes {Xn = A} is bounded. If (4.6) does not hold, then, in general, the regions n~ are not bounded, and the problem becomes more delicate. In fact, for unbounded solutions, one can construct examples (for which (4.6) is not satisfied) where the conclusion of theorem 4.2 fails. In an earlier paper, S. Angenent [4] had proved the same result - under the same assumptions- as in theorem 4.1 for the particular case of a half space. For a general geometry, it is an open problem to know under what more general assumptions on for 1p do the conclusions of Theorem 4.1 hold. 5. HALF SPACES

Consider now the case of a half space: n = {x = (xl> ... 'Xn) E 01." 'Xn > 0}. In this setting, one is interested in two types of properties: monotonicity with respect to Xn and symmetry - which in this case refers to one-dimensional symmetry, that is, that u is a function of Xn alone. Notice that these two properties are the exact analogues of the properties stated in theorem 2.1 if one thinks of the half space as a limiting case of a sphere tangent at the origin to the plane Xn = 0 and whose center goes to infinity. The function f will always be assumed here to be (globally) Lipschitz continuous: 01.+--+ 01. Let us start with the monotonicity property.

Theorem 5.1. In the half space n = n

.JR+, suppose that u is a solution of (1.1}-(1.2),

= 2 or when n ~ 3 that f(O) ~ 0. Then the function u satisfies au > 0 m. nu. -a Xn

It is still an open problem to know whether this property holds in general in dimension n ~ 3 in case /(0) < 0. Under the additional hypotheses that u is bounded and that /(0) ~ 0 in all dimensions (also in dimension two), Theorem 5.1 was first proved by E.N. Dancer

[20].

Let us now turn to symmetry. In [5] we prove the following:

Theorem 5.2. Suppose that u is a bounded positive solution of (1.1)-(1.2) in a half space with M =supu. n Then if f(M) $ 0, the function u is symmetric, i.e., u monotonic, that is, Uz,. > 0 inn. Furthermore, f(M) = 0.

=

u(xn), and it is also

Actually, in [5], we consider more general equations of the type ~'IL

+ g(xn, u) =

0,

'IL

> 0 inn.

(5.1)

40

H. Berestycki

Assuming that g is Lipschitz, that t _... g(t,u) is nondecreasing in t, that /(u) .~~~ g(t, u) exists and is Lipschitz continuous in u and, lastly, that g(t, M) S 0 'r:Jt, we prove symmetry - and monotonicity - of solution u of (5.1} and {1.2}. Tehrani [41] has treated a more general form of (5.1}:

.:lu + g(xn, u, IVul)

= 0.

It should be noted that the condition that u be bounded is important. Indeed, one can find counter-examples to symmetry for unbounded solutions. For instance, in the half plane fl = {(xlt X2)i X2 > 0}, the function U{Xlt X2) = X2ez1 is a positive solution of .:lu - u = 0 with u = 0 on the boundary. In [6], we have obtained still another symmetry result in low dimensions using a different approach. Here is one of the main results.

Theorem 5.3. In the half space fl = .IR~ with n = 2 or 3, let u be a bounded positive solution of {1.1)-(1.2}. In case, n = 3, assume further that /{0) ;?: 0 and that f E C 1 • Then, u is symmetric, that is, u = u(xn)· The proof of this result combines the monotonicity result theorem 5.1, the previous symmetry result, theorem 5.2 to which we reduce it together with a result about Schrodinger operators in the plane. It is not known whether this result holds in general. 6.

ONE-DIMENSIONAL SYMMETRY IN ALL OF SPACE

The question of one-dimensional symmetry is also natural for monotone solutions in all of space. One such problem arises in a conjecture of De Giorgi [21]. It is about solutions of problems of the type

.:lu+u-u3 =0

in

.IR".

{6.1}

Suppose that u satisfies -1 < u < 1 and that ::,. > 0 in all of ./Rn. Then the conjecture is to know whether u has one-dimensional symmetry. That is, are the level sets of u parallel hyperplanes? The answer is known in dimension two owing to works by Modica and Mortola [32] and a recent paper by Ghoussoub and Gui [23] which proves this result in full generality in the plane. They essentially derive the following.

Theorem 6.1. In the plane .IR2 , consider a bounded solution u E L 00 {./R2 ) of the equation

.:lu + /(u}

=0

in all of .IR2 ,

(6.2}

where f is an arbitrary continuously differentiable function. Suppose in addition that u is monotonic in some direction, say

au

>0

axl-

in JR2 •

(6.3)

Then u is a function of one variable only, that is, there exists a and b such that u

= u(ax1 + bx2 ).

{6.4}

Qualitative Properties of Positive Solutions of Elliptic Equations

41

The proof uses some spectral results for Schrodinger operators (see also a proof in [6]). Some results related to this problem appear in [17]. Related results about one-dimensional symmetry - and cases where it fails - are to be found in the papers [10, 15, 26]. 7. INFINITE CYLINDERS In our paper [8] we take up another class of unbounded domains: infinite cylinders or, more generally, product domains of the form

n = IRn-]

X

w,

where w is a bounded smooth domain in JR1 •

We denote the variables in n by (x, y), X E mn-;' y E w c JR.i. It is not assumed that u is bounded. In fact, we establish that solutions u of (1.1)-(1.2) have at most exponential growth, that is, u(x, y) $ Ce"lzl

\:lx E mn-J, 'rly E w.

(7.1)

Assuming either that j 2: 2 or that j = 1 with /(0) 2: 0, we show that if w is convex in some direction, say Yll and symmetric with respect to the hyperplane y 1 = 0, then any solution u of (1.1)-(1.2) is symmetric about that hyperplane and decreases in y 1 for y 1 > 0. In [8] we also consider more general operators. The case /{0) < 0 and j = 1, that is, when w is an interval, say w = (0, h), has been considered separately in [6]. It turns out to be somewhat particular. We prove the same symmetry result in this case in dimension two. The case of higher dimensional slab like domains of the type f2 = mn-l X (0, h) with n 2: 3 is still open. For other qualitative properties in infinite cylinders for travelling waves and fronts (such as monotonicity, uniqueness), I refer the reader to the papers [12, 13, 42]. A result about the symmetry of water waves has also been proved by Craig and Sternberg in [19]. 8. OVERDETERMINED PROBLEMS A well-known result of Serrin [38] concerns the determination of the shape of domains for which there exists a solution of the following overdetermined problem:

flu+ /(u) =0 u au av

and

u>O

=0

on

=k

on

in

an, an,

n,

(8.1) (8.2)

(8.3)

where k is a constant. Using the method of moving planes, Serrin proved that if n is a bounded and smooth domain, then it is a sphere [38]. Very recently, this type of problem has been considered in the the setting of unbounded domains by W. Reichel [35, 36] and Aftalion and Busca [1]. The most recent work in this direction is by B. Sirakov [40], who has obtained a general result which I will now describe.

H. Berestycki

42

Consider the problem

V · (g(IVui)Vu) + /(u, IVul) = 0, { Boundary Conditions (BC), where G C /Rn is a bounded set such that

G

u

2:: 0'

in

mn \G

(8.4)

mn \ G is connected and

= i=l u" Gi,

(8.5)

where k is an integer and Gi are bounded C2-domains such that G; n(i; = 0 fori :F j . The elliptic operator and the nonlinearity are supposed to satisfy the following assumptions:

g E C 2 ([0, oo)), g(s)

>0

and

(sg(s))'

> 0 for all s 2:: 0,

(8.6)

the function f(u, v) : (0, oo) x (0, oo) -+ IR is locally Lipschitz continuous and f is a non-increasing function of u for small positive values of u and v. Note that the Laplace operator as well as the mean curvature operator satisfy the assumption. The overdetermined boundary conditions (BC) in this context are of the following type:

u =eli> 0

and

av. 811

= Oi ~ 0

on aGi ' i

= 1, ... 'k,

(8.7)

where the eli, ai and {3 are constants. Furthermore, it is assumed that

Vu(x), u(x) -+ 0

as lxl -+ oo.

(8.8)

(No assumption needs to be imposed on Vu iff does not depend on IVul.) The main result of Sirakov is the following.

Theorem 8.1. Under the above assumptions, if u is a solution of (8.4} satisfying (BC}, then k = 1, and G is a ball centered at some point x 0 E /Rn, u is spherically symmetric about the point x 0 and u is decreasing away from x 0 • This result has applications in potential theory. Sirakov also derives this type of results in bounded domains of the type G with overdetermined conditions on the boundary an. It extends earlier results of Alessandrini ((2]), Willms, Gladwell and Siegel ((43)) in bounded domains and of W. Reichel ((35], (36], (37)), A. Aftalion and J. Busca (1] in the case of exterior domains.

n\

REFERENCES

(1) A. Aftalion and J. Busca, Radial 1ymmetrtt of overdetermined boundo'lfvolue problem~ in ezterior domain.t. Arch. Rational Mech. Anal., to appear. (2) G . Alessandrini, A IJI77lmetrtf theorem for conderuer1. Math. Methods Appl. Sc., 15 (1992), p.315-320. (3) A.D. Alexandrov, A charocteriltic propertJI of the ~pheru. Ann. Mat. Pura Appl., 58 (1962), p. 303-354. (4) S.B. Angenent, Uniquenu1 of the 1olution of a 1emilinear boundo'lfvolue problem. Math. Ann., 272 (1985), p. 129-138.

Qualitative Properties of Positive Solutions of Elliptic Equations

43

[5) H. Berestycki, L. Caffarelli, and L. Nirenberg, Sf1'71metry for elliptic equatioru in a half !pace. In Boundary value proble~ for partial differential equatioru and application&, volume dedicated to E. Magenes. J.L. Lions et al. ed., RMA Res. Notes Appl. Math., 29, Masson, Paris (1993), p. 27- 42. [6) H. Berestycki, L. Caffarelli, and L. Nirenberg, Further qualitative properties for elliptic equation& in unbounded domairu. Annal. Scuola Norm. Sup. Pisa, to appear. (7] H. Berestycki, L. Caffarelli, and L. Nirenberg, Monotonicity for elliptic equation& in unbounded Lipschitz domairu. Comm. Pure Appl. Math., 50 (1997), p. 1089-1112. (8] H. Berestycki, L. Caffarelli, and L. Nirenberg, Inequalitiu for !econd order elliptic equation& with application& to unbounded domairu, I. Duke Math. J ., 81 (1996), p. 467--494. (9] H. Berestycki, L. Caffarelli, and L. Nirenberg, Uniform edimate! for regularization of fru boundary proble~ in Analy!i.t and Partial Differential Equation&, C. Sadovsky ed., M. Dekker, New York (1990), Lecture Notes in Pure and Applied Math., 122, p. 567-619. (10] H. Berestycki, F. Hamel, and R. Monneau, One dimeruional !11f71metry for some travelling fronts proble~ in all of space. Preprint. (11] H. Berestycki and L. Nirenberg, On the method of moving plane! and the !liding method. Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), p. 1-37. (12] H. Berestycki and L. Nirenberg, 7hwelling fronts in cylinder!. Ann. Inst. H. Poincare, Anal. Non Lineaire, 9 (1992), p. 497- 572. (13] H. Berestycki and L. Nirenberg, Some qualitative propertie8 of !olutions of semilinear elliptic equatioru in a cylinder. In Analy!i.t Et Cetero, P. Rabinowitz et al. ed., Academic Press, New York (1990), p. 115-164. (14] H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operotor! in generol domains. Comm. Pure Appl. Math., 47 (1994), p. 47-92. (15] A. Bonnet and F. Hamel, Existence of a non planar solution of a !imple model of premU:ed Buruen ftamu . Preprint Lab. Analyse Numerique, Univ. Paris VI, 1997, SIAM J. Math. Anal., submitted. (16] F. Brock, Habilitation, Univ. of Leipzig, 1998. (17] L. Caffarelli, N. Garofalo, and F. Segala, A grodient bound for entire solution& of quO!i-linear equation& and its coruequences. Comm. Pure Appl. Math., 47 (1994), p. 1457-1473. (18] W. Chen and C. Li, ClO!!ijication of !olutions of some nonlinear elliptic equation&. Duke Math. J ., 63 (1991), p. 615-622. [19) W. Craig and P. Sternberg, S11f71metry of fru-!urjace flow!. Arch. Rational Mech. Anal., 118 (1992), p. 1- 36. (20] E.N. Dancer, Some notes on the method of moving planu. Bull. Austral. Math. Soc., 46 (1992), p. 425--434. (21] E. de Giorgi, Convergence proble~ for junctionab and operator!. Proc. lnt. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978. Pitagora Ed., 1979, p. 131-188. (22] M. Esteban and P.L. Lions, Existence and non existence re!ults for !emilinear elliptic proble~ in unbounded domairu. Proc. Roy. Soc. Edinburgh, 93A (1982), p. 1- 14. (23] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and !orne related proble~, Math. Ann., 311 3 (1998), p. 481--491. (24] B. Gldas, W. M. Ni, and L. Nirenberg, S11f71metry and related propertiu Ilia the marimum principle. Comm. Math. Phys., 6 (1981), p. 883- 901. (25] B. Gidas, W. M. Ni, and L. Nirenberg, S11f71metry of positive !olutioru of nonlinear elliptic equation& in Rn. Math. Anal. Appl. Part A, Advances in Math. Suppl. Studies, 7 A (1981), p. 369-402. (26] F. Hamel and R. Monneau, Uniquenus of conical 8haped !olutioru of an elliptic bidimeruionnal equation, work in preparation.

44

H. Berestycki

[27) P. Hess and P. Polatik, S'/lfflmetry and convergence properties for non-negative solutioru of nonautonomous reJJction-diffusion problems. Proc. Roy. Soc. Edinburgh, 124A (1994), p. 573587. [28) B. Kawohl, Rearrnngemenu and convexity of level set& in partial differential equatioru, Lecture Notes in Math., Springer-Verlag, Berlin, 1985. [29) Congming Li, Monotonicity and s'f1111metry of solutioru of fully nonlinear elliptic equatioru on unbounded domairu. Comm. Partial Differential Equations, 16 {1991), p. 585 615. [30) Yi Li and Wei-Ming Ni, Radial S'/lfflmetry of positive solutioru of nonlinear elliptic equatioru in JR." . Comm. Partial Differential Equations, 18 {1993), p. 1043-1054. [31) H. Matano and T . Ogiwara, Monotonicity and convergence in order-pre6erving systeffl6 in the presence of S'/lfflmetry, in preparation. [32) L. Modica and S. Mortola, Some entire solutioru in the family of nonlinear Poisson equation. Boll. Uni. Math. Ita!., 5 {1980), p. 614-622. [33) T. Ogiwara and H. Matano, Stability analysis in order-pre6erving systeffl6 in the presence of S'/lfflmetry. Proc. Roy. Soc. Edinburgh, to appear. (34) T. Ogiwara and H. Matano, Monotonicity and convergence result& for pseudo-trovelling waves for parobolic equatioru in temporolly or spatially periodic media, in preparation. (35) W. Reichel, Radial s'f1111metry for elliptic boundary value problem6 on exterior domairu. Arch. Rational Mech. Anal., 137 {1997), p. 381- 394. (36) W. Reichel, Radial s'/lfflmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined probleffl6 on exterior domairu, Z. Anal. Anwendungen, 15 {1996), p. 619-635. [37) W. Reichel, Radial6'/lfflmetry by moving planes for semilinear elliptic BVP's on annuli and other non-convex domairu. In Progress in POE's: Elliptic and Parabolic Problems, Pitman Res. Notes, Vol. 325; C.Bandle et al., eds. p. 164 182 . (38) J . Serrin, A S'/lfflmetry theorem in potential theory. Arch. Rational Mech. Anal., 43 {1971), p. 304 318. [39) J. Serrin and H. Zou, S'/lfflmetry of ground states of quasilinear elliptic equatioru. Preprint. (40) B. Sirakov, S'/lfflmetry in elliptic overdetermined problem6 and two conjectures in potential theory.

?reprint Lab. Ananlyse Numerique, Univ. Paris VI.

(41) H. Tehrani, Doctoral Thesis, Courant Institute, 1994. (42) J .M. Vega, On the uniqueness of multidimeruional travelling front& of some semilinear equations. J. Math. Anal. Appl., 177 {1993), p. 481-490. (43) N.B. Willms, G.M.L. Gladwell and D. Siegel, S'/lfflmetry theoreffl6 for some overdetermined Angew. Math. Phys., 45 {1994), p. 556-579. boundary-value probleffl6 on ring domairu.

z.

H. Berestycki, Departement de mathematiques, Universite Pierre et Marie Curie 4, Place Jussieu, 752230 Paris CEDEX, France

Nonlinear Boundary Stabilization of the Wave Equation R.

BEY,

J .-P.

LOHEAC, M. MOUSSAOUI

Abstract. We investigate the problem of the boundary stabilization of the wave equation in a bounded domain. Many authors have proved the exponential decrease of the energy function, with linear or nonlinear feedbacks, under restrictive geometrical assumptions. Here we obtain similar results by only assuming that the boundary of the domain is sufficiently smooth. Keywords:

mixed problems, singularities, controllability, stabilization

Classification:

35L05, 35Q72, 93B03, 93B07, 93C20, 93Dl5

INTRODUCTION

Let n be a bounded open set of Rn (n ~ 3) such that its boundary an satisfies in the sense of Necas (11]:

an is of class C2 •

(1)

Given x a point of an, wr denote by v(x) the normal unit vector pointing outward of

n.

Let Xo be in Rn. For x in Rn, we define m(x) by: m(x) following subsets anN and anD of an by:

=x -

Xo

as well as the two

anN= {x E an I m(x).v(x) > 0} j anD= an\ anN = {x E an I m(x).v(x) :50}.

(2)

We have anD u anN = chosen such that

an and anD nanN

=

0. Furthermore, we assume that Xo is

meas(anv) :f. 0, meas(anN) :f. 0;

r =anD nanN is a C3-manifold of dimension n there exists a neighbourhood w of r such that an n w is a C3 -manifold of dimension n - 1 . Let

g :

2j

(3)

lR -+ lR be a function such that

g is non-decreasing;

g E C0 •1 (R) ;

g(O)

=0 .

(4)

R. Bey, J.-P. Loheac, M. Moussaoui

46

We consider the following problem: u 11 - ~U = 0, in 0 X (0, +co); u = 0, on an0 x (O,+co); : : = -(m.v)g(u1) , on

anN X (0, +co);

(W)

u(O) = u0 , in 0; u'(O) = u 1 , in 0, where u' = 8ul&t and u" = fPul &t2 • Under reasonable assumptions on (u0 ,u1 ), this problem is well-posed. We define the following Hilbert spaces: V = {v E H 1(0) I v = 0, on anv}; H = V x 1 2(0). We introduce a nonlinear operator A in H:

V(A) = {(u, u) E v X vI ~u E 12(0); :: A(u, u) = ( -u, -~u), 'v'(u, u) E V(A) .

= -(m.v)g(u), on anN};

One can show that A is a maximal monotone operator in H. So for some initial data (u0 , u 1 ) in V(A), problem (W) has one and only one (strong) solution u satisfying u E W 1·""(Rt.;V) and

~u E 1""(Rt.;12 (0)).

The energy function of this solution is given by E(t) =

~L 1Vul2 (x, t) dx + ~L lu'l 2 (x, t) dx,

'v't

~ 0.

Furthermore, one can prove that V(A) is dense in H. Thus, for every t in Rt, the function (u0 ,u1 ) .-+ (u(t),u(t)) can be uniquely extended as a continuous semi-group S of contractions in H. For (u0 , u 1 ) in H, one may define the weak solution u of (W) which satisfies (u, u') E C(Rt; H). Exponential decrease of the energy function has been obtained under restrictive assumptions. For the linear case, the reader can see [9], [4], [8], [7]. These results have been generalized in [10] and [1]. The nonlinear case has been addressed by many authors (see, for example, [3], [5]) under restrictive geometrical conditions (mainly, anv nanN = 0). We extend here the study of [1] by using a method close to [5]: we only assume (1)-(4) and a less restrictive geometrical hypothesis (5) which is given below. This hypothesis is satisfied when 0 is a convex set and Xo is an exterior point of 0 . Under (3), f can be locally seen as a C3 -submanifold of anN of codimension 1. This allows us to define, in the tangent space, the normal unit vector pointing outward of anN ("from anN to anv"). This vector will be denoted by r(x). As in [1], we will assume that For almost every

X

E

r' m(x).r(x) 5 0.

(5)

Nonlinear Boundary Stabilization of the Wave Equation

47

Under assumptions (1)-(5), we obtain a regularity result concerning strong solutions of (W), proposition 1. Then by using results given in (6J, theorem 2 yields the decrease of the energy function E. MAIN RESULTS

In (5J, the stabilization result has been obtained by using a Rellich's identity. This is possible because of the regularity of the solution. Here, with weak geometrical assumptions, the solution of (W) is not regular enough and we are not able to use this identity. Nevertheless, the following proposition will be sufficient to get a similar stabilization result. Proposition 1. -Assume (1)-(4). For every (u0 ,u1 ) in D(.A), the solution u of (W) satisfies (m.11)1Vul 2 E L 1 (an). With further assumption (5), we have 2ln ~u(m.Vu)dx

~ (n-2)fn1Vul 2 dx+2/8fl~(m.Vu)da -/8fl(m.II)IVul 2 dCT.

Proof.

We only have to notice that, under (4), for v in H112 (an), g(v) belongs to Hll2 (an). And since m.11 belongs to C1 (an), (m.11)g(v) E H112 (an). So for every (u, u) in D(.A), we have 8u 2 ~u E L (!l) and Bll = -(m.11)g(u), on anN. Since -(m.11)g(u) belongs to H112 (an), there exists some u E H2 (!l) such that

u= Then, u -

0' on

u belongs to V

an

and

:~ =

-(m.ll)g(u)' on

an

0

and satisfies

8(u-ii)

Bll

= 0, On anN .

Our proposition can be immediately deduced from results of (1J. We now present our main result for the stabilization of problem (W). Theorem 2. - Assume (1)-(5) and suppose that there are p E N* and a positive constant C such that lg(x)l ~ CMin(lxl, lxl"),

'v'x E R.

(6)

Then for every E H, there exists T > 0 such that the energy function of the solution of (W) satisfies (u 0 ,

u 1)

ifp > 1 :

E(t) ~ Ct2J(l-p},

ifp = 1 :

E(t)

'v't

> T;

~ E(O) exp ( 1 - ~)

,

'v't >T;

where, in the first case, C depends on the initial energy E(O); in the second case, C is independent of the initial data.

48

R. Bey, J.-P. Loheac, M. Moussa.oui

Proof

Proposition 1 is an essential tool of this proof where we follow [5]. We firstly prove the result for strong solutions of (W) by applying some results of [6] (theorems 8.1 and 9.1). The result is extended to weak solutions by using a density argument. The reader will find details of proof in [2]. Remark. For the 2nd case, our result remains valid if n is convex or C2 • It is also valid for a polygonal domain n provided that, at each vertex of the boundary condition changes, the angle of is less than 1T.

an

an where

REFERENCES

[1] R. BEY - J .-P. LOHEAC - M. MoussAOUJ. Singularitiu of the solution of a mixed problem, applications. CNRS UMR 5585, preprint 269 (1997), to appear. [2] R. BEY - J.-P. LOHEAC - M. MouSSAOUJ. Stabiliaation non lineaire de /'equation des ondu. CNRS UMR 5585, preprint 282 (1998), to appear. [3] F . CONRAD - B. RAo. Decay of solutions of the wave equation in a star-shaped domain with non-linear boundary feedback. Asymptotic Analysis, 7, pp 159-177 (1993). [4] P. GRISVARD. Controlabilite uacte du solutions de /'equation des ondu en presence de singularitu. J. Math. Pures Appl., 68, pp 215-259 (1989). [5] V. KOMORNIK. On the nonlinear boundary stabilization of the wave equation. Chin. Ann. of Math., 14B:2, pp 153-164 (1993). [6] V. KOMORNJK. Ezact controllability and stabilization; The multiplier method. Masson-John Wiley, Paris (1994). [7] V. KoMORNIK- B. RAo. Stabiliaation frontiere d'un systeme d'equations des ondu. C. R. Acad. Sci., t. 320, Serie I, 69, pp 833-838 (1995). [8] V. KoMORNJK - E. ZUAZUA. A direct method for the boundary stabilization of the wave equation. J . Math. Pures App., 69, pp 33-54 (1990). [9] J. LAGNESE. Deca11 of Solutions of Wave Equations in a Bounded Region with Boundary Diasipation. Arch. Rational Mech. Anal., 43, pp 304-318 (1971). [10] M. MoussAOUI. Singularitis des solutions du probleme me/e, controlabilite ezacte et stabiliaation frontiere. ESAIM Proceedings, Elasticite, Viscoelasticite et Controle optimal, Huitiemes Entretiens du Centre Jacques Cartier, pp 157-168 (1996). [11] J . NECAS. Les methodes directes en theorie des equations elliptiques. Masson, Paris (1967).

R. Bey, J.-P. Loheac, M. Moussaoui, Equipe d'Analyse Numerique de Lyon Saint-Etienne, C.N.R.S. U.M.R. 5585 Ecole Centrale de Lyon, departement M.l, B.P. 163, 69131 ECULLY Cedex, FRANCE Email: Rabah . Bey~ec-lyon.fr, Jean-Pierre.LoheacGec-lyon.fr, Mohand.Houssaoui~ec-lyon . fr,

http://numerix.univ-lyonl . fr/publis/

Shape Derivative of Sharp Functionals Governed by N avier-Stokes Flow S.

BOISGERAULT,

J.-P.

ZOLESIO

Abstract. The shape analysis of the Navier-Stokes equation has already been considered in the literature. Classical techniques, such as the Implicit Function Theorem, may be used to show that some functionals, the drag for example, are shape differentiable. However, this property relies on results established for the basic regularity of the pressure and the velocity fields. Many other criteria of physical interest are out of this scope; we consider here the shape analysis of such functionals, for example, the (total) force exerted by the fluid on a body or the moment of these forces. The velocity and pressure fields u and p are assumed to be solutions of the stationary incompressible Navier-Stokes equation -li~U + (Du]u + Vp = I in n with the boundary condition ulr = 0, r = on. These new results are based on the so-called speed method which allows us to "bring back" vector fields from a perturbed domain to the initial one while preserving the divergence-free property. Regularity results are established for that correspondence and used to define and show some properties of the shape derivative u1 and of the boundary shape derivative '-'t· Keywords:

shape optimization, shape gradient, Navier-Stokes equation

Classification:

35Q30, 49J20

1. INTRODUCTION We study the shape differentiability of functionals of the solutions (u,p) of the Navier-Stokes equation in flo which is not defined for the minimal regularity of the solutions, for example because high-order derivatives or integrals on submanifolds appear in the expression of these functionals. In the framework of the Speed Method (section 3.1), some perturbed sets [s ~--+ fl,) are associated with the initial set fl 0 ; the velocities u, solutions of the Navier-Stokes problem in the moving sets fl, are associated with fields u' in the fixed set flo (section 3.2) which are solutions of the Transported Navier-Stokes Equation (section 3.3). The definition of this equation uses the functional spaces introduced in section 2. The Implicit Function Theorem is then used to prove some regularity of [s ~--+ u'] for the desired spatial regularity of u (section 3.4). This result allows us to show the existence of the shape derivatives of u and p for a given spatial regularity, and then through the development of a Tangential Calculus, their boundary shape derivatives (section 4). Theses objects are then intensively used to obtain the explicit form of the shape

S. Boisgerault, J.-P. Zolesio

50

gradient of the force and the moment of the forces applied by a fluid on a part of its boundary (section 5).

2.

THE

N AYlER-STOKES

EQUATIONS

The Navier-Stokes problem in an open and bounded set n as -v~u + [Du)u + Vp = divu = 0 u= 0

I

c lR3 is classically written

in n (1.1) } inn (1.2) on an, {1.3)

{1)

where v is the kinematic viscosity, u the velocity of the fluid, p the pressure and I the force. The study of this equation requires functional spaces built as subspaces (to deal with the fluid incompressibility {1.2) or the boundary value {1.3)) or quotients of Sobolev spaces (to handle forces defined up to a gradient): for any integer m ~ 1, we define V"'(n) = {u E H"'(n; Jtl) n HJ(n; R3 ), divu =

o}

(2)

endowed with the Hm(n; R3 ) norm and for any integer m ~ -1, we set wm(n) = Hm(n; JRl)/{Vp, p E H"'+l(O; JR)}

(3)

endowed with the quotient norm. The linear continuous mappings i : V"'(O) -+ Hm(n; Jltl)

and

1r :

H"'(O; JRl) -+ wm(n)

{4)

are respectively the canonical injection and the quotient mapping.

Remark 1. Strictly speaking, we did not uniquely define the mappings i and 1r, but only some sequences {im)m>l and {7rm)m>-1· It is obvious that when u E Hm(n), for any integer m', 1 ~ m' ~-m, im(u) = im'(u), so i is uniquely determined. For 1r, we must first notice that for any -1 ~ m' ~ m, the mapping 7rm{f) E Wm(n) t-+ ?rm'{!) E wm'(n) is well defined {that is, does not depends on the choice of!) and is a linear continuous injection, so we may identify wm(n) with a subspace of wm' (n). With this convention, 7rm{J) = ?rm'{!) when the two expressions make sense, and 1r is also well defined. Moreover, 1r{!) E wm(n) if and only iff may be decomposed as 1 = g + Vp, g E Hm(n; JR3 ) and p E L2 {0; R) {and not necessarily p E Hm(n; R)): the spaces wm(n) characterize the best possible regularity of 1 "up to a gradient" {whatever its regularity is). Let V(O; JR3 ) be the set of functions of C""(Jll; R3 ) whose support is compactly included inn. We recall that ifO is Lipschitz, V(O) = {v E V(O;Rl),divv = 0} is dense in V 1(0). Consequently, Proposition 1. The spaces 'v'IEH- 1(0;R3 )

w- 1 (0)

and V 1 (0)' are isomorphic. Moreover, 1r{!)=O 'v'vEV1(0), =O

'v'v

E V{n),

< 1. v >= o.

{5)

Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow

Proof: The linear continuous operator I E n- 1 (0; 1R3 ) clearly onto and IE n- 1{0; 1R3 ) belongs its kernel iff Vv E V 1(!l),

.....

51

llvi(O) E V 1 (!l)' is

= 0 3p E £ 2(0; R), I= Vp 1r{f) =

0

(see [10] for the first equivalence), which proves the first part of (5) and that V 1 (!l)' ~ H- 1{0; R3 )/{Vp, p e £ 2 (0; R)} = w- 1 (0). The second equivalence follows by density. o The operators 1r and i will be used to show some regularity results for some (one-variable or two-variable) mappings from vm(n) to wm' (0) which are built on mappings from Hm(n; JR3 ) to Hm' (0; R3 ) (as it is shown in the following definition) whose regularity is known. Definition 1. For any integer n and mapping T : (H 1 (0; JR.l))" -+ define the mapping t': (V 1 (0))"-+ w- 1 (0) by

n- 1 (0; JR3 ),

t' = 1r o To (i ® i ® ... ® i).

we

(6)

In particular, this construction is applied for the operators A and B defined by

V~(n)xP(n)=

i

Du·· Dcpdx, < B(u, v), cp

>V~(O)xP(n)= i[Du]v · cpdx

{7)

for u E H 1!,.,(0) and v E L~oc(O). These operators, used in the variational formulation of (1), have the following regularity: Proposition 2. For any integer n ;::: 1, A is a continuous mapping H"(O; 1R3 ) -+ nn- 2 (0; :Rl). B is a continuous mapping H 1 (0; JR.l) x H 1 {!l; 1R3 ) -+ H- 1 {0; 1R3 ) and Hn(Q; JR3 ) X Hn(Q; JR.l)-+ nn-l(Q; JR.l) for any integer n:::: 2. Sketch of the proof: The regularity of A is classical. The one of B is a consequence of the continuity of the trilinear form

(u,v,w) e H,.(n;R) x Hb(!l;R) x Hc(n;R) t-t

i

uvwdx

for nonnegative real numbers a, b, c such that a+ b + c > 3/2 (see [4]). o Proposition 1 shows that the usual variational formulation of the Navier-Stokes problem is equivalent to the equation

v.Au + B(u, u)

= 1r{f).

(8)

Moreover, the regularity of the Stokes equation combined with iterated evaluations of the regularity of (Du]u shows that if n is of class cr, r = max(2, m + 2), and I E nm(n; R3 ), m :::: -1, any solution u of the Navier-Stokes equation is in nm+2(n; JR.l). 3.

TRANSPORT

3.1. The Speed Method. In this section, we consider a hold-all D which contains the set 0 0 filled by the fluid and a (time-dependent) vector field V defined on D which is used to define the perturbed sets n. based on !lo: each point X E !lois continuously

52

S. Boisgerault, J.-P. Zolesio

transported by the ODE defined by the field V. The parameter which controls the amplitude of the deformation is denoted by s. Precisely, the hold-all D is assumed to be an open bounded set of class Ole, k ~ 1 and s is in a (possibly infinite) interval I C R such that 0 E I. The vector field V is assumed to be an element of the set _E'l.k(I, D) (or simply _E'l.k) defined by

E"•1 (I, D)= {V E C"(I; C1 (D; R')) I 'Vs E I, V(s) n = 0 on 8D}, 0

(9)

where n denotes the unitary outer normal to D. Then we may define the mappings t-+ T,(V) (or simply T, when there is no possible confusion on the vector field) as the solution of

'Vs E I,



and also the perturbed sets s ......

(10)

= V(s) oT, and To= ld

n. by n, =

T,(no).

(11)

We recall the following result which can be found in [9], [11]: Proposition 3. Let V E .E'l•1 (I, D), n ~ 0, k ~ 1, be a given vector field. Then i) 'Vs E I, T,(V): D-+ D and D-+ D are one-to-one mappings. ii) s t-+ T,(V) E C"'+ 1(I; C 1 (D; D)) and s t-+ [T,(V)]- 1 E C"'(I; C 1 (D; D)). iii) 'Vs E I, 'Vx E D, DT,(x) is invertible, and the mappings s t-+ DT, and s t-+ [DT,)- 1 are in C"'+l(I; C1 - 1 (D; Jl3x3)). As a first consequence of this proposition, the family of perturbed sets has its boundary regularity preserved for v smooth enough: if no is of class cr' r $ k, then for any s E I, n, is also of class cr. Remark 2. In order to make the future calculations easier, we define 'Y· = det(DT,) and C, = 'Y; 1DT,.

{12) .E"•1 ,

Notice that, as a simple consequence of the Proposition 3, for V E n ~ 0, k ~ 1, 'Vs E I, 'Y; 1 = det([DT,)- 1) exists and, moreover, 'Ya and 'Y; 1 E C"'+l(I;C1 - 1(D;R)). As 'Yo = 1, by continuity, 'Vs E I, 'Ya > 0. Obviously, we also have the inversibility of C, for any s E I and the regularity C, and 0; 1 E C"'+l(I;Cic-1(D;RJx3)). 3.2. Correspondence between vector fields. The diffeomorphisms T, defined in the previous section are used to build a correspondence between the mappings defined on the open and bounded set no and those defined on n,. As for any s E I, T, and T,- 1 are Lipschitz, the mapping [v t-+ voT,J is an isomorphism between HJ(n,;R) and HJ (no; R) (see [8]) and the equation D[v o T,) = [Dv o T,]DT, holds. Lemma 1. Assume that no is an open and bounded subset of R_3 and that V E E'l•2 • Then the mapping T, defined between functions no-+ R3 and n, -+ JR3 by T,(u) o T,

= C, u,

is an isomorphism between V(n0 ) and V(n,).

{13)

Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow

53

Proof: As for all s E /, C, and C; 1 are C 1 (D; R3 x 3 } (see remark 2}, T, is an isomorphism between HJ(Oo; JRl) and HJ(S1,; ({3). Moreover, for any u E HJ(S10 ; JRl) and cp E HJ(n,; R3 }, we have

( div(T,(u)}cpdx=- ( T,(u)·Vcpdx=- ( (T,(u)oT,)·(VcpoT,}"y,dx

k.

k.

~

and as V(cp o T,) = DT,•(Vcp o T,), ( div(T,(u)}cpdx=- ( (C; 1T,(u)oT,) · V(cpoT,}dx= ( (divu}cpoT,dx=O.

lo.

lr~o

lno

The mapping cp t-+ cp o T, being an isomorphism between HJ(S1,; R) and HJ(S10 ; JR) , divT,(u) if and only if divu = 0 which achieves the proof. o Remark 3. As 1', : HJ(S10 ; R3 ) -+ HJ(S1,; JR3 ) is an isomorphism, so is its adjoint

1': : n-l(f!,; JR3) -+ H-l(f!o; JR3).

3.3. The Transported Navier-Stokes equations. In the sequel, the objects (operators, vector fields, duality brackets) associated with the perturbed set n. will be noted with a subscripts. Many of them are used to defined objects (by different means) associated with the initial set 110 ; they are noted with the superscript s. Precisely, we define the following correspondences: • Velocity fields u, E V 1(S1,) (solutions of the Navier-Stokes equation in 11,) and test functions v, E V 1 (S1,) are associated with the fields u' E V 1 (0o) (resp. v' E V 1 (S10 )) by: u' = T; 1 (u,) (resp. v' = T; 1 (v,)). • Force fields j, E H- 1 (11,; R3 ) (or in H - 1(D; JRl)) are transported in f' E H- 1 (0o; JRl) (or in n- 1 (D;JR3 )) defined by:

!' = T,(J,).

• The (linear and bilinear) operators A, and B, from H 1 (S1,;1R3 ) to n- 1 (S1,;JR3 ) defined by (7} are associated with A' and B' linear and bilinear from H 1 (S10 ; 1R3 } to H- 1 (S10 ;R3 } by:

A' =

T. o A o T, and B' = T. o B o (T, ® T,).

(14}

With these notations, we have Theorem 1. Assume that V E £0•2 and that f E H- 1 (D; 1R3 } . The field u, E V 1 (S1,) i.s a solution of the Navier-Stokes equation in 11, if and only if u' E V1 (110 } i.s a solution of

vA'u' + B'(u', u')

= 1r(j').

(15)

The operators A' and B' satisfy for any u, v in H 1 (S10 ; Rl) A'u = c:div {D[C,u]'y; 1C; 1 (C:)- 1 ) and B'(u,v) = c;o(C,u) · v 3

and for f E L (D; lR 2

},

we

{16}

have

f' = [DT,}*(f o T,).

(17}

54

S. Boisgerault, J.-P. Zolesio

Remark 4. The pressure is not explicitly transported in theorem 1. Nevertheless, if p, E L2 (0,; R)/R is a pressure solution of the Navier-Stokes problem in 0, that is if vA,u, + B.(u, u,) + 'ilp, = J, then it can be verified that vA'u' + B'(u', u') + 'il(p, o T,) = j'.

It is therefore natural to define the transported pressure p' by p•

= p, oT,.

(18)

The operators involved in the equation (16) have the following regularity:

Proposition 4. Assume that V E f:O.m+l, m ~ 1. Then i) ' 0. Many tangential calculus formulas are shown easily with the use of the equations (22) to (23) for smooth functions and then extended by density. In particular, we will need the identity divr(yv)

= ydivrv + Y'ry · v

(24)

which is true for any y E C 1 (fi;R) and v E Hi+E(f2;JR3). Proposition 5 (Integration by part on r). For all (y,v) E H 1 (f;R) x H 1 (f;JR3) we have

i

Y'ry · vdr = -

i

ydivrvdr +

Lemma 2. Assume that u E HJ(O; R3 ) Du

Moreover, div u = 0 in

i ~ty(v

n H 2 (0; R3 ).

= [Du)nn•

on

· n)dr.

(25)

Then we have

r.

(26)

n (that is, u E V2 (0)) implies that [Du)n · n

= 0 and [Du]"n = 0

on

r.

(27)

Proof: Let y E HJ(O; R) n H (0; R) . As Ylr = 0, Y'rY = 0 and the equation (22) leads to Y'ylr = ~n. This r~ult, used for y = ui, 1 ~ i ~ 3, proves the first part of the lemma. The second part is proved as follows: we use the decomposition (22) on the boundary. As ulr = 0, divr u = 0 and we conclude that [Du]n · n = 0 on r. As [Du] = [Du]nn•, [Du]• = nn"[Du]• and [Du]•n = n(n"[Du]"n) = n(n"[Du)n) = 0. o 2

4.3. The boundary shape derivative. We may now define the boundary shape derivative of a mapping n 1--t y(O) which is shape differentiable in Hm, m 2::: 1 (resp. in m;::: 0): it is the element of nm-i(r;R) (resp. ofcm(r;R)), defined by

em,

Y~ =

Ylr - Y'rY · V(O).

(28)

58

S. Boisgerault, J.-P. Zolesio

Obviously, using the equations (20) and (22), we also have

y~ =

y'lr + :: (V(O) · n).

(29}

As the material derivative and tangential gradient which appear in formula (28) only depend on the values of yon the boundary r of n, we may more generally define the boundary shape derivative of a mapping r H y(r) such that there is an extension of y in Hm(n; JR) (resp. C"'(O; R)) which is shape differentiable. This is the way the boundary shape derivative of n may be defined (via a shape differentiable (fl extension of n in n) and we have (see (2)) n~ = -Vr(V(O) · n).

(30)

For two mappings y and z, respectively, shape differentiable in H and in the product is shape differentiable in H 1 and its boundary shape derivative is given by 1

C1,

(yz)~ = (y)~z + y(z)~.

(31)

Proposition 6. Assume that y is shape differentiable in H 1 • Then

! (i.

y(r ,) dr)

i y~ + i ~~:y(V(O)

~=O =

dr

. n) dr.

(32}

4.4. PDE satisfied by u'. Adjoint Equation. The assumption A2 implies that the linearized Navier-Stokes problem with a right-hand side in H- 1 (flo; JR3 ) and homogeneous Dirichlet boundary condition has a unique solution u E V 1 (n0 ) . The operator L: V 1 (f20 )-+ w- 1 (flo) involved in abstract formulation L(u) = 1r(g) of this equation being an isomorphism, the adjoint operator L* : V 1 (flo) -+ w- 1 (n0 ) is also an isomorphism (we made the identification w- 1(n0 ) ~ V 1 (flo)'). The PDE form of the adjoint equation L*(Tl) = 1r(g) is -11A11 + (Du]*Tl- (D11]u + V( = g for 11 E V 1 (n0 ) and g E H- 1 (n0 ; IR3 ). The well-posedness of the two problems

1}

-vAv + (Du]v + (Dv]u + Vq = 0 {-11A11 + (Du)*Tl- (D11]u + V( = 0 divv=O 2) div77=0 { vir = h 11lr = h

(33)

for h E H 112 (r; Jil) follows directly from the existence of g E V 1 (flo} such that Ylr = h if h satisfies the compatibility condition

no

i

h n dr = 0 0

(34}

(remember that is connected). Moreover, for both equations, and from the regularity of u implied by A2, the solutions v and 17 corresponding to a boundary condition h E H 312 (r; R) are in H 2 (f2 0 ; R). The problems (33.1} and (33.2} are used intensively in the calculations of the shape derivatives because of the following result: Proposition 7. The couple (u',p') is the only solution (v, q) of the linearized NavierStokes equation at u {99.1} with the boundary condition vir= -(Du]n(V(O} · n).

Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow

59

Proof: First we notice that [Du)n(V(O) · n) · n being zero on the boundary (see (27)), the compatibility condition (34) is satisfied. Moreover, for any 'P E V(!l0 ; ~), for any lsi being small enough, I{) E V(!l,; lR3 ) and any solution u, of the Navier-Stokes satisfies equation in

n.

{ ( -v~u. + [Du,)u, + Vp, ln. 1

f) · 'P dx =

=

lv{ ( -v~U. + [DU,]U, + V P, -

f) . 'P dx

0,

3

2

where [s 1-+ U,] E C (/; H (D; IR )) and [s 1-+ P,] E C 1 (/; H 1 (D; R)) are the extensions considered in the equation (21). The differentiation of this equation with respect to s gives

L(-v~U' +

[DU']U + [DU]U'

+ V P') · 'P dx =

0

which implies that u' and rl are solutions of (33.1) . The boundary condition is proved as follows: as ulr = 0, for any 1P E C 00 (D; Rl),

f U·!pdf=O lr. using equation (32), we differentiate this equation for s + [Du)n(V(O) · n), we obtain

u~ = u'lr

i

= 0.

As (u · 'PYr

= u~ · 'P and

(u' + [Du]n(V(O) · n)) ·!pdf= 0

which implies that u'lr = -[Du]n(V(O) · n). o The calculation of the shape derivative of the functional considered in the next section also requires the following equations, given for divergence-free fields v and 11 in H 2 (!l;IR3 ) and u E V 1 (!l) by the repeated use of Green's formula.

In -ll~f1·

v dx =In 11 · (

-v~v) dx + 11

i[Du]•f1· vdx = In 11 · [Du]vdx and

In

V( · vdx

5.

=

i

((v · n) elf and

In In

i

([Dv]n · 11- [D11]n · v) elf

(35)

fo

(36)

-[D11]u · v =

11· Vqdx

=

i

11 · [Dv]udx

(11 · n)qdf.

(37)

SHAPE DERIVATIVE OF SHARP FUNCTIONALS

Geometrical Setting: In this section, the boundary r of n is the union of two disjoint two-dimensional submanifolds r a and r fJ · Let F and M be, respectively, the resultant of the forces and the moment of the forces exerted by the fluid on a piece r a of the boundary. The vectors F and M are given by F=

f

Jro

(-u(u)n+pn)df and M=

f

Jro

xx(-u(u)n+pn)df

(38)

60

S. Boisgerault, J.-P. Zolesio

with u(u) = v([Du]• + [Du]). Our aim in this section is to prove that F(n) and

M(fl) are shape differentiable functionals and to find the expression of their Eulerian

Derivative. We recall (see [9]) that a functional J(n) is shape differentiable at no if for any V regular enough, s t-t J(n,) is differentiable at s = 0 and this derivative (the Eulerian derivative) dJ(no; V) is linear continuous in V.

Theorem 4. Under the hypotheses AI with m = 2 and A2, both F and M are shape differentiable at no. For a given e E IR3 , let TJ be the solution of the adjoint equation {33.2} with the boundary conditions TJir.. = e and 11lr11 = 0 {resp. TJir.. = e x x and 11lr11 = 0}. Then the Eulerian Derivative of Fe= F · e {resp. Me= M ·e) is given by dFe(Oo; V) =

resp.

dMe(no; V) =

{ (! · e)(V(O) · n) df + v { [D71]n · [Du]n(V(O) · n) df (39)

lr..

r (x

lr..

lr

X

I+ v[Du]n

X

n) . e (V(O) . n) df

+v l[DTJ]n · {Du]n(V(O) · n) df.

(40)

Let r E C""(D; R) be a function such that r = 1 in a neighbourhood of r 0 and r = 0 in a neighbourhood of r fJ· We set YF = re and YM = re x x. The values Fe and Me may be studied as special cases of the functionals J(n) = J 1 (n) - J 2(n), where J 1 (n) = lpn·gdf, J2(u) = l u(u)n · gdf and g satisfies div g = 0 in a neighbourhood of rand the compatibility condition (34). It is obvious that with g = YF. J(n) = Fe. Moreover, as (x x ( -u(u)n + pn)) · re = ( -u(u)n+pn) ·(rex x), J(n) =Me with g = YM · The two additional properties come easily from the equations dive = 0 and div (e x x) =rot(e) · x- rot(x) · e = 0. Shape Derivative of J 1 (n): using the shape differentiability of p, g and n, and proposition 6, we obtain

dJ1 (no; V) = l

~(g · n) df + l

p(g · n)~ df + fr~~:(pn · g)(V(O) · n) df.

As pj. = p'lr+ ~(V(O) ·n) (see equation (29)), we have dJ1 (n; V) =A+ l r/(g·n) df with A= l

( : (g · n)(V(O) · n) + p(g ·

n)~ + ~~:(pn · g)(V(O) · n)) df.

• As g~ = g'lr + [Dg]n(V(O) · n) (straightforward adaptation of (29) to the vectorial case) and n~ = -Vr{V(O) · n) (equation {30)), we have (g · n)~ = g~ · n + g · nr = ([Dg]n · n)(V(O) · n)- g · Vr(V(O) · n). Using the integration by parts formula on the boundary (25), we get - l pg · Vr(V(O) · n) df = l divr{pg)(V(O) · n) df

-l~~:(pn · g)(V(O) · n) df

Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow

and finally, since div g A

= 0 on r'

i (: i

=

61

(g · n) + p[Dg]n · n + divr{pg)) (V(O). n) df

div (pg)(V(O) · n) df

=

i

Vp · g(V(O) . n) df.

• We set h = (g • n)n. The compatibility condition (34) being satisfied, we may introduce TJh the unique solution of the adjoint equation (33.2). We have then

i

p'n · gdf =

=

=

i

(p'fJt) · ndf

In div {p'TJt) dx In Vp' · TJt dx (as div fJt = 0) In (v~u'- [Du]u'- [Du1u) ·Tit dx.

Using equations (35) and (36) we come to

i

p'n · g df

=In (v~fJt - (Du]*fJt + [DfJt]u) · u' dx +

As li~TJt - [Du]*TJt + [DTJt]u

11

i

((Du']n ·Tit - [DfJt]n · u') df.

= V(~, equation (37) yields

In (ll~fJt- [Du]*fJt + [DTJt]u) · u' dx = LV(t · u' dx = i (t(u' · n) df = 0

due to u'lr = -(Du]n(V(O) · n) and [Du]n · n = 0 (see Lemma 2). On the other hand, (Du'Jn · TJt = (g · n)[Du'Jn · n and on the boundary, 0 = div u' = divr u' + (Du'Jn · n . Accordingly, we have by integration by parts on r and because u' = u~ 11 i(Du']n

= -11

i

= 11

i

Vr(g · n) · u' df.

(41)

(Vp · g + v([DTJt]n- Vr(g · n)) · [Du]n) (V(O) · n) df.

(42)

· TJt df

(g · n) divr u' df

• Finally, since u'lr = -[Du]n(V(O) · n), we end up with dJt(flo; V) =

i

Shape Derivative of J 2 (!l): we notice that lemma 2 implies that u(u)n = v[Du]n. The shape differentiability of n, g and u (in H 2 for the latter) implies that dJ2 (110 ; V) =

i

v([Du]n

i

·g)~ df +

i ~ev([Du]n

As ([Du]n·g)r = (Du]rn·g+[Du]nr · u+[Du]n·g~ and [Du]r we have

B

=

lv

dJ2(!l; V) = B +

((g · n)[Du1n ·

n+

· g)(V(O) · n) df.

= [Du'Jir+[D2 u]n(V(O) ·n),

v[Du']n · gT df with gT = g- (g · n)n

and

(n*(D2 u]n · g + (Dg]n · (Du]n + ~~:(Du]n · g)(V(O) · n)) df.

S. Boisgerault, J.-P. Zolesio

62

• The first term of B has already been calculated (see equation {41)) . The second is involved in the decomposition of the Laplace operator on the boundary (23). As ~ru = 0 and [Du]u=O on r, 11n•[D2 u]n = -11~~:[Du]n + Vp- Jon r . Therefore,

B =

i

((Vp-!) · g + 11[Dg]n • [Du]n- 11Vr(9 • n)) (V(O). n) di'.

• We seth= 9T· The compatibility condition {34) being satisfied, we may introduce 112, the unique solution of the adjoint equation {33.2). The equation {35) yields 11 i[Du']n · gT di'

=In (-11~112 • u' + 112 • (11~u'))

dx + 11 i[D112]n · u' di'.

Using -11~'72 = -[Du]•112 + [D112]u- V( and 11~u' = [Du]u' + [Du']u + Vp', and then the equations {36) to {37), we come to 11 i[Du1n · 9T di'

=

i

({u' · n) df +

i

(112 • n)p' df' + 11 i[D112]n · u' df'

and the two first terms of the right-hand side vanish. • We finally have

dJ2(flo; V) =

i

((Vp-!) · g + 11[Dg]n • [Du]n- IIVr(g · n)) (V(O) · n) df'.

{43)

Shape Derivative of J{fl): with the expressions {42) and {43), we come to

dJ(fl0 ; V)

=i

(! · g)(V(O) • n) + 11 i[D(17- g)]n · [Du]n(V(O) · n) di',

{44)

where 17 = 771 + 112 is the unique solution of the equation {33.2) with the boundary condition TJir = g. This expression easily leads to formulas {39) and {40) of theorem 4 with the choices of g = 9F and g = 9M· 6.

CONCLUSION

The natural extension of this work is the study of the shape derivative of functionals that require even more regularity than the one considered in section 5. For example, for the functionals that characterize the uniformity of the pressure on a given body, the shape differentiability of pin H 1 {which corresponds tom = 2) is not sufficient. Notice also that with a slight adaptation, the case of non-homogeneous boundary conditions, which appear for example in the study of a body which moves in a fluid with a constant velocity, also fits in the preceding framework. REFERENCES

[1) J . A. Bello, E. Fernandez-Cara, J . Lemoine, and J. Simon. The differentiability of the drag with respect to the variations of a lipschitz domain in a navier-stokes flow. SIAM J. Control Optim., 35(2):626-640, 1997. [2) Desaint, F.R. and Zolesio, J .-P. Manifold derivative in the Laplace-Beltrami equation. J. 1'\mct. Anal., 151(1):234-269, 1997. [3) L. C. Evans and R. F. Gariepy. Mea~~ure Theory and Fine Propertiu of.FUnctiom. Studies in Advanced Mathematics, 1992.

Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow

63

(4] C. Foias and R. Temam. Structure of the set of stationary solutions of the navier-stokes equations. Comm. Pure Appl. Math., 30:149-164, 1977. (5] V. Girault and P.-A. R.aviart. Finite Element MethiHU for Nauier-Stokes Equatioru, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1986. (6] J . G. Heywood. Remarks on the possible global regularity of solutions of the three-dimensional navier-stokes equations. In Progreu in theoretical and computational fluid mechanics. 3rd winter !chool, Pa8eky, Czech Republic. , volume 308 of Pitman Res. Notes Math. Ser. 308, pages 1-32. Harlow, Essex: Longman Scientific & Technical, December 1994. (7] P.-L. Lions. Mathematical topics in fluid mechanics. Vol. 1: /ncompreuible model6. Clarendon Press, Oxford, 1996. (8] J. Neeas. Les methodes directes en theorie des equatioru elliptiques. Masson et Ci•, 1967. (9] J. Sokolowski and J.-P. Zolesio. Introduction to Shape Optimization. Shape Seruillity Analy.ti8, volume 16 of Springer Series in Computational Mathematic.t. Springer-Verlag, 1992. (10] R. Temam. Nallier-Stoke.t Equatioru. Elsevier Science Publishers B. V, North-Holland, 1984. [11] J.-P. Zolesio. Identification de Domaines par Deformations. These de Doctorat d'Etat, Nice, 1979.

S. Boisgerault, J.-P. Zolesio, CMA, Ecole des Mines de Paris- INRIA, 2004 route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex, France. Email: Sebastien. BoisgeraultGsophia. inria. fr,

Jean-Paul.ZolesioGsophia.inria.fr

Second-Order Shape Derivative for Hyperbolic PDEs J . CAGNOL, J.-P. ZoLEsiO Abstract. In this paper we study the second-order shape derivative for the solution to the wave equation with Dirichlet boundary conditions. We first prove the shape derivative exists for this problem. Specific complications arise when differentiating with respect to the domain hyperbolic PDEs; this requires an exclusive approach using the hidden regularity described in [7]. This study has been started in [1] and announced in [2). We present here a new result where the data are no longer defined on the whole D and then restricted to the subsets of D but rather defined differently on each domain of D. Moreover, the differentiability is improved under stronger regularity of those functions. The Dirichlet boundary condition is now nonhomogeneous which was also necessary to consider the secondorder shape derivative. Keywords: hyperbolic partial differential equations, shape differentiability, domain optimization Classification:

35L05, 65Kl0, 49-99

1. INTRODUCTION

1.1. Shape and Material Derivatives. We consider a bounded domain D in RN and a family O~c of open domains n in D whose boundary r = 00 is a C" manifold oriented by the unitary normal field n outgoing to n. Throughout this paper we assume k ~ 2. Let T be a nonnegative real and I = [0, T] be the time interoal. We note Q =]O; T[xn the cylindrical evolution domain and I: =]O,T[xr the lateral boundary associated with any element n of the family O~c. Let £1c be the set of V E C([O, S]; C"(D, RN)) with (V, nao} = 0. For any V E Et we consider the flow mapping T,(V); we have V(s)(x) = (f.T,) o T,- 1 (x). For each s E [0, S[, T, is a one-to-one mapping from D onto D such that T0 = I, (s,x) t-+ T,(x) belongs to C 1 ((0; S[, C"(D; D)) with T,(oD) = oD and (s, x) t-+ T,- 1 (x) belongs to C([O; S[, Ck(D; D)). Such transformations were studied in [3] and [4] where a full analysis of the situation was given. The family O~c is stable under the perturbations n t-+ n,(V) = T,(V)(fl). We denote by Q, the perturbed cylinder ]0; T(xfl,(V), r, = 00, and I:, =]0, T(xr, the perturbed lateral boundary. We consider a map defined on the family O~c

(!,go 8, ~~'· t/J) : o1c-+

U (L (Q) x H (I:) x H (n) x L (n}), 2

neo.

1

1

2

Second-Order Shape Derivative for Hyperbolic PDEs

65

verifying /(0) E L2 (Q), g(r) E H 1 (E), cp(O) E H 1 (0), .,P(O) E £ 2 (0) with possible compatibilities conditions described later. Let K be a coercive symmetric matrix with coefficients into L 00 ( / , W 2•00 (D)) n W 1•00 ( / , L 00 (D)). To each element 0 E O~o we associate the solution y = y(O) of the generalized wave problem

{ffy- div (KVy) = f on Q y = g on E, y(O) = cp on 0,

o1y(O) = .,P

on 0. }

(1)

For any V E Ct and s E [0; S] we set y, = y(O,) E L2 (Q,). Following [1], [2], [9], [6], [10] the mapping 0 .-.+ y(O) is said to be shape differentiable in L2 (1, Hm(D)) if there exists Y in C 1 ([0; SJ, £ 2 (1, Hm(D))) such that Y(s, ·, ·)lq. = y(O,) and HY(s)- Y(O))- a.Y(O) -+ 0 in £ 2 (/, Hm(D)) ass-+ 0. Then a.Y(O, ·, ·)lq. which is the restriction to Q of the derivative with respect to the perturbation parameter s at s = 0, is independent of the choice of Y verifying those properties. We define, as well, the weak shape differentiability replacing the convergence in £ 2 (/, Hm(D)) by a weak convergence in that space.

Definition 1 (shape derivative). The shape derivative is that unique element y'(O; V) = ( :

uS

y) I

E L (Q). 2

a=O (t,z)EQ

In this paper we shall study the shape differentiability for the Dirichlet-wave problem

(1) under shape differentiability assumptions on the data f(O), g(f), cp(O) and .,P(O). Those functions depend on the domain as opposed to simple restrictions to 0 of functions defined on D. This is required to do the second-order shape derivative. In some weaker situations the solution y(O) will be only weakly shape differentiable in L 1 (1, H- 1(D)). The definition is the same with L 1 (I, H- 1(D)) instead of 2 L (I,Hm(D)) . Following [6], [9], [10] the boundary data g is shape boundary differentiable in L 1 (I, H"(f)) if there exists G in C 1 ([0; S], L 1 (1, HI>+4(D))) such that a':.. G(s) = 0 on E, and G(s, ·, ·)1~:. = g{f,} on E,. The restriction toE of the derivative with respect to the perturbation parameter s does not depend of the choice of the mapping G verifying those properties ( cf. [6]}. Similarly one defines the weak boundary shape differentiability.

Definition 2 {shape boundary derivative). The element

I

9~(r; v) = (~a) OS •=O (t,z)EE is the shape boundary derivative of g.

Definition 3 (material derivative). The element y(O; V) is the material derivative of y in L2 (I, Hm(D)) (resp. weakly in L2 (I, Hm(D))) if it is the limit in L2 (I, Hm(D)) (resp. weakly in L2 (1, Hm(D))) of~ (y(O,) o T,- y(O)) ass tends to 0. Let V and W be two autonomous vector fields in £~o.

J. Cagnol, J.-P. Zolesio

66

Definition 4 (Second-Order Shape Derivative). y"(O; V, W)

= (y'(O; V))'(O; W).

Definition 5 (Second-Order Boundary Shape Derivative). g~(r; v, W)

= (gHf; V))Hf; W).

1.2. The Results. Let m and q be two integers with q ~ 1 (the subsequent results the spaces being obtained could be extended to positive reals m and q with q ~ through an interpolation; however, for the sake of shortness we shall restrict the study to integers indices only). Let ('Hm,9 ) be the hypothesis concerning the regularity of the data: I belongs to L 1 (1, Hm(o)), a~m) I belongs to L 1 (1, £ 2 (0)), g belongs to HV+l(E), r.p belongs to nm+l(O) and 1/J belongs to nm(O) with compatibility conditions and suppr.p 0, Ve e]O; e•[, 3o e]1 -

ollw'IIL""(t,HJ(n)) ~ 2a, +

Jb.

.,.,, 1 + f1[, Vs E [0; E[,

and ollo,w'IIL""(f,L2(0)) ~ 2a, +

Lemma 2. When s is in the neighborhood of 0 we have a, MG and Mb are independent of s.

~

MG and b,

Jb,. ~

M6 where

Proposition 2. llw'IIH•(Ql is bounded and 3w E L 00 (1, HJ(n)) n W 1•00 ( / , L2 (n)) such that wn; _. w weakly in H 1 (Q) {thus wn; ~ w in L 2 (Q)). Proposition 3. The function w belongs to Z 1(I, n) and is unique. For each sequence (sn) of corollary 2 there is a function w. We prove the above proposition showing w is the solution of a well-posed problem. Lemma 3. For any sequence (sn), w(O) = (Vt.p, V(O)) + t.p1 • Lemma 4. There exists 3 E L 2 (Q) such that for any sequence (sn), Pw Moreover, as a consequence of corollary 2 we have w

P(w) = 3 on Q { w = 0 on !:, w(O) = (Vt.p, V(O}) + t.p' on where

EK

= !(DV(O).K +

(V f, V(O})

= 0 on !:.

n, o,w(O) =

= 3.

Hence

(Vt/J, V(O))

+ t/J'

on

n,

•(DV(O).K)) and 3 is given by

+ f' + div(V(O))div(KVy)- div(((divV(O)).K- 2eK + DK.V(O))Vy).

We refer to [7, theorem 2.1] and [7, section 4] to prove that w is unique and belongs to Z 1 (I,n).

Second-Order Shape Derivative for Hyperbolic PDEs

Proposition 4. Under (1lt), (1lD and k

69

= 2 there exists iJ e Z 1 (I, 0) such that

y'- y ...... iJ weakly in L 2 (1, H 1 (D)) ass-+ 0 s the material derivative iJ is solution to the problem P(iJ) = 3 on Q { iJ = 0 onE, iJ(O) = (Vtp, V(O)} +'{I on 0,

Corollary 1. Under (1lt), (1lD and k c e]o; 1]

81y(O) = (Vt/J, V(O)) + '1/J' on 0 .

= 2 there exists iJ e Z 1 (I, 0) such that for any

y•- y -+ iJ strongly in L2(I, H 1-E(D)) ass-+ 0. s Let us enhance the differentiability for greater values of m.

Proposition 5. Let m ~ 2 be an integer and assume (1l.n), (1l:,.), k differentiation takes place in L 2 (I, Hm- 1 (D)).

= m.

The shape

Proof. Let us consider em(I, Q)

=

{8 E

zm+ 1 (I, r!) s.t. 8h; = 0, 1

8(0)

= ~o.

8t8(0)

= ~b

and

P8 E L (I,Hm(D)), a~ml(P8) E L (I,L (D))} and the norm

1

2

II · lie"' defined by

ll8llem = ll8llz...+l(IO) + IIP811Li(f •H"'{O)) + lla~m)(P8)11 Ll(f,L2(0)) · '

Let

w be defined by w:

(0; S[xe(I, 0) -+ £ 1 (/, £ 2 (0)) (s, 8) t--+ P8- f o T, + (P,- P)y'

we have D 9 w = P, that function is continuous on (0; S] x e. Moreover, according to (7, theorem 2.2], D,w(O, 8) is an isomorphism between em(/, 0) and {! E L 1 (I, Hm(D)) I a~m)(P8) E L 1 (I, L 2 (D))}. Let 8 E e(I, 0) then w(s,8+8) - w(O, 8) = P(8+8) - foT,+(P, - P)(8+8) -P8+ I hence w(s, 8 + 8) - 'l!(O, 8) = P8- (! o T, -f)+ (P, - P)(8 + 8) since f E £ 1 (/, Hm(O)) we have ~(loT,- f)-+ V f.V(O) in L 2 (I, Hm- 1 (S1)). Moreover, (P, - P)(8 + 8) = o(s, 8); hence

w(s, 8 + 8) - 111(0, 8) L 2 (I,

= sVf. V(O) + o(s, 8, P8)

Hm- 1 (0)).

and V f .V(O) E Therefore W is differentiable at (0, 8) in L 2 (I, Hm- 1 (D)). The implicit function theorem applies: if 8 E t: satisfies W(s, 8) = 0 then 8 is a function of s in the neighborhood of 0 and 8,8 exists. Since lll(s, y•) = Py'- f oT,+P,y' -Py' = P,(y,oT,)- f oT, = 0 the differentiation

takes place in L2 {I, Hm- 1 (D)).

o

70

J. Cagnol, J.-P. Zolesio

2.2. Shape Derivative with Strong Regularity of the Data. Let ~m be an extension operator from ym(n) onto Hm(D) and 'B!, = (J H 'Bm(OoT,) oT.- 1. For the sake of shortness we will also denote by ~m the extension id ® ~m· The same remark applies to ~:,.. Proposition 6. H~:,.y,- ~mY)-+ Y in L2 (1 x D) ass tends to 0.

Proof. This proposition comes from the decomposition

~:,.y,- ~mY= ~:,.y,- ~mY'+ ~m

(y'- Y) .

s s s The use of the previous sections for the second term of the right-hand side gives the result. 0

Proposition 7. y' = Ylq is the shape derivative of y. Corollary 2. y' = iJ- (Vy, V(O)) on Q. We denote by y~ the shape derivative in u. (y' = y~.) Lemma 5. Assume 9 E C 1 ([0; S[, L 1(1 x D)), we note 8,

e.(x, t) dx dt) = f 8,9(0, x, t) dx dt s }q. •=0 }q Proposition 8. P(y') = f' on Q.

: (f

= 9(s, ·, ·) and 8 = 80 ,

+ f

IE

then

fJ(x, t) (V(O), n) df dt.

Corollary 3. The shape derivative y' is solution to

P(y') = f' on Q y' = -~ (V(O), n) onE,

y'(O) = r.p' on n,

8ty'(O) =

t/1

on n.

}

(4)

If y is shape differentiable at any n E Ot in any domain field V, then y'(O; V) = ;f;YI{o}x/xn. we have ;f;YI{cr}x/xO.. = y'(O.,(V), V(u)) = y~ andy~ is solution to P(~) = f~ on

{ y~

Q.,

= -~V(u).n.,

on E.,,

y~(O) = rp'., on

n.,,

8ty~(O)

= t/1.,

on

n.,.

2.3. Shape Derivative with Weak Regularity of the Data. In this section, we suppose (1£0 ), (1£~) and k = 2. According to [7, theorem 2.1) and [7, section 4) we have ~ e L2 (E); thus, using [7, theorem 2.3) and [7, section 4) on (4) we obtain that y' exists, is unique, and belongs to ZO(I, n). Let us prove y' is the shape derivative of the solution to (1). Since~ (V(O),n) E L 1 (/,Hm- t(r)), [7, theorem 2.1) and [7, section 4) say that (4) has a unique solution y' defined in ZO(I, n). Hence the shape derivative may continue to exist even though the propositions to prove the existence of iJ do not apply anymore. Remark 1. It should be noticed that y' continues to exist while iJ does not exist anymore. From [7, theorem 2.1] and [7, section 4), the solution to (1) exists, is unique and belongs to C(I, L2 (D)). Let y be this solution. We denote by (J a function in L""(I, L2 (0)). Let us introduce h(s) = JQ. y,(J dx dt and h(s) = JQ. y~(J dx dt.

Second-Order Shape Derivative for Hyperbolic PDEs

71

Lemma 6. The function h is absolutely continuous. More precisely h(s) = h(O)

J; h(u) du.

+

This lemma may be proved using a mollifier sequence and the result of the previous section.

Proposition 9. ~:..v·;~m!l ~ y' in L 1 (I, H- 1 (D)) as s tends to 0. We now suppose (} E L 00 ( / , H 1(0)). To prove proposition 9, we are going to prove that h is differentiable. Since lemma 6 applies, we can do so by proving lim.-.0 h(s) = ii(O). That will be done using the next 4 lemmae.

Lemma 7. Let A, E Z 1 (/, 0) be the solution to P(A,)

= (}

on Q, A,(T) = 0 on 0,,

A,= 0 onE,, then h(s) =

JQ. J;A, dx dt + JE. t-~ (K n, n,) (V(s), n,)

Lemma 8. Let W(s) h(s)

=

8,A,(T)

k

= 0 on 0,

(5)

df, dt.

= 4(K,n,n) ii•DT,- 1nii- 1DT,- 1V(s) oT,

J; oT,A' dxdt+

}

then

L([B(y'a:A·)r- [:;f- [~:r)

(W(s),n)dr dt

Lemma 9. The functions t-+ JE(J!.8') 2 (W(s), n) df dt is continuous in 0 for(}= y, (} = A and (} = y + A. Proof. Let W(T) = 0; this request is not restrictive since T can be taken as large as we want. Following [5) we use the extmctor technique. By density, 8,8• and 'iJ(}' are continuous as s -+ 0. Moreover, • DT,- 1W(O} and 'il(DT,- 1 • DT,- 1 )W(O) are bounded. 0

Lemma 10. The function s t-+ JQ. J; o T,A' dx dt is continuous in 0. The differentiability L 1 (/, H- 1(0)) may be improved when /' = 0. Assume 9 e L00 ( / , L2 (0)) then A, the solution to (5), exists and belongs to ZO(I, 0). Because

f' = 0 we get

h(s) =

f ~· : · (Kn,n,) (V(s),n,) JE. • •

df, dt.

The continuity is a consequence of lemmae 8 and 9 that does not require a better regularity of 9. This proves h is continuous in 0, hence

Proposition 10. Assume f' = 0, (1lo), (1£~) and k = 2 then the solution to (1} is weakly shape differentiable in L 1 (/, L 2 (D)). Moreover, y' E zm(/, 0) andy' is solution

to {3).

J. Cagnol, J.-P. Zolesio

72

3.

NoNHOMOGENEous DIRICHLET

B.C

We consider the problem with a nonhomogeneous Dirichlet boundary condition (1). We consider a function G of definition 2.

Theorem 3. Assume (1tm,9 ), (1£:,.,9 ) and k = max{2,min{m,q}}. The solution to (1) is • weakly shape differentiable at s = 0 in L 1 (1, H- 1 (D)) if m = 0 or q = 1. • weakly shape differentiable at s = 0 in L 1 (1, L 2 (D)) if (m = 0 or q = 1} and

f'

= o, 9r = o.

• weakly material differentiable at s = 0 in L 2 (1, H 1 (D)) if m = 1 and q = 2. • strongly shape differentiable and strongly material differentiable at s = 0 in £2(J, Hmin{m-1,q-2}(D)) if m ~ 1 and q ~ 2. Moreover,

y' E zmin{m,q} (1, f!)

furthermore, y' is solution to

P(y') = f' on Q = gj. - ~ (V(O), n} on E,

y'

y'(O)

= 1{1

on n,

81y'(O)

= 1/1

on n.

}

(6 )

Proof. The function u = y- G(O) is solution to

P(u) = f- PG(O) on Q { u = 0 on E, y(O) = rp - G(O) on

n, 81y(O} = t/J - 81G(O)

on

n.

Since G(O) E Hq+J(Q) we get PG(O) E H9-l(Q) thus f -PG(O) E £ 1(1, Hmin{m,q-U). If m ~ 1 and q ~ 2 we also have

8~miD{m,q-i})(f- PG(O)) E L1(J,L2(S1)) and (f- PG(O))' E L1(J,HmiD{m-t,q-fl). If m ~ 2 and q ~ 3, we have o~min{m-t,q- 2})(/- PG(O))' E £ 1 (1, £ 2(0)) as well. Moreover, rp - G(O) E H•+1 (S1) and t/J - 81G(O) E H 9 (S1); therefore theorem 2 gives the shape differentiability of u. Furthermore, u' belongs to zmin{m,q}(J, n) and is solution to

P(u) = !'- (P(8,GI,=o)) on Q u' = -:;: (V(O),n) onE u'(O) = rp'- 8,GI,=O(O) on n, 8tu'(O)

}

= 1/1- 8,8tGI,=o(O)

on

n

(7)

using 8,(PG) = P(8,GI,=o) and u' = y'- 8,GI,=O, (7) yields (6). Furthermore [7] and (6) give y' E zmin{m,q}(J,f!). When min{m,q} E [0; 1[, !' = 0 and gj. = 0, proposition 10 applies and gives the weak shape differentiability in £ 1 (1, L 2 (D)). 0 Theorem 1 is a consequence of theorem 3. The only remaining difficulty lies in the characterization of y". Indeed one has to compute the boundary shape derivative of gj. - ~ (V(O), n). This problem is solved using the following lemma ( cf. [6]).

Second-Order Shape Derivative for Hyperbolic PDEs

Lemma 11. We have (Vy)~(r; W)

73

= V(y'(O; W))lr + D 2 y.n (W, n) n~

= - V r (W, n) .

Remark 2. Assuming enough regularity on the data, one can iterate the method presented here to prove the solution to {1} is shape analytic. REFERENCES

[1] John Cagnol and Jean-Paul Zolesio. Hidden shape derivative in the wave equation. In Peter Kall, Irena Lasiecka, and Mike Polis, editors, S11stem Modelling and Optimization, pages 42- 52. Addison Wesley Longman, 1998. Detroit, Michigan. [2] John Cagnol and Jean-Paul Zolesio. Hidden shape derivative in the wave equation with dirichlet boundary condition. C. R . Acad. Sci., Paris, 11erie I, 326(9):1079-1084, 1998. Partial Differential Equations. [3] Michel C. Delfour and Jean-Paul Zolesio. Structure of shape derivatives for non smooth domains. J. Funct. Anal., 104, 1992. [4] Michel C. Delfour and Jean-Paul Zolesio. Shape analysis via oriented distance functions. J. Funct. Anal., 123, 1994. [5] Michel C. Delfour and Jean-Paul Zolesio. Hidden boundary smoothness for some classes of differential equations on submanifolds. ln I. Lasiecka and S. Cox, editors, Joint Summer Research Conference on Optimization Methods in PDE'11, vol. 209, p. 59 and 73. Contemporary Mathematics, AMS Publications, Providenoe, R.I., 1997. [6] Fabrice R. Desaint and Jean-Paul Zolesio. Manifold derivative in the laplace-beltrami equation. J. Funct. Anal., 151(1):234 and 269, 1997. [7] I Lasiecka, J.-L. Lions, and R. Triggiani. Non homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pure Appl., 65, 1986. [8] I. Lasiecka and R. Triggiani. Recent advances in regularity of second-order hyperbolic mixed problems, and applications. DvnamiC/1 reported. Exposition~~ in dvnamical 111111tem~~ . New 11eriu., 3:104, 162, 1994. [9] Jan Sokolowski and Jean-Paul Zolesio. Introduction to Shape Optimization. SCM 16. SpringerVerlag, 1991. [10] Jean-Paul Zolesio. Introduction to shape optimization and free boundary problems. ln Michel C. Delfour, editor, Shape Optimization and Free Boundariu, volume 380 of NATO ASI, Serie11 C: Mathematical and Ph1111ical Science&, p. 397 and 457, 1992.

J. Cagnol, Centre de Matbematiques Appliquees, Ecole des Mines de Paris, INRIA, 2004 route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex, France J.-P. Zolt~sio, also CNRS, Institut Non Lineaire de Nice, 1361 route des lucioles, 06560 Valbonne, France

An Axiomatic Approach to Image Interpolation F.

CAO,

v.

CASELLES, J.-M. MOREL,

c.

SBERT

Abstract. We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The Absolute Minimal Lipschitz Extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range. Keywords: image interpolation, degenerate elliptic problems, viscosity solutions

1. INTRODUCTION

Our purpose in this paper will be to discuss possible algorithms for interpolating scalar data given on a set of points and/or curves in the plane. Our main motivation comes from the field of image processing. A number of different approaches using interpolation techniques have been proposed in the literature for "perceptually motivated" coding applications [5, 18, 24]. The underlying image model is based on the concept of ''raw primal sketch" [19]. The image is assumed to be made mainly of areas of constant or smoothly changing intensity separated by discontinuities represented by strong edges. The coded information, also known as sketch data, consists of the geometric structure of the discontinuities and the amplitudes at the edge pixels. In very low bit rate applications, the decoder has to reconstruct the smooth areas in-between by using the edge information. This can be posed as a scattered data interpolation problem from an arbitrary initial set (the sketch data) under certain smoothness constraints. For higher bit rates, the residual texture information has to be separately coded by means of a waveform coding technique, for instance, pyramidal or transform coding. In the following we assume that a set of curves and points is given and we want to construct a function interpolating these data. Several interpolation techniques using implicitely or explicitely the solution of a partial differential equation have been used in the engineering literature [4, 5, 6]. In the spirit of [1], our approach to the problem will be based on a set of formal requirements that any interpolation operator in the plane should satisfy. Then we show that any operator which interpolates continuous data given on a set of curves can be given as the ~osity solution of a degenerate elliptic partial differential equation of a certain type. The examples include the Laplacian operator and the minimal Lipschitz extension operator (2], [16] which is related to the work of J. Casas [5, 6].

An Axiomatic Approach to Image Interpolation

75

Our plan is as follows. In Section 2 we introduce a formal set of axioms which should be satisfied by any interpolation operator in the plane and derive the associated partial differential equation. In Section 3 we discuss several examples of interpolation operators relating them to the set of axioms studied in the previous section. We shall concentrate our attention in a special one, the so-called AMLE model, giving existence and uniqueness results for the associated PDE. Its numerical analysis is given in Section 4. In Section 5, we give a generalization of the AMLE model when the data are singular. We conclude that the AMLE cannot be used in reconstructing occluded objects. Finally, in Section 6 we display some experimental results obtained by using the previous model. Although the whole theory will be developed in JR?, there are very few alterations in order to extend it to !Rn. In this short note, we do not give any proof but the interested reader may find them in [8] and [9]. 2. AXIOMATIC ANALYSIS OF INTERPOLATION OPERATORS

We begin by recalling the definition of a continuous simple Jordan curve. Definition 1. A continuous junction r : [a, b] -+ JR? is a continuous simple Jordan curve if it is one-to-one in (a, b) and r(a) = f(b). By Alexandroff Theorem, such a curve surrounds a bounded simply connected domain which we denote by D(f). Let C be the set of continuous simple Jordan curves in JR2 • For each r E C, let F(f) be the set of continuous functions defined on r. We shall consider an interpolation operator as a transformation E which associates with each r E C and each r.p E F(f) a unique function E(r.p, f) defined in the region D(f) insider satisfying the following axioms: (Al) Comparison principle:

E(r.p, f) :5 E(t/J, r) for any r E C and any r.p, t/J E F(f) with r.p :5 t/J.

(A2) Stability principle: E(E(r.p, r) lr•. ~)

= E(r.p, r) Iocr•)

for any r E C, any r.p E F(f) and r' E C such that D(f') ~ D{f). This principle means that no new application of the interpolation can improve a given interpolant. If this were not the case, we should iterate the interpolation operator indefinitely until a limit interpolant satisfying (A2) is attained. For the next principle, we denote by SM(2) the set of symmetric two-dimensional matrices. (A3) Regularity principle: Let A E SM(2), p E JR2 - {0}, c E lR and Q(y)

=

A(y-x y-x) ' 2

+ < p, y -

(where < x, y >= E~=l X;y;). Let D(x, r) = {y E IR2 boundary. Then

E(Q 18 o(s.rl•8D(x, r))(x)- Q(x) r2

12

-+

F(A

:

x > +c

IIY- xll :5 r} and aD(x, r) its

, p, c, x

)

as r -+ 0 +

{1)

F. Cao, V. Caselles, J.-M. Morel, C. Sbert

76

where F: SM(2) x JR2 - {0} x JR. x JR2 -+ JR. is a continuous function. This assumption is much weaker than what it appears to be. Indeed, assume only that given A,p,c,x, we can find a C2 function u such that D2u(x) Du(x) p, u(x) = c, such that the differentiability assumption (1) holds (with u instead of Q). Then, arguing as in Theorem 1 in [9] it is easily proven that (1) holds for all C 2 functions and in particular for Q. Together with these basic axioms, let us consider the following axioms which express obvious independence properties of the interpolation process with respect to the observer's standpoint and the grey level encoding scale.

=A,

=

(A4) Translation invariance:

E(rhcp. r- h)

= ThE(cp, r),

2

where Thcp(x) = cp(x +h), hE JR , tp E .1"(r), r E C. The interpolant of a translated image is the translated interpolant. (A5) Rotation invariance:

E(Rcp, Rr)

= RE(cp, r),

where Rcp(x) = cp(R x), R being an orthogonal map in JR?, cp E interpolant of a rotated image is the rotated interpolant. 1

.r(r), r

E C. The

(A6) Grey scale shift invariance: E(cp + c, r) = E(cp, r) + c for any r E C, any tp E .1"(r), c E JR. (A7) Linear grey scale invariance:

E(>.cp, r)

= >.E(cp, r)

for any). E JR.

(A8) Zoom invariance: E(6~cp.>.- 1 r)

= o~E(cp,r),

where 6~cp(x) = cp(>.x), ). > 0. The interpolant of a zoomed image is the zoomed interpolant. Axioms (A1), (A3) and (A4) to (A8) are obvious adaptations from the axiomatic developed in [1). As we shall always assume (A4) and (A6), F only depends on its first two arguments. Let us write G(A) = F(A, e1 ), A E SM(2), e1 = (1, 0). Then G is a continuous function of A. Given a matrix

A=(::).

let us write for simplicity G(a, b, c) instead of G(A). Then using an argument similar to the one developed in [1) we prove

77

An Axiomatic Approach to Image Interpolation

Theorem 1. ([9]) Assume that E is an interpolation operator satisfying (Al)- (A8). Let cp E C(f), u = E(cp, f). Then u is a viscosity solution of 2

G (Du

(,~:I' ~~: 1 ) ,D u (,~:I'~~) ,D u (~~· ~~)) = 0 2

2

in

D(r) (2)

In addition, G(A) is a nondecreasing function of A satisfying G(..XA) = ..XG(A) for all A E JR.

We do not give the notion of the viscosity solution explicitly for the general model

(2) and we refer to [9]. We shall specify this notion when needed for each concrete example below. From now on, we shall assume that the interpolation operator E satisfies (Al) - (A8). Using the monotonicity of G, we can reduce the number of involved arguments inside G.

Proposition 1. i) If G does not depend upon its first or its last argument, then it only depends on its last (resp. its first) argument. In other terms, if G(o, {J, -y)

= G(o, {J),

then

G = G(o) = oG(l),

if G(o, {J, -y)

= G({J, -y),

then

G

o, {3, 'Y E JR.

= G(-y) = -yG(l).

ii) If G is differentiable at 0 then G may be written as G(A) nonnegative matrix.

= Tr(BA)

where B is a

Proposition 1 i) is due to the fact that A--+ A(v,v-1-) (v E R 2 , lvl = 1, v..L being the vector obtained by rotation of 1r /2 of v) is not a monotone operator with respect to A. Thus if we assume that G is differentiable at (0, 0, 0) then we may write Equation (2) as 2

aD u

(

2 ( Du Du..L) 2 (Du..L Du..L) Du Du) IDul' IDul + 2bD u IDul' IDul + cD u IDul' IDul

= 0,

(3)

where a, c ;::: 0, ac - b2 ;::: 0 which is the same as to say that the symmetric matrix B with diagonal coefficients a and c and cross term b is nonnegative. 3.

Example 1. Given

r

EXAMPLES

E C and cp E F(f) we consider E 1 ( cp, f) to be the solution of ~u

ulr

=0 = 'P·

in D(r)

(4)

The operator E 1 satisfies all axioms (Al) - (A8) above. Just mention that the regularity axiom follows from the mean value theorem for the Laplace equation on a disk. It corresponds to the function G(A) = -Tr(A), that is G(a, b, c) = -(a+ c).

F. Cao, V. Caselles, J.-M. Morel, C. Sbert

78

We recall that this operator does not permit to interpolate points (see introduction). A more general situation is given by the so-called p-Laplacian

div(IVui"- 2Vu) Ulr = cp,

=0

in D(r)

(5)

where p ~ 1. Formally, after dividing by jVul"-2 , the above equation can be written as 2 (p- 1)Du

Du Du ) 2 ( Du.l. Du.l.) jDui'IDul +Du IDui'IDul =O

(

(6)

which is contained in the family of equations (3) with a = p- 1, b = 0, c = 1. As it is known, the value of u can be fixed at point if and only if p > 2. This corresponds to the case a > c. By specifying radial solution in a ball, it can be seen that, unless c = 0, which corresponds to the case p = co, the gradient of u can be unbounded. The case of p = co will be our next example.

Example 2. Our next example is more interesting and will be discussed in detail. Given a domain 0 with E C and cp E F(an) we consider ~(cp, an) to be the viscosity solution of

an

2

D u Ul.m

(Du Du) IDul' IDul = cp.

=0

. m n,

(7)

We consider Equation (7) in the viscosity sense. Given u E C(O) we say that u is a viscosity subsolution (supersolution) of (7) if for any '1/J E C 2 (0) and any x 0 local maximum (minimum) of u- '1/J inn such that D'f/J(x0 ) f. 0 2

D '1/J(xo)

(

D'f/J(xo) D'f/J(xo) ) ID'f/J(xo)l' ID'f/J(xo)l

~ 0 (~ O).

A viscosity solution is a function which is a viscosity sub- and supersolution. Equation (7) was introduced by G. Aronsson in [2] and recently it has been studied by R. Jensen [16]. Indeed, in [2] the author considered the following problem: Given a domain n in /Rn, does a Lipschitz function u in n exist such that sup IDu(x)l ~sup IDw(x)j, xEfl

:rei'!

for all (2 ~ n and w such that u - w is Lipschitz in 0 and u = w on fJO. If it exists, such a function will be called an absolutely minimizing Lipschitz interpolant of wj 80 inside n or AM LE for short. Notice that the above definition, if it defines uniquely u, immediately implies the stability of AMLE in the sense of (A2). Then it was proved in [2] that if u is an AM LE and is C2 in n, then u is a classical solution of

D2 u(Du, Du)

=0

in

n.

(8)

Later Jensen [16] proved that if u is an AMLE, then u solves (8) in the viscosity sense. Moreover, the viscosity solution is unique. We shall use the viscosity solution formulation of Equation (8). Given u E C(n) we say that u is a viscosity subsolution

79

An Axiomatic Approach to Image Interpolation

(supersolution) of (8) if for any

u-'1/J inn

'1/J

E C:Z(n) and any

x0 local maximum (minimum) of

D 2.,P(x0 )(D.,P(xo), D.,P(xo)) ~ 0 ($; 0). A viscosity solution is a function which is a viscosity sub- and supersolution. Then Jensen proved (16] a comparison principle between sub- and supersolutions of Equation (8) together with an existence result for boundary data in the space of functions Lipa(n) which are Lipschitz continuous with respect to the distance dn(x, y). We denote by d0 (x,y) the geodesic distance between x andy, i.e., the minimal length of all possible paths joining x and y and contained in n [16]. Observe that u is a viscosity subsolution (supersolution, solution) of (7) if and only if u is a viscosity subsolution (supersolution, solution) of (8). From this follows the corresponding comparison principle for solutions of (7).

Theorem 2. Assume that v is a subsolution and w a supersolution of (7) (equivalently of (B)). If v lao, w lanE Lipa(n) then sup(v- w) zen

=

sup (v- w).

(9)

ze80

Theorem 3. Given g E Lip8 (n), u is the AM LE of g into solution of (8) with u lao= g.

n if and only if u

is the

Let us stateR. Jensen's existence result for (7) in a way that makes explicit the fact that we are able to interpolate a datum which is given on a set of curves and points. Let us consider a domain n whose boundary an = o1n u ~n u ~n where 81 n is a finite union of rectifiable simple Jordan curves,

where C1 are rectifiable curves homeomorphic to a closed interval and

Ban= {x1 : i = 1, ...,N} is a finite number of points. The boundary data to be interpolated is given by a Lipschitz function r.p1 on 8 1n, two Lipschitz functions r.p~+• r.p~- on each curve C1, which coincide on the extreme points of C1, i = 1, ... , m and a constant value u1 on each point X;' i = 1' ... , N. We shall denote by the same curve where we take into account the direction of the normalvt(x), v;(x) = -vt(x), x E C1 as in Figure 1. When we write u 1ct = r.p~+ as in the next theorem we mean that u(y) --+ r.p~+(x) as y--+ x if< y, vt(x) >< 0, x E C, and similarly for ulc- = r.p~-·

c:' c,-

c,

j

Theorem 4. Given n, r.p11 r.p~+• r.p~_, u;, i

= 1, ... , m, j = 1, ..., N,

exists a unique Lipschitz viscosity solution u of

as above then there

80

F. Cao, V. Caselles, J.-M. Morel, C. Sbert

c~:l' ~~:I)= 0

2

D u uiB1n = '1'.1 ulc+ = cp~+

{10)

' Ufc:- = 'P~-• u(~i)

n

inn

i = l, ...,m

= Ui,

i

= l, ...,N.

c,

FIGURE 1.

General domain of interpolation.

This result enables us to define the following interpolation operator. Given cp E Lip8 (n), let ~{cp, on) be the viscosity solution of {7). Then

Theorem 5. The opemtor E2 defined above satisfiea axioms {Al)- {AS). For numerical reasons it is interesting to study the asymptotic behavior of the evolution problem corresponding to Equation {7). Then, under certain smoothness assumptions on the boundary data, we prove that the solution of the evolution problem converges to the solution of Equation {7). Let us consider the evolution equation :

2

= D u

c~:l' ~~: 1 )

in {O,+oo) x fl

{ll) u{O, x) = Uo(x) x E fl u{t,x) = cp(x) (t,x) E {O,+oo) X where we suppose that Uo(x) = cp(x) for all X Eon. We say that u E C{[O, +oo) X n) is a viscosity subsolution of {11) if u{O, x) = Uo(x), u(t, x) = cp(x) for all (t, x) E {0, +oo) X 8f2 and for any t/J E 02({0, +oo) X fl) and any (to, Xo) local maximum of u- t/J in {O,+oo) x fl

on,

01/J

Ot (to , xo)

2

(

Dt{J(to, xo)

Dt/J(to, xo) )

~ D t/J(to,xo) IDt/J(to,xo)l' IDt/J(to,xo)l '

{12)

if Dt/J(to, xo) =F 0 and

0:: (to, Xo) ~ suplui~ID2t{J(to, xo)(v, v),

if Dt{J(t0 , x0 ) = 0. Similarly we define a viscosity supersolution. A viscosity solution is a viscosity sub- and supersolution.

An Axiomatic Approach to Image Interpolation

81

Theorem 6. Let n be a generalized domain in IR? as described above. Suppose that 810, Ct, Ci-, i = 1, ... , m , have a bounded curvature; the initial condition Uo(X) and the boundary data rp(x) have bounded second derivatives. Then there exists a unique continuous viscosity solution u( t, x) of {11) such that u(t) is Lipschitz for all t > 0 with uniformly bounded Lipschitz norm. Moreover, u(t, .) -+ u 00 , where u 00 is the unique viscosity solution of {10).

Geometric interpretation of Equation (8) Proposition 2. Let u be C 2 and satisfy D 2u(Du, Du) = 0. We define a gradient line as a curve x(t), t E (a, b) such that Du(x(t)) =F 0

and

x'(t)

Du

= IDul (x(t)).

Then there is a constant C such that for every t E (a, b)

IDul(x(t)) =C. Corollary 1. Viscosity solutions of {7) are not necessarily G2. For example, let u be defined on the boundary of the square [0, 1]2 by u(O, 1) = 1 = u{1,0), u(O,O) = 0 = u(1,1) and u is affine on each side of the square. Let u 0 the AMLE of u. Assume by contradiction that u 0 is C2. By symmetry of the datum and uniqueness, Uo = ~· Thus, there is some pointy on the segment L joining {0, 1) and (1, 0) such that DUo(Y) =F 0, and therefore a neighborhood of yin which DUo =F 0. By symmetry again, DUQ(y) is parallel to L. Thus, a segment of L containing y is a gradient line. By Proposition 2, on this segment we have DUQ(z) = DUQ(y). So we conclude that the maximal segment is the whole line L. This is a contradiction with u(O, 1) = u{1, 0) . D

a. D

fiJ

(0,1)

(1,1)

(0,0)

(1,0)

FIGURE 2. A pointy on the diagonal with DUo(Y) =F 0.

Example 3. Our next example is concerned with the curvature operator. Let n be a domain in JR.2 with Lipschitz boundary and rp be a Lipschitz continuous function on 00. We consider the equation D u(

fv~, fn~) = o

Ulan=

'P·

2

in n,

(13)

F. Cao, V. Caselles, J.-M. Morel, C. Sbert

82

B

FIGURE

3. No viscosity solutions of (13) exist in this domain.

We consider solutions of (13} in the viscosity sense, which can be defined in the same way as solutions of (7) in Example 2. This model cannot be used as a model for interpolating data because of the following facts a) There is no uniqueness of viscosity solutions of (13). b) There are no viscosity solutions of (13) for general smooth curves an and boundary data r.p E .1"(00). This is easily deduced from techniques developed in [7). Indeed, let and

n=

D(O, 1), cp(x) = AlX~ + A2X~, Al > A2· Since on 8D(O, 1}, X~+ X~ = 1

the functions

u1(x1,x2)

= r.p ( .j1-~,x2),

u2(x., x2)

= r.p ( x., .jt - ~)

are two viscosity solutions of (13} in D(O, 1) with the same boundary data.

Theorem 7. There are no viscosity solutions of {13} for general smooth curoes and boundary data r.p E .1"(00).

an

The proof is based on simple heuristic arguments. A solution u of (13) is also a static solution of the evolution problem

8v

(Dv.l. Dv.l.)

IDvl, IDvl in (0, +oo) v(O, x) = u(x) inn v(t, x) = r.p(x) (t, x) E (0, +oo) X 00, at = D2v

X

n

(14)

which means that all level lines of u are moving by mean curvature. This is impossible unless the level lines of u are straight lines. In general, this is not possible as can be seen in Figure 3. Figure 3 depicts a nonconvex smooth domain with boundary data r.p such that r.p increases when we go from A to B along the boundary in the clockwise direction and then decreases symmetrically when going from B to A.

An Axiomatic Approach to Image Interpolation

4. A

GENERALIZATION OF THE

AMLE

83

IN THE PLANE

The results proved by Jensen assume that the boundary data are at least continuous. This assumption is obviously not satisfied when we deal with real images. Indeed objects in a scene are occluded by other ones and these occlusions create discontinuities (edges). What does the AMLE model become when trying to extend singularities? Can edges be preserved? Heuristically, the AMLE model is elliptic and thus regularizing and edges should be lost. Thanks to the generalization below, this pessimistic statement will be confirmed: The AMLE is a bad edges interpolation operator. For this task some interpolation operators are being studied (21]. On the other hand, the AMLE is very efficient for interpolation between level lines, as shown in the numerical experiments. We start with the following heuristic argument. Let n be the unbounded domain of JR2 between two semi-lines crossing at the origin. We set rp constant on each semi-line. There exists a unique affine function of the argument interpolating rp in n. We call it interpolating spiral. It is also easy to check that the spiral is a twice differentiable AM LE in n. Hence, it is reasonable to look for a solution with the same behavior in the neighborhood of each singularity of the boundary value. More precisely, we can prove Proposition 3. The spiral is the unique bounded interpolant between two lines.

Let us now give the generalization. Let n be a bounded domain in JR2 and rp be defined on an with possibly a finite set Z c an of discontinuity points. We assume that out of Z, rp is uniformly Lipschitz continuous and that if z E Z, then an admits at z two semi-tangential directions. Note that rp admits a limit on both sides of any point of Z. For z E Z, we shall also denote by az the interpolating spiral between the two tangent directions at z. Definition 2. We say that u E USC(O\Z) (resp. LSC(ft\Z)) is viscosity subsolution (resp. supersolution) of D 2 u(Du, Du) = 0 if and only if

n

1. for all x E and t/J E C 2 (0), if u- t/J attains a local maximum (resp. minimum) 2 at x then D t/J(Dt/J, Dt/J) ~ 0 (resp. :50} . 2. Let z E Z and {xn)neN such that limxn = z and lim 1 1 = r E S 1 • Then limsupu(xn) :5 az(r) {resp. ~ az(r)).

;::=:

We say that u is solution if and only if it is both sub and supersolution.

We have the following generalized maximum principle. Proposition 4 (Maximum Principle). Let u and v bounded sub and supersolutions of {8). Assume that on an\Z, u :5 v. Then inn, we also have u :5 v .

By constructing approximated solutions, we can prove the following results. Theorem 8. Let rp be defined on of {8) such that u = rp on an\Z.

an as above.

Then there exists a unique solution u

Consequences of this result will be clear in numerical experiments.

84

F. Cao, V. Caselles, J.-M. Morel, C. Sbert 5. NUMERICAL ANALYSIS OF THE AMLE MODEL

We shall use the AMLE model studied above as the basic equation to interpolate data given on a set of curves and/or a set of points which may be irregularly sampled. Thus, we discretize the equation 2

D u

c~:l' ~~:,) = 0.

It is easy to see that there is a relation between iterative methods for the solution of elliptic problems and time stepping finite difference methods for the solution of the corresponding parabolic problems. Because of that and thanks to Theorem 6, we study the equation

: =D

2

u

c~:l' ~~:,),

with corresponding initial and boundary data as in (ll). Using an implicit Euler scheme we transform this evolution problem into a sequence of nonlinear elliptic problems. Thus, we may write the following implicit difference scheme in the image grid u!n.+l) = u!n.> '" '"

+ ~tD2u\n.+l) '"

D (n+ll u,J

nu 0 and o are two given real constants. Our problem is: Find u and p such that -11/l. U

+

curl(u- ofl. u)

div u = 0 in 0

XU+

,

Vp= f

u = g

on

in 0

1

an .

By introducing the auxiliary variables

z = curl(u - oll. u) ,

z = (0, 0, z),

£ 2 (0),

and looking for a solution u such that z is in this problem can be split into a generalized Stokes equation and a transport equation -11/l. U

+Z

X U

+ Vp = f

in 0

(1.10)

1

(1.11)

divu = 0 in 0, u = g II Z

with g . n = 0 on

+ Cl U • V Z =

II curl U

an ,

+ Cl CUrl f

(1.12) in 0

1

(1.13)

with (u,p, z) E H}(O) x L~(O) x £ 2 (0). This formulation, which was introduced by Ouazar in [29] in view of the discretization, is equivalent to the original problem.

D. Cioranescu, V. Girault, R. Glowinski, L.R. Scott

104

By using the Leray-Hopf's lifting (cf. [25]) of the tangential boundary value, and

discretizing (1.13) by a standard Galerkin method, we easily prove the existence of one

solution of {1.10)-(1.13) for all the above data. This solution satisfies the estimates

+ ialllcurlfii£2(0)) + C2llgiiHI/2(Bn),

(1.14)

+ C7llgiiHI/2(Bn) + CsluiHI(O)IIzll£2(0),

(1.16)

llzll£2(0) $ Ct(llfll£2(0)

IIPII£2(0) $ C6llfll£2(o)

where C; are constants that depend on

11

but not on a.

However, it is much more difficult to prove that all solutions of (1.10)-(1.13) satisfy these bounds. This is equivalent to proving that the transport equation {1.13) has a unique solution for a given u, and it then shows that all solutions z of (1.13) satisfy the Green's formula

L(u·'Vz)zdx=O.

an

Clearly, as z is only in L 2 (n), this formula requires a proof. Since the boundary of is not smooth, it is not a direct consequence of [11] and it deserves some discussion. Here the quantity U·~d. where dis the distance to plays a crucial part. In [20], we prove that if 00 is a curvilinear polygon of class C1•1 (cf. [23]), then u·~d E L2 in a neighborhood of 00. This allows one to truncate any function cp in W 1"'(n) in such a way that the truncated function, say cpe, satisfies

an,

(1.17) This has several interesting consequences; one of them is the above Green's formula that we can state here more precisely. Let

Xu= {z E L 2 (n); u · 'V z E L 2 (n)}, where u E W. Then if an is a curvilinear polygon of class C1•1, we have

(1.18)

'VzEXu, L(u·'Vz)zdx=O. The proof of (1.18), found in [20], uses an argument of [12]. As an application, (1.18) yields the additional a priori estimate II au · 'V zll£2(0) $

11

.../2iuiHa(O)

+ laillcurl fll£2(0) .

(1.19)

It also allows one to show that each solution of (1.10}-(1.13) converges strongly, as a tends to zero, to a solution (u0 ,p0 ) of the Navier-Stokes equations: -II

Ll u

0

+ curl u 0 X div u 0

=0

u

0

in

+ v p0 =

n '

u0

0

f ' z = curl u 0

=g

on

an .

in

n'

Some Theoretical and Numerical Aspects of Grade-Two Fluid Models

105

2. SOME FINITE ELEMENT SCHEMES

The difficulties encountered in solving grade-two fluid models theoretically are amplified when solving them numerically, because of the high order of derivatives involved. However, the two-dimensional case does bring a major simplification, namely, by choosing appropriate schemes, all the numerical analysis can be performed without having to derive a uniform W 1•"" estimate for the discrete velocity. Of course, several schemes can be proposed to discretize this problem; the first two schemes we present here satisfy two criteria: existence of a solution in a polygonal domain without smallness restriction on the data, and error estimates. To derive error estimates in the transport equation, we find that we need either discrete velocities with exactly zero divergence, or we must discretize the transport equation by an adequate upwind scheme. In the first case, following [32] and [22], this leads us to work with triangular finite elements of degree at least four in each triangle. In the second case, triangular finite elements of degree at least three in each triangle are sufficient. Finally, the third scheme follows a least-squares method, for which we have yet no convergence analysis. From now on, we assume that the domain n has a polygonal Lipschitz-continuous boundary, so it can be entirely triangulated. Let h > 0 be a discretization parameter and let 7i. be a regular family of triangulations of 0, consisting of triangles K with diameter hK and with maximum mesh size h (cf. [7], [5]).

In the first scheme, we discretize the stream-function of u. For the sake of briefness, we assume that n is simply connected; then each function v E W (resp. v E V) has a unique stream-function I{J E H 2 (n) n HJ(n) (resp.

1. It is shown that ll is a projection and it preserves individually homogeneous boundary conditions. Furthermore, n has the following approximation property, for any numbers p and l such that 1 $ p $ oo and 1 + 1/p < l $ r + 1 if p > 1, or 2 $ l $ r + 1 if p = 1, and any integer m such that 0 $ m $ l:

V/ E W 1.P(n), VK E 7i.,

I/- ll/lw"'·P(K) $

C h~mll/llw'·P(SK),

(2.1)

Then we define the where S K is an adequate macro-element containing K. approximation operator Ph E C(W; Wh) by Ph(v) = curl(lli{J), where I{J is the stream-function of v. The approximation properties of Wh are deduced from (2.1); in

106

D. Cioranescu, V. Girault, R. Glowinski, L.R. Scott

particular, there exists C independent of hK, such that Vv E W, VK E 7i., llv- Ph(v)IILP(K)

:5 C h~"llr.ollwa+2I•·•(SK).

(2.2)

To simplify, we assume that the boundary data g is the trace of a known function r E W, but the method below is independent of the particular lifting r. We discretize u in wh and z in the standard finite element space (2.3) for an integer k 2: 1. Then our first scheme reads: Find llh in with Zh in Z11, such that Vvh E vh' II(V u/1, v vh) + (zh

u,. = g,. = P,.(r)IM Vf)h E Zh, II (zh, 811)

U/1, vh)

X

wh and Z/1 = (0, 0, Zh)

= (f, vh)'

(2.4) (2.5)

on 00,

+ o (uh · V Z11, 811) =II (curl llh, 811) + o (curl f, 811).

(2.6)

The trace-preserving properties of P11 and (2.2) permit to construct a Leray-Hopf's lifting of g", for h small enough, say h :5 ho; in tum this allows us to show the existence of a solution of (2.4)-(2.6) for all data and for all h :5 ho. All solutions of (2.4)-(2.6) satisfy the analogue of (1.14) and (1.15) with constants independent of h. With these uniform estimates and (1.18), we prove in [22] that u,. and z11 converge strongly to u and z, respectively. These results carry over to the pressure. Establishing error estimates is more delicate because (2.6) does not give directly a bound for u,. • V z11. In [22], we tum this difficulty by associating with u" and u the auxiliary transport equation: Find ( = ((h) E L 2 (0) such that 11 (

+ o u,. · V (

=

11 curl u

+ o curl f .

(2.7) 2

Since W11 is contained in W, (2.7) has a unique solution ( E L (0) for any u" in W" and any u such that curl u belongs to L 2 (0). In addition, we have the following upper bounds: llu11 · V((- z)IIL2(0) :5 ll(uh- u) · V ziiL2(0), II.t/J = (F(x), DR( A, L)t/J), t/J E L 2 (H, J.&). Now by Corollary 2.4 it follows that IIT>.II < 1. Therefore A E p(K) and we have

R(A, K) Consequently

= R(A, L)(1- T>.)-1 •

Large Asymptotic Behaviour of Kolmogorov Equations in Hilbert Spaces

115

so that K generates a strongly continuous semigroup. Finally tA(H) is a core for K since it is a core for L. o The following result is proved in (7J.

Proposition 3.3. Under the assumptions of Proposition 9.2, there exists a unique invariant measure 11 for the semigroup Pt. t ~ 0. Moreover 11 is absolutely continuous with respect to J.l. and the Radom- Nikodym derivative o := d11/dJ.t is nonnegative and belongs to W 1•2 (H, J.t). We now can give a characterization of the adjoint K• of K under the following additional assumption.

Hypothesis 3.4.

(i) F is continuously differentiable and bounded together with its derivative. (ii) There exists a continuous and bounded function F1 : H-+ H, such that

F 1 (x)

= (Q- 1x,F(x)),

x E Q(H).

The following result is proved in (5J.

Proposition 3.5. Assume that Hypotheses 2.1, 2.5 and 3.4 are fulfilled. Then the adjoint K• of K in L 2 (H, J.t) is given by K•rp = L•rp- (F(x), Drp) + (F1 (x)- div F) rp, rp E D(L•).

(3.2)

We notice that the density o is the solution to K•o

= 0.

Let us notice that, setting t/J = - logo, then t/J is a solution to the following HamiltonJacobi equation,

(3.3) 4. CONSTRUCTION OF THE TRANSITION SEMIGROUP ON L2 (H, 11)

In this section we assume- besides Hypotheses 2.1 2.5, 3.4- the following:

Hypothesis 4.1. We have (4.1)

Under this assumption it follows that L 2 (H, J.t) is included, with continuous embedding, in L 2 (H, 11). We prove now an integration by parts formula.

Proposition 4.2. For any rp, t/J E tA(H), we have

JH D,.rp .,Pd11 = - JH rp D,.'ljJdll + ;,.

k

x,.rp .,Pd11-

JH rp t/J D log od11.

(4.2)

Giuseppe Da Prato

116

Using this result it is easy to see that Dh is closable for any h E N. Thus space

W 1·2(H, v) can be defined as usual.

We consider now the linear operator in L2 (H, v),

Kor.p = Lr.p + (F(x), Dr.p}, r.p E £A(H). The following result can be proved by using the integration by parts formula (4.2) and recalling that - log a is a solution to the Hamilton-Jacobi equation (3.3); see [5}. Proposition 4.3. Assume that Hypotheses 2.1, 2.5, 3.4 and 4.1 are fulfilled. Then for any r.p E £A(H) the following identity holds:

k Kor.p r.pd11 = -~ k

2

IDr.pl dll.

(4.3)

By (4.3) it follows that K 0 is dissipative in L 2 (H, v) and consequently is closable. We denote by K" its closure. Proposition 4.4. Assume that Hypotheses 2.1, 2.5, 3.4 and 4.1 are fulfilled. Then K" is m - dissipative. Moreover D(K") C W 1.2(H, 11) and for any r.p E D(K") identity (4.3) holds. Finally the adjoint (K")* of K" in L 2 (H, v) is given by

(K")*r.p = L"r.p + (A 1x- F(x)

+ Dloga(x), Dr.p},

r.p E D(L*) .

(4.4)

Proof: We first prove that D(L) C D(K"). In fact let r.p E D(L) and let {r.pn} C £A(H) be such that Then we have also

Kor.pn-+ Lr.p+ (F,Dr.p}, in L 2 (H,J.I.). Since L2 (H, J.l.) C L2 (H, 11) it follows that r.p E D(K") as required. Now by Proposition 3.2 it follows that (A- K")D(L) = L2 (H, J.l.). Since L 2 (H, J.l.) is dense in L2 (H, v), K" is m-dissipative. Now let r.p E D(K") and let {r.pn} C £A(H) be such that

0 we have by Proposition 5.1 Un -+ u in £ 2 (0, T; W 1•2 (H, p.)). Since !IDUn(·)ll is decreasing, then IIDu(·)ll is decreasing 11-a.s. and the conclusion follows arguing as before. o We can prove now strongly mixing of 11. From now on we set 7jJ = {cp, l}Lz(H,u)·

Theorem 5. 7. Let cp E

L 2 (H, 11).

Then

lim P,cp = rp, weakly.

t-++oo

(5.12)

Proof: Let {tt} t +oo and 1/J E L 2 (H,11) be such that u(tt) converges weakly to 1/J as k -+ +oo. By Corollary 5.6 it follows that liwt-++oo Du(tt) = 0. Since D is a closed operator in L 2 (H, 11) it follows D,P = 0. Therefore 1/J = rp. The conclusion follows now

from a standard argument. o

Corollary 5.8. Assume, besides the assumptions of Theorem 5. 7, that K" is symmetric. Then for any cp E L 2 (H, 11) we have lim Pt'{J

t-++oo

= ;:p in L2 (H, 11).

(5.13)

120 Proof: We have

Giuseppe Da Prato

IIP,rpll2

Therefore, letting t tend to infinity,

=(P'ltrp, rp}L2(H,v)·

lim IIP,rpll 2

t-+00

= (~)

2



Now the conclusion follows from Theorem 5.7. o REFERENCES

(1) BOGACHEV V. I. , R0CKNER M. & SCHMULAND B. (1996) Generalized Mehler 1emigroup1 and application~, Probability and Related Fields, 114, 193-225. (2) CANNARSA P. & DA PRATO G. (1993) On a Functional Analy1il approach to parabolic equatiom in infinite dimemiom, J. F\mct. Anal., 118 (1), 22-42. (3) CERRA! S. (1996) Elliptic and parabolic equatiom in R" with coefficient& having poll/flomial growth, Comm. Partial Differential Equations 21 (1 & 2), 281-317. (4) DA PRATO G. (1997) Regularity re1ult& for Kolmogorov equatiom on L 2 (H,Io') 1pace1 and application~, Ukrainian Mathematical Journal, m.49, (3), 448-457. (5) DA PRATO G. (1998) Non gradient perturbation~ of Ormtein-Uhlenbeck 1emigroup1, SNS, Preprint n.12. (6) DA PRATO G. & ZABCZYK J . (1992) Stocha~tic equation~ in infinite dimemiom. Encyclopedia of Mathematics and its Applications, Cambridge University Press. (7) DA PRATO G. & ZABCZYK J. (1996) Ergodicit11for Infinite Dimemional S11stem6. London Mathematical Society Lecture Notes, n.229, Cambridge University Press. (8) FUHRMAN M. (1995) Anal11ticity of tramition 1emigroup1 and clo1abilit11 of bilinear fo'rml in Hilbert 1pace1, Studia Math., 115, 53-11. (9) GOZZI F. (1995) Regularity of 1olutiom of a 1econd order Hamilton-Jacobi equation and application to a control problem, Comm. Partial Differential Equations, 20, (5& 6), 775-826. (10) HAS'MINSKII R. Z. (1980) AIJimptotic 1tabilit11 of differential equatiom, Sijthoff & Noordhoff. [11) LUNARDI A. (1997) On the Ormtein-Uhlenbeck operator in L 2 1pace1 with re~pect to invariant mea~ure1, Trans. Amer. Math. Soc., 349, 155-169. (12) LUNARDI A. Schauder theorem~ for linear elliptic and parabolic problem~ with unbounded coefficient& in Rn, Studia Math., 128 (1998), no. 2, 111-198. (13) MA Z. M. & ROCKNER M. (1992) Introduction to the Theoru of (Non Summetric) Dirichlet Fo'rml, Springer-Verlag. Giuseppe Da Prato, Scuola Normale Superiore, Piazza dei Cavalieri 7, 52126 Pisa, Italy

On Existence Results for Fluids with Shear Dependent Viscosity - Unsteady Flows J.

FREHSE,

J.

MALEK, M. STEINHAUER

Abstract. New existence results for the system describing unsteady motion of incompressible fluids with shear dependent viscosity are presented. The nonlinear elliptic operator, which is related to the stress tensor, satisfies the p coercivity condition and is strictly monotone. We prove global existence of weak solutions if 2 p > ~.:;> for the space periodic problem; and for no-stick boundary conditions. Keywords: Non-Newtonian fluid, shear dependent viscosity, unsteady flow, space-periodic boundary condition, weak solution, almost everywhere convergence Classification:

35005, 35Q35, 76A05, 76099

1.

INTRODUCTION

The unsteady motion of an incompressible fluid with a constant density 1 in a bounded domain n s;;; JR.d, d ;::: 2, is described by the following system of partial differential equations:

a;; +v~~:: -divTE=-V7r+f,

(1.1)

divv=O,

where v = (v1 , ••• , v11) is the velocity field, 71" represents the pressure and f denotes the vector of given body forces. All functions are evaluated at (t, x) E QT, where QT = (O,T) X n, T E (O,oo). We denote the symmetric velocity gradient by D(v), i.e., D(v) HVv + (Vv)T). If the fluid is a fluid with shear dependent viscosity, then, by definition, the stress tensor TE at (t, x) E QT is given by a tensorial function of D (v); it means

=

TE(t, x) = T(t, x, D(v(t, x))).

(1.2)

If the fluid is in addition homogeneous, then (1.2) reduces to TE(t, x) = T(D(v(t, x))).

Fluids with shear dependent viscosity represent one important class of non-Newtonian fluids, consisting of those fluids that have the ability to capture phenomena called shear thinning or shear thickening. All variants of power-law fluids that enjoy significant attention among engineers and physicists fall into this category. Typical examples are given by the formula T(D(v)) = 2 Vo (~-to+ ID(vW) 1

l!j!

D(v),

We suppose that the density is equal to 1, for simplicity.

v0 >

0, 1-'o ;::: 0,

(1.3)

122

J. Frehse, J. Malek, M. Steinhauer

where pis a real parameter. If p = 2 in (1.3), then the fluid is Newtonian; note that then (1.1) leads to the Navier-Stokes equations. If p E [1, 2}, then the model (1.3} belongs to shear thinning fluids, while for p > 2 it captures shear thickening effects. We refer the interested reader to (9], where a detailed physical introduction and further references can be found. In this paper we will present new results on global-in-time existence of a weak solution to the space periodic problem: to solve (1.1)-(1.2) such that v, 1r are periodic in the variables Xi with period Li ,

v(O, x)

= vo(x) inn= (0, Ll) X

••• X

(0, L,).

(1.4)

Here, v 0 is a given periodic, divergence-free function; Li E (0, oo ), i = 1, 2, .. . , d. We also assume that f is periodic and J0 fdx = 0. The existence result presented below holds also for the following "no-stick" boundary conditions: to solve (1.1)-(1.2) in bounded domain 0 with Lipschitz boundary such that2

(v)n = v. n

= 0 and (To).,. = 0 at

an

(0, T)

X

an,

v(O, x)

= Vo(x) in n.

(1.5)

Here, v 0 is a given divergence-free function satisfying v 0 • n = 0 at the boundary. We will assume that (for p > 1) the tensor T = (Ti;)f.;=l• having continuous components, satisfies the conditions of p-coercivity of polynomial growth of order p -1 and of strict monotonicity. This means 3cl

> 0,

3c2

E L 1 (Qr): T(t, x, '1) · '7 ~ell 'II" -I{Jl(t, x) 1{)1

\f (t, x) E Qr, '7 E ~;!

(1.6)

eE ~;! ,'1 i: (.

(1.8)

> 0,

1{)2 E L~(Qr): IT(t, x, '1)1 ::5 c2l'll"- 1 + 1{)2(x)

and

(T(t, x, '1)- T(t,x, e))· ('1- e) > 0 \f (t, x) E Qr, '7,

ur:;::.

Here, denotes the space of symmetric real d x d matrices. It is worth noticing that if p ~ 2 then (1.3) fulfils (\f '7, E

e ur:;::. ,'7 #: ()

(T('I) - T(()) · ('1- e) ~ Cl'l- El",

(1.9)

1'1 - El2 (T( ) - T(c)) . ( - c) > C(p- 1) " ,. " "' (#Ao + 1'112 + IEI2 )1? .

(1.10)

while for p E (1, 2)

By Problem (Sh V)per we mean the problem of finding a solution to (1.1), (1.2), (1.6)(1.8) and (1.4), while Problem (Sh V).tick stands for the problem of finding a solution to (1.1), (1.2), {1.6}-(1.8} and {1.5}. There are essentially two different methods that were applied to prove the global existence of weak solutions without restricting the size of initial data and/or the 2 The subscripts (-),. and (·)., denote the normal and tangential components of a vector (·); n represents the outer normal to an.

On Existence Results for Fluids with Shear Dependent Viscosity- Unsteady Flows ... 123

length of the time interval for Problem {Sh V),_. The first one, developed by 0. A. Ladyzhenskaya and J. L. Lions (independently of each other) (cf. [4], [5]), is a combination of compactness and monotonicity arguments, which reveals to be applicable to Problem (Sh V),_ (or Problem {Sh V},ticJc or the Dirichlet problem) if p

>

3dt2

dt2 •

The second (regularity) method assumes the existence of a scalar potential to T 8 , which enables to use the second energy inequalities to estimate a certain fraction of the W 2•2-norm (or W 2 "'-norm) versus the W 1•2-norm, integrated over time. This method is, for example, applicable to (1.1), (1.3) (however, JJo has to be positive if p > 2) and yields existence if p > d~2 for Problem {Sh V),_ (see the monograph [8], or also [1], [9]). The same result holds also for the Cauchy problem as presented in [11]. If p ~ 2 and d = 3 the existence result for the Dirichlet problem is performed in [7]. In two dimensions, this method works for a large range of p's, namely for p > 1 for Problem (Sh V),_, and for p > for the Dirichlet problem; see [8]. We wish to remark that the main contribution of this second method does not lie in the existence theory, but in new regularity and uniqueness results (we refer to [8] or [7] for details). In this paper, we incorporate the third, so-called truncation (or capacity) method and we show the existence of a weak solution if p > 2 ~rt_:;> for Problem {Sh V),_ and Problem (Sh V),tick. The method is based on a construction of a special bounded test function and strongly relies on the strict monotonicity ofT. Comparing the results of the second and third method, we see that the new results are worth, in higher dimensions, d ~ 3. Of course, the assumptions on T are slightly more restrictive compared to the first method; on the other hand, they are more general compared to the second method. Nevertheless, examples (1.3) satisfy all of them. In the papers [3] and [12], the third method was applied3 to the steady problem, and the existence of weak solutions was shown for p ~ d~l • (In [12}, the limiting case is not included.) Our aim was to find modifications that are needed to analyze the evolutionary case. The main difficulty is due to missing information on 1r which is overcome by constructing special divergence-free truncated function. The extension of the method to the Dirichlet problem up to p > 2 ~:;> is open.

t

The introductory section is completed by a list of notation and some useful inequalities. After that, in Section 2, we define the notion of weak solutions to (1.1) and formulate our main theorem. Approximations are introduced in Section 3. The proof of the main theorem is presented in Section 4. We use standard notation for function spaces: If q > 1, then W 1,q(O) denotes the Sobolev space of scalar-, vector- or tensor-valued functions in £9(0) having first derivatives also in £9{0).

3 A different variant of this method bas been used to solve different elliptic and parabolic problems before. Let us mention for example (2] and references cited there.

J. Frehse, J . Malek, M. Steinhauer

124

Next, we define the spaces 8 9 and V : (i) for Problem {Sh V),..,. we set

S9

= {~; ~. E W 1•9 (.J.Rd), ~. periodic ,

L~~

d.x

= 0, i = 1, 2, ... , d};

V := {c); ~i E C 00 (.J.Rd), ~; periodic }. (ii) for Problem (Sh V),tici: we put 8 9 := {c);~, E W 1•9 (0), c) · n = 0 at 00}; V :=

{c) ; ~, E

C0 (0)}.

Finally, we define (in both cases) Has the closure ofV with respect to the £ 2 (11)-norm, and Vq u E 8 9 ; div u = 0}. By V9" we mean the dual space to Vq; the brackets (-, ·} represent this duality. Finally, if T E (0, oo), I (0, T), and X is a Banach space, then £9(I; X) denote the Bochner spaces, q E [1, oo]. Besides usual summation convention, we keep the convention that K represents a fixed constant that maximizes all a priori estimates used in the text, while cis a generic constant that does not depend on "important" parameters and can change from line to line. If g, hare vector-valued functions then (g,h) J0 g;h;dx and g1h; E £ 1 (11), while for tensor-valued functions fl, the symbol (fJ, (} denotes fo fl;;{;; d.x and 1 Tli;{;; e L (n). The Korn inequality is used during the proof. It says (cf. [10]): If p > 1 and 0 has Lipschitz boundary, then there exists a constant c3 such that

={

=

=

e

IIVvllp $ c3IID(v)llp

'Vv E W~"'(11).

2. DEFINITION OF WEAK SOLUTIONS AND MAIN THEOREM

Definition 1. Let vo E H, f E V (I; V,"), tf = ~· We say that v E V(I; V,.) n L 00 (I; H) is a weak solution to the Problem (Sh V),_ (or Problem {Sh V),tid:) if

1T [-

(v(t),

a~t)) + (vk a;;:> ~(t)) + (T(D(v(t))), D(c))(t))] dt I

=

1T

{f(t), ~(t)) dt + (v0 , c)(O))

'V~ E C0 (-oo, T; V).

(2.1)

With this definition we are ready to formulate our main theorem.

Theorem 1. Let p > 2 ~.:21 ). Then there exist weak solutions v to Problem (Sh V),_ or Problem (Sh V),tid: . In the next section we will study properties of weak solutions to an approximate problem. The proof of Theorem 1 is given in Section 4. 3. APPROXIMATIONS AND THEIR PROPERTIES We will use approximations to (1.1) which are based on fl"mollification of the convective term. Let w E C 0 (Jil:d) be a usual mollification kernel with support in

On Existence Results for Fluids with Shear Dependent Viscosity- Unsteady Flows ... 125

Bt(O)

w.,(x)

={x

E

JRd : lxl < 1} and J.Rdwdx

= ~w(~) and put v., =v * w.,.

= 1.

For every 'T/ > 0 (small) we set

Then we look for solutions v = v" of the problem

a;;+ (v,)t~- div (T(D(v))) = -V1r + f,

v(O,x) = v 0 (x)

divv

= 0,

(3.1)

inn and either (1.4) or (1.5) holds.

Lemma 1. For p ~ 2~.:21 ) and every 17 > 0 there exists a weak solution v" to (3.1) such that

~"

v" E C(l; H) n V'(J; V,.),

(3.2)

ELY(J;V,.•)

and the identity

1T (IIY· Then cl>o is a divergence-free function and keeps our boundary conditions. Therefore +o E V'(/; V,) n L'' (QT). {Let us remark that this is not true for the Dirichlet problem; see also [6}, pp.83-86, for related comments.) In particular, we can test by +0 in {3.3). Moreover, as Vq is orthogonal to a:;', we have

cl>(t)

In

sup

II+IJy$1

11To oiiY $ cii+IIY· Let us remark that to prove (3.8) we need slightly more information on integrability of a:;'. For example, knowing that there are potentials to T", where T" approximate T locally in (3.3)) that E L 2(QT), and uniformly, then we can easily check (testing by (3.8) easily follows. We will not go into more details, but we wish to say that the typical example (1.3) posesses a potential; consequently, to verify L 2-integrability of for example (1.3) is an easy exercise (see [7] for more details). o

a:;

a:;

a:;

A straightforward consequence of the uniform estimates {3.4) and {3.7) is the following lemma.

Lemma 2. Under the assumptions of Lemma 1 there exists a sequence T/k k ~ oo) and v E L 00 (/; H) n V'(/; V,) such that vk = v"• satisfy D(v") _... D(v), v"

~v

v" ~ v

Vv" _... Vv

weakly in V(I; V'{!l)),

strongly in Lr(I;L9 (!l))

0 (as

{3.9)

' K. For i = 1, 2, ... , N, define iteratively L; = L~_ 1 and set Ef•m {(t,x) E QT; L~ $ lvl{t,x)- v"'(t,x)l $ L;}. For fixed f.,m the sets Ef•m are disjoint. Therefore 1 lll•mll 1,st·"' $ K due to the uniform boundedness of gt.m. Then certainly there exists i0 (t, m) such that

=

E;:

On Existence Results for Fluids with Shear Dependent Viscosity- Unsteady Flows ... 127

For each l, m, however, i 0 e {1, 2, .. . , N}. Then necessarily there is a subsequence {vn} ofvk such that io(l,m} is the same, i.e. i 0 is independent of l,m. Denote the corresponding LiG by L and Ef;,m by Et,m. The proof is complete. o Further, we introduce a double-sequence {lltt,m}t,meN by

(3.13}

=

Then lltt,m = 0 on Q~m {(t,x) E QT : lvi- vml 2:: L} and lllft,ml ~ L otherwise. Thus, lltt,m e L""(QT)nlJ'(/; 8 11 ) and by Lebesgue's Dominated Convergence Theorem and (3.10} lllltt,mllr,QT --+ 0 for all r E [1, oo)

as l, m --too.

Further we observe that for given c E (0, 1} lldiv q,t,mii11,Qr ~ 2c.

{3.14}

Indeed, if x denotes the characteristic function of the set QT \ Q~m, then div q,t.m

= (vi L

vm)i IJ(vi - vm)r (vi - vm)r . IJxi lvl - vml X

Thus, by (3.12}, the definition of Et,m and L we see lldivlltt,mllp,QT

~ IIV(vt- vm)±lvi - vmiii,,QT\Q~"'

=IIV(vt- vm)±lvl- vmlllp,(QT\Q~,.)\E'·"' ~LK

+ IIV(vt- vm)±lvl- vmlllp,E'·"'

+ c ~ 2c.

(3.15)

Next, we consider an auxiliary problem (for almost all t E /, s' = P(d+1i.~Wd+l))

-t::.ht,m = -div q,t.m in

IJhl,m

n, ---all = 0 on 00,

or ht,m periodic;

(3.16)

The existence of ht,m satisfying (3.16) is standard. Set finally

41t,m

= lltt,m _

V ht,m.

(3.17)

Note that 41t,m E LP(/; V,.) n L''(QT) (div4lt,m = 0, 41t,m · n = 0 at (0, T) x 00 for Problem {Sh V),tid:, and 41t,m is periodic for Problem {Sh V),_ ). In addition, from (3.16} we have in both cases for l, m sufficiently large (3.18)

J. Frehse, J. MAlek, M. Steinhauer

128

4.

PROOF OF THEOREM

1

We are going to show that the following condition holds

VB e(O, 1} Ve0 e (0, 1} 3 mo e IN Vl, m;:: mo 0 < I=

1T

(T(D(vt)) - T(D(vm)), D(vt- vm))' dt < e0 •

(4.1)

Disposing of (4.1} and assuming (1.8} (or one of the stronger conditions (1.9} or (1.10)} we see that

D(vn) -+ D(v)

almost everywhere in QT.

(4.2)

To show that v is a weak solution to (1.1} is standard (see [3]} making use of (4.2} and (3.10). In order to prove (4.1) we substract the weak formulation (3.3) for vl from the weak formulation for vm and take ~t,m defined in (3.17) on the place of the test function. We obtain the following identity (X denotes the characteristic function of the set Qr \ Q~m)

1T(~ - ~m, + 1T = 1T (~ - ~m' + 1T + 1T wt,m} dt

(T(D(vt))- T(D(vm)), D(vt- vm)X) dt

Vhl,m} dt

(T(D(vt)) - T(D(vm)), D(Vht,m)) dt

(4.3)

(T(D(vt))- T(D(vm)}, D(vt- vm) lvl ~ vml X) dt

+

1 T

0

avm

ayL

((v""')A,- - (v71,)A,- , ~t,m) dt 8 Xt 8Xt

=I1 + I2 + Ia + I4.

Before estimating I 11 I 2 , I 3 , I4 let us first observe that the first term equals H(vl - vm)(T) - H(vl - vm)(o), where H is a nonnegative primitive function to wt,m. As H(vl- vm)(o) = 0, we see that the first term is nonnegative. Next, integration by parts leads to / 1 = 0 (see also (3.8) page 7). Further, by (3.16}, (3.15) and the uniform £ 1-bound for gl,m

II3I :5 clll•mii1,QTIIhl,mll2,p,QT :5 eKe. I 2 is estimated in the same way as div wt,m in (3.15) by using that the domain of integration is Qr \ Q~m. Therefore II2! :5 ct. Finally, I 4 is small due to (3.6) and (3.18). Then (4.1) follows from the proved estimates of the second term in (4.3) by the Holder inequality and (3.10}2, cf. [2]. o

Remark. Instead of estimating Cauchy sequences vl- vm, one can work more directly with vm- v. This brings only minor changes; see [3] for details.

On Existence Results for Fluids with Shear Dependent Viscosity- Unsteady Flows ... 129 REFERENCES

(1] Bellout, H., Bloom, F., NeCa.s, J., Young Measure- Valued Solutions for Non-Newtonian Incompressible Fluids, Comm. Partial Differential Equations 19 (11 & 12) (1994), 1763-1803. (2] DalMaso, G., Murat, F., Almost everywhere converyence of gradients of solutions to nonlinear elliptic systems, Nonlinear Anal. TMA 31(3-4) {1998), 405-412. [3] Frehse, J., Malek, J., Steinhauer, M., An Existence Result for Fluids with Shear Dependent Viscosity-Steady Flows, Nonlinear Anal. TMA 30 {1997), 3041-3049. (4] Ladyzhenskaya, 0. A., On some new equations describing dynamics of incompressible fluids and on global solvability of boundary value problems to these equations, Ttudy Steklov's Math. Institute 102 (1967), 85-104. [5] Lions, J. L., Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris {1969). (6] Lions, P.-L., Mathematical Topics in Fluid Mechanics, Volume 1, Incompressible Models, Oxford Science Publications, Oxford University Press, Oxford, (1996). [7] Malek, J ., NeCa.s, J., Ruzicka, M., On weak solutions to a class of non- Newtonian incompressible fluids in bounded three-dimensional domains. The case p ~ 2., Advances in Differential Equations (accepted), Preprint SFB256 No. 481. [8] Malek, J., Neeas, J., Ruzicka, M., Rokyta, M., Weak and Measure- valued Solutions to Evolutionary Partial Differential Equations, Applied Mathematics and Mathematical Computation 13, Chapman and Hall, London {1996). [9) Malek, J., Rajagopal, K. R., Ruzicka, M., Existence and Regularity of Solutions and Stability of the Rest State for Fluids with Shear Dependent Viscosity, Math. Models Methods Appl. Sci. 6 (1995), 789-812. [10) NeCa.s, J., Sur les normes equivalentes dans w:(n) et sur la coercivite des formes formellement positives, Les presses de 1' Universite de Montreal {1966). [ll] Pokorny, M., Cauchy problem for the Non-Newtonian Viscous Incompressible Fluid, Appl. Math. 41(3) {1996), 169- 201. (12) Ruzicka, M., A Note on Steady Flow of Fluids with Shear Dependent Viscosity, Proceedings of the Second World Congress of Nonlinear Analysts, Part 5 (Athens, 1996). Nonlinear Anal. 30(5) {1997}, 3029- 3039.

J. Malek was partially supported by the grant GACR no. 201/96/0228 and by grants of the SFB 256 at the University of Bonn. J. Frehse, M. Steinhauer, Institute of Applied Mathematics, University of Bonn, Beringstr. 4-6, 53115 Bonn, Germany;

J. Malek, Mathematical Institute, Charles University, Sokolovska 83, 186 75 Prague 8, Czech Republic

Stability of Propagating Fronts in Damped Hyperbolic Equations TH.

GALLAY, G. RAUGEL

Abstract. We consider the damped hyperbolic equation £uu + Ut

=

U.z:

+ F(u) ,

x E R,

t;?::

0, u E R,

where£ is a positive, not necessarily small parameter. We assume that F(O) = F(l) = 0 and that F is concave on the interval (0, 1]. Under these assumptions, our equation has a continuous family of monotone propagating fronts (or travelling waves) indexed by the speed parameter c;?:: c.. Using energy estimates, we first show that the travelling waves are locally stable with respect to perturbations in a weighted Sobolev space. Then, under additional assumptions on the non-linearity, we obtain global stability results using a suitable version of the hyperbolic Maximum Principle. Finally, in the critical case c c., we use self-similar variables to compute the exact asymptotic behavior of the perturbations as t -+ +oo. In particular, setting £ = 0, we recover several stability results for the travelling waves of the corresponding parabolic equation.

=

Keywords: damped hyperbolic equations, travelling waves, stability, asymptotic behavior, self-similar variables Classification: 35840, 35835, 35830, 35L05, 35C20

1. Introduction

Mathematical models for spreading and interacting particles or individuals are very common in chemistry and biology, especially in genetics and population dynamics. If the spatial spread of the particles is described by Brownian motion, these models usually take the form of reaction-diffusion equations or systems for the population densities [10,27]. Depending on the precise form of the interaction, such systems exhibit interesting solutions like propagating fronts or travelling waves, whose existence and stability properties have attracted a lot of attention in recent years

[32].

It can be argued, however, that diffusion is not a realistic model of spatial spread for short times, because particles performing Brownian motion can move with arbitrarily high speed, and the directions of motion at successive times are uncorrelated. This drawback can be eliminated by replacing the diffusion process with a velocity jump process which is more satisfactory for short times and has

Stability of Propagating Fronts in Damped Hyperbolic Equations

131

equivalent long-time properties [17,22,31]. In one space dimension, this procedure leads to damped wave equations instead of reaction-diffusion systems [6,19,20,33]. Under the same assumptions as in the parabolic case, the damped hyperbolic equations also have travelling wave solutions [18] with analogous stability properties [14,16]. The aim of this paper is to review some of these stability results in the simplest case of a scalar equation with a non-linearity of "monostable" type. We thus consider the damped hyperbolic equation cUtt

+ Ut

=

u 1111

+ F(u)

,

(1.1)

where y E R, t E R+ and e- is a positive, not necessarily small parameter. We assume that the non-linearity FE C2 (R, R) has the following properties: (H1)

F(O)

= F(1) = 0,

=

F'(O)

> 0,

F'(1)

0 and becomes smaller when s increases. It is thus natural to look for the the smallest value of s > 0

133

Stability of Propagating Fronts in Damped Hyperbolic Equations

for which the origin in (1.5) is linearly stable in Z. This is most conveniently done by setting w(x, t) = e-...,w(x, t) and studying the equation for w, namely: EWtt

+ (1 + 2t:cs)Wt- 2cCW:ct =

w.,., + (c- 2s)w., + (F'(h)- cs + s 2)w +N(h,e-•"'w)e-•"'w2



(1.9)

A straightforward computation in the Fourier variables shows that the origin in (1.9) is linearly stable in H 1 x L2 only ifF' (0) -cs+s2 $ 0. In fact, this condition can easily be inferred from the coefficient of w in (1.9). Therefore, the largest perturbation space Z for which we can expect stability of the front h is obtained by choosing s

= 21 (c- y'c2 -&.).

(1.10)

Note that this value corresponds to the decay rate of has x ~ +oo, since h(x) ""'e_.., if c > c. and h(x) ""' xe-•"' if c = c. [1]. Thus, if (1.10) holds, the translations h(x+xo)-h(x) do not belong to the perturbation space H!. In view of the translation invariance of Eq.(1.1), this is of course necessary to obtain an asymptotic stability result. In the sequel, we always assume that the condition (1.10) holds. In this paper, we present three different results which show that the travelling wave h is stable with respect to perturbations in Zc or in a subspace of it. In Section 2, we give a local stability result valid for all c ~ c. and all E > 0. Using appropriate energy functionals, we show that if (w(O), we(O)) is sufficiently small in Zc, then the solution (w(t), Wt(t)) of (1.5) stays in a neighborhood of the origin in Zc and converges to zero as t ~ +oo in a slightly weaker norm. In the critical case c = c., we also give an estimate of the convergence rate. These results have been proved in [14] in the particular case where F(u) = u- u 2 • Their proofs can be adapted, with minor changes, to cover the general case of a non-linearity F satisfying (H1). Section 3 is devoted to global stability results. We first recall the Maximum Principle for hyperbolic equations [29] in a version adapted to our problem. Then, under the additional assumption that F'(u) be strictly negative for u ~ 1, we show that the travelling wave h is stable with respect to "large" perturbations in Zc, provided some positivity conditions are satisfied. Furthermore, if 1 + 4e:F'(1) ~ 0 and F"(u) $ 0 for u ~ 0, we obtain linear upper and lower bounds for the solutions of Eq.(1.5), as well as a decay rate in time for the quantity llp.w(t)IILoo· The proofs of these results can also be found in [14] if F(u) = u- u 2 • In Section 4, we restrict our analysis to the critical case c = c., and we study the long-time behavior of the solutions (w, we) of (1.5) in a slightly smaller function space. In particular, we show that

a

w(x,t) = t3/2 h'(x)cp*

(x VJ'tC.'-* 'i+?) +o(t-3 2) 1

I

t~+oo,

134

Th. Gallay, G. Raugel

where a E R and I()• : R -+ R is a universal profile. In the parabolic case E = 0, this asymptotic expansion has been obtained by Gallay [13] using the renormalization group method (5] combined with resolvent estimates. We follow here a simpler and fairly different approach based on self-similar variables and energy estimates only. We refer to [16] for the detailed proof. To conclude this introduction, we would like to point out the striking similarity between the stability results presented here and the corresponding statements in the parabolic case. This is an illustration of the more general fact that the long-time behavior of solutions to damped hyperbolic equations such as (1.1) is essentially parabolic [15). A similar phenomenon can be observed in the context of hyperbolic conservation laws with damping [21,28).

2. Local Stability of the Travelling Waves In this section, we show that the travelling wave h is stable with respect to sufficiently small perturbations in the space z~. Our main result is: Theorem 2.1. Assume that (H1) holds, and let Eo > 0, c ~ c.. Then there exist constants 6o > 0 and Ko ~ 1 such that, for all 0 < E :5 Eo, the following result holds: for all (l()o, 'Pl) E z~ such that II('Po, 'Pt)llz. :5 6o, there exists a unique solution (w, We) E C0 ([0, oo), Z~) of (1.5) with initial data (w(O), Wc(O)) = ('Po. 1()1). Moreover, one bas

ll(w(t), wc(t))llz.

and lim (llw(t)lln'

C-++oo

:5 Koii('Po, 'Pt)llz. ,

(2.1)

t ~0 ,

+ ll(p.w(t)):.:IIL2 + IIP.wc(t)IIL2)

= 0,

(2.2)

where p.(x) = 1 + e•:.: and sis given by (1.10). Sketch of the proof. We follow the lines of the proof of Theorem 1.1 in [14). Let

N 1 (h)

= fo

1

= ~fo (1-u) 2 P'(h+uw)du. 1

F"(uh)du,

Given 0 < E :5 Eo and (1.5) satisfying

c

~

(AI)

N 2 (h,w)

c., we assume that (w, we) E C0 ([0, T], Z~) is a solution of llw(t)llu!

:5 6 , t E [0, T) ,

for some (sufficiently small) 6 > 0. As in (1.9), we set w(x, t) = e-•:..w(x, t). To control the behavior of w on [0, T), we introduce two families of energy functionals:

Eo(t) =

Et(t) =

~

L

L

(Ewl + w~- w2(h.Nt(h) + 2wN2(h, w)))

((i+ecs)w2 +ewwc) dx,

E2(t) = aoEo(t) + Et(t) ,

dx,

Stability of Propagating Fronts in Damped Hyperbolic Equations

135

where a 0 = max(2e, 1/(2c2)), and

£o(t) = £1(t) = e2(t)

k L{(l~

(ewl +

w~ + s 2w 2 -

w 2(h.N1(h) + 2wN2(h, w))) dx,

ecs)w2 + EWWt) dx,

= alt'o(t) + £1 (t) + a2(Eo(t)E2(t)) 1' 2

I

where a1 = max(2E,9/(2c2)) and 6,a2 are positive constants. Note that the functionals Ei, £i control the behavior of the perturbation w "ahead" and "behind" the front, respectively. Arguing as in [14, Section 2], one can show that if t5, (J and a2 1 are sufficiently small, then the functions Eo, E 2 , £ 2 are non-negative and satisfy the differential inequalities

dEo() dt t 5

1 2 2(c2 -c.) Eo(t) ,

dE2 dt(t) + E 0 (t)

2

d! (t) + aa£2(t) ::::; Cl(Eo(t)E2(tWI 2 , for some aa, C 1

t

0,

=:::;

e [0, T] ,

> 0. In addition, there exists a constant C2

~

(2.3)

1 such that

where the function Ill : R+ --+ R+ is defined by lli(K)

=

sup

IF'(u)l .

{2.5)

O~u$1+K

Combining the estimates Eqs.(2.3), (2.4), we obtain the bound

ll(w(t), we(t))llz. 5 Coll(w(O), we(O))IIz. (1 + lll(llw{O)IILoo W' 2

,

t E [0, T], (2.6)

where Co is a positive constant depending only on e0 , c and F. Since the Cauchy problem for Eq.(1.5) in Zt: is locally well-posed [14], this proves global existence of the solution (w, we) provided the right-hand side of {2.6) is smaller than the quantity t5 appearing in {AI). Then, the differential inequalities {2.3) imply that E0 (t) and £2 (t) converge to zero as t-+ +oo, thus proving (2.2). 0

Remarks.

1) In the critical case c =c., it follows from (2.3) that Eo is non-increasing in time, and that tE0 (t) + t 112 &2 (t) --+ 0 as t -+ +oo. In particular, we have

136

Th. Gallay, G. Raugel

This decay rate is not optimal; see Section 4 below. 2) Combining (2.1), (2.2), (2.7), we obtain lim llp.w(t)IIL""

e-++oo

= 0 if c >c. , and

lim t e-++oo

1 4 ' llp.w(t)IILoo

= 0 if c =c•.

(2.8) 3) All the estimates in the proof of Theorem 2.1 are uniform in E forE E (O,e0 ]. In particular, taking the limit £ -+ 0 everywhere, we obtain a proof of the local stability of the travelling wave h for the corresponding parabolic equation. 4) If c =c., the stability condition F'(O) -c8+82 = (8-c./2) 2 $ 0 implies 8 = c./2, hence (1.10) is the only possibility. On the other hand, if c > c. and 8 is chosen so that F'(O) - C8 + 8 2 < 0 (in contrast to (1.10)), one can show using the same spectral estimates as in the parabolic case that the origin in (1.5) is exponentially stable in ZE. The fastest decay rate in time for the perturbations is obtained if we set 8 = s(e), where

s(e) =

i

1 + 4£F'(O) 1 +ec2

3. Global Stability of the Travelling Waves Throughout this section, we assume that the non-linearity F satisfies (H1) and (H2)

F'(u) $ -1-' < 0 ,

u~1,

for some 1-' > 0. Under this additional assumption, it is known in the parabolic case that the front h is stable with respect to large perturbations in H! satisfying a positivity condition. This property is a consequence of the classical Maximum Principle for parabolic equations. In this section, we show a similar global stability result for e > 0 using a hyperbolic Maximum Principle. Our starting point is the following crucial observation. Let 0 < e $ e0 , c ~c., d E (0, 1], and assume that (w, we) E C0([0, TJ, ZE) is a solution of (1.5) satisfying, instead of (A1), (A2)

w(x, t) ~ -(1- d)h(x) ,

x E R,

t E (0, Tj .

In other words, we assume that v(x, t) = h(x) + w(x, t) ~ dh(x). Then it can be shown (see [14) in the case F(u) = u- u 2 ) that the a priori estimates (2.3), (2.4), (2.6) still hold for the solution (w, we), with constants C0 , C 11 C 2 depending only on e0 , c, d and F. Thus, we can remove the smallness condition (A1) in the proof of Theorem 2.1 provided we are able to show that the positivity condition (A2) is satisfied for all times. This in turn can be obtained from the Maximum Principle under appropriate assumptions on the initial data.

Stability of Propagating Fronts in Damped Hyperbolic Equations

137

3.1. The Hyperbolic Maximum Principle Motivated by (1.4} or (1.5}, we consider the hyperbolic operator L with constant coefficients Lu = U:z:., + 2ecu.,, - euu + cu., - Ut • (3.1} Assume that l : R x [0, T) -+ R is a continuous function such that

l(x, t)

,

~ ~

where T is a positive number and

~

x ER ,

t E [0, T) ,

(3.2}

E R satisfies

1 +4e~

~

(3.3}

0.

The following result is a consequence of the Maximum Principle given by Protter and Weinberger (see [29, Chapter 4, Theorem 1) or [14, Appendix A, Theorem A.1)}.

Theorem 3.1. Let e > 0, c > 0, and assume that the conditions (3.2) and (3.3) are satisfied. If (u(x, t}, u,(x, t)) belongs to C0 ([0, T], Hl~c(R) X L~(R)), with U:z::z: + 2eCU:z:t- euu in L~oc(R x (0, T}}, and if

(L+l(x,t))u(x,t) ~ 0, u(x, 0) :5 0 , eue(x, 0} - ecu:z:(x, 0) +

1

a.e.(x,t) E R x (O,T),

(3.4}

'V x E R ,

(3.5}

u(x, 0) :5 0 ,

2

a.e. in R ,

(3.6}

then u(x, t) :50 for all (x, t) E R x [0, T). As an application, we define ford E [0, 1], K

~

0 the function

We have the following result:

Proposition 3.2. Assume that (Hl), (H2) hold. Let e let K be a non-negative constant such that

> 0, c ~c., dE [0, 1) and (3.8)

For some T > 0, assume that (w, we) E C0 ([0, T], Zc) is a solution of (1.5) with initial data (cp0 , cp 1 ) satisfying

cpo(x)

~

-(1- d)h(x) ,

xER ,

(3.9)

138

Th. Gallay, G. R.augel

e!fl(z) ~ ec(!fo(z) + (1-d)h'(z))-

1

2(!fo(z) + (1-d)h(z)) ,

a.e. in R .

(3.10)

Suppose moreover that w(z,t)

Then

~

K,

(z,t) E R x [O,T].

w(z, t) 2: -(1 - d)h(z) ,

(3.11)

(3.12)

(z, t) E R x [0, T] .

Proof. Without loss of generality, we may assume that F'(u) 2: 0 for all u ~ 0. Let u(z, t) = dh(z) - v(z, t), where v(z, t) = h(z) + w(z, t). Since v(z, t) is a solution of (1.4), it is straightforward to verify that (L + l)u(z, t) = F(dh(z))- dF(h(x)), where L is defined in (3.1) and

l(

) _ F(v(z,t))- F(dh(z)) z, t v(x, t)- dh(z) ·

In view of (H1), one has F(dh(x)) - dF(h(x)) 2: 0 for all z E R. Furthermore, since v(z, t) ~ 1 + K by (3.11), we have l(z, t) 2: !. = Ac~(K). Indeed, this inequality follows immediately from (3.7) if v(z, t) 2: dh(x); in the converse case, we observe that l(z, t) 2: min(F'(dh(x)), 0) 2: Ac~(K). Thus (3.8) implies (3.3), and the conditions (3.9), (3.10) are nothing else as the hypotheses (3.5) and (3.6). Therefore Theorem 3.1 shows that u(z, t) = dh(x)- v(z, t) ~ 0 for all (x, t) E R x [0, T], which 0 is (3.12).

Remark. Theorem 3.1 suggests the definition of a partial order in H1~(R) x L~oc(R) as follows. We say that (!fo, !ft) ~ (,Po, tPt) if !fo(x) ~ t/Jo(z) , e!ft(z)- eC!f~(z) +

~!fo(x) ~

xER ,

et/Jt(z)- ect/J~(z) + ~t/Jo(z)

a.e. in R;

see (3.5), (3.6). Then, if (!fo, 1ft) ~ (,Po, tPt), the solution of the linear hyperbolic equation (L + l)u(z, t) = 0 satisfying u(z, 0) = !fo(z), ut(z, 0) = !fl (x) stays for all t e R+ below the solution of the same equation with initial data (,P0 , ,P1 ). This order has the property that we can write any (lfo, !fl) E H1~ x L~oc as the sum of a "positive" part (~Pri, ~Pt) 2: 0 and a "negative" part (~Po, If}) ~ 0. This decomposition is unique if we impose that (~Pri, ~Pt) = 0 whenever (!fo, !fl) ~ 0 and (~Po, tp}) = 0 whenever (tpo, ~Pt) 2: 0. The formulas for (tp~, tpf) are given by ~Pri(z) = sup(O, tpo(z)),

1

1 2

~Pt(z) = c(tpt)'(x)- e- ~Pri(x) + sup(O, (~Pt- ctp~ + e tpo)(x)),

2

(3.13)

Stability of Propagating Fronts in Damped Hyperbolic Equations

139

and

lf'o (x) = inf{O, !f'o(x)), If'I (x)

= c{tpiJ)'(x) -

1

c 2

lf'o (x) + inf{O, (!f'l -

1 ctp~ + c tpo){x)).

~~

2

Remark that if {tpo, !f'l) E ZE, then {rp~ , rpf) E ZE. Moreover, it can be verified that lrpr(x)l :5 lrp1(x)l + clrp0(x)l a.e. in R. 3.2. A General Global Stability Result Combining the a priori estimates of Section 2 and the Maximum Principle, we are now able to state and prove our main stability result:

Theorem 3.3. Assume that (Hl), (H2) hold, and let co > 0, c ~ c., d E {0, 1]. There exists a constant Co ~ 1 such that, for all 0 < c :5 co and all K > 0 satisfying (3.15)

the following result holds: If K. > 0 is such that CoK.(1

+ llf{K.)) 112 < K

,

{3.16)

where 'lf is defined in (2.5), then for any (rpo, fPl) E ZE satisfying the inequalities (3.9), (3.10) and the bound ll(rpo, fPl)llz. :5 K., there exists a unique solution (w,wt) E C0 {[0,oo),Zt) of(1.5) with initial data {tpo,!f'l)· Moreover, one has ll(w(t),wt(t))llz. :5 K,

w{x,t)

~

-{1- d)h{x) ,

{3.17)

for all x E R, t E R+, and (2.2), (2.7) hold. Remark. The constant Co in (3.16) is the same as in {2.6). Sketch of the proof. By Proposition 3.2, the solution (w, Wt) with initial data {tp0 , tp 1 ) satisfies w(x, t) ~ -{1-d)h(x) as long as w(x, t)::; K. On the other hand, in view of {3.16), the a priori estimate {2.6) implies that w(x, t) :5 ll (w(t), Wt{t))llz. < K as long as w(x, t) ~ -{1 - d)h{x). Combining these facts and using a contradiction argument, we show that the solution is globally defined and satisfies {3.17) for all times. The differential inequalities {2.3) then imply {2.2), (2.7). 0 Remarks. 1} The relations (3.15), {3.16) imply that K {hence K.) can be chosen very big if c is sufficiently small. In this case, Theorem 3.3 shows that the travelling wave h is stable with respect to large perturbations, provided the positivity conditions {3.9), (3.10) are satisfied. Conversely, if cis large, then K (hence K.) has to be very small, and Theorem 3.3 reduces to a local stability result similar to Theorem 2.1.

140

Th. Gallay, G. Raugel

2) As a simple example, consider the non-linearity F(u) = u - u 2 (14] which satisfies {Hl), (H2). Then Ad(K) = -(d + K), and the condition (3.15) reads 1 - 4e(d + K) 2: 0. Also w(K) = 1 + 2K, hence the hypothesis ll(cpo, IPt)llz. ~ K. can be replaced by li(cpo, IPt)liz. ~ CK(l + K)- 113 , where Cis a sufficiently large positive constant. 3.3. Further Stability Results if 1 + 4eF'(l) ;::: 0 The previous results are still incomplete for at least two reasons. First, if c > c., they fail to give a decay rate of the perturbations as t-+ +oo; see (2.8). Next, they do not provide any global existence result if d = 0, that is, if v(.x, 0) ;::: 0. Assuming that the non-linearity F satisfies the stronger assumption

(H3)

F"(u) ~ 0 ,

u ;::: 0 ,

we can give a partial answer to both questions when 1 +4EF'(l) ;::: 0. Indeed, in this case, the Maximum Principle allows us to compare the solution w(.x, t) of (1.5) with solutions of the linear equations

= tii.,., + ctii., + F' (h )tii ,

(3.18)

etiiu + tii, - 2Ectiizt = tii.,., + ctii., + F' (O)tii .

(3.19)

etiiu + tii, - 2ectii.,,

and

We denote by (tii(t), tii,(t)) the solution of (3.18) with initial data (cp 0 , cpt), and by (tii±(t), wr(t)) the solution of (3.19) with initial data (cpt' cpf), the "positive" and "negative" parts of (cpo, IPt) defined in (3.13), (3.14). Following the arguments in (14, Section 4), we obtain our last stability result: Theorem 3.4. Assume that (Hl), (H3) hold and that 1+4EF'(1);::: 0. Given c;::: c., d E (0, 1), there exists a constant C 3 (c) ;::: 1 such that, for all K;::: 0 satisfying 1 + 4EAt (K) 2: 0 ,

the following result holds: for any (cpo, IPt) E Ze satisfying (3.9), (3.10) and

there exists a unique solution (w, w,) E C0 ([0, oo), Ze) of (1.5) with initial data (cpo, cp1). Moreover, we have

-(1 - d)h(.x)

~

w(.x, t)

~

K ,

(3.20)

Stability of Propagating Fronts in Damped Hyperbolic Equations

141

and w-(x,t):::; w(x,t):::; w(x,t):::; w+(x,t)'

for all x E R, t E R+· Finally, if d

> 0 and 1- 4eF'(O) > 0, one has

1 4

lim t 1 (llp.w(t)lltoo t-++oo

+ llw(t)llu• + llwe(t)llv)

= 0·

Remark. If ( ~Pri, tpt} = 0, i.e. if the initial data are non-positive, then (3.20) with K = 0 shows that the solution w(x, t) remains non-positive for all times. 4. Asymptotic Expansions in the Critical Case c = c. In this section, we restrict ourselves to the critical case c = c., and we consider perturbations of the travelling wave h in a strict subspace of Z.:. Using self-similar variables and energy estimates, we are able to compute explicitly the long-time asymptotics of the perturbations as t -+ +oo. In particular, we recover the results obtained by Gallay [13] in the parabolic case e = 0. Following Kirchgii.ssner (25], we consider solutions of (1.4} of the following form: v(x, t)

= h(x) + h'(x)W ( x, 1 +te~)

,

i.e. we set w(x, t) = h'(x)W(x, t/(l+e~)). Then W satisfies the equation r1Wu +(1-v'Y(x))We- 2vW,t = W,,+'Y(x)W, +h'(x)W2N(h(x), h'(x)W) , (4.1)

where "1 =

(1

+ ec~) 2

'

II=

ec. 1 +e~'

h"(x) 'Y(x) = c. + 2 h'(x) ,

xeR.

In (4.1) and in the sequel, the second argument of the function W is simply denoted by t, instead of t/(1+e~). From [1] we know that the travelling wave h (with c =c.) satisfies

where a 1 > 0, a 2 E R, and /3 = i(-c. + ,j~- 4F'(1)} > 0. Similar asymptotic expansions hold for the derivatives h', h", hence asx-+-oo, asx-++oo,

(4.2)

142

Th. Gallay, G . Raugel

where "'f- = c.+ 2(3 = 2y'F'(O)- F'(1) and zo = (a2/a1 - 2/c.). The hypothesis (HI) on F also implies that i(x) < 0 for all x E R. To study the long-time behavior of the solution W of (4.1), we use the scaling variables or self-similar variables defined by

{=

X

~

vt+to

,

-r = log(t+to) ,

for some t 0 > 0. These variables have been widely used to study the long-time behavior of solutions to parabolic equations, in particular to prove convergence to self-similar solutions [4,8,9,12,24]. In [15], it has been shown by the authors that these variables are also a powerful tool in the framework of damped hyperbolic equations. Following [15], we define the rescaled functions U and V by

V({,-r) = e5~/ 2 Wt({e~l 2 ,e~- to),

U({,-r) = e3~12w({e~l 2 ,e~- to),

(4.3)

or equivalently

W(x, t)

=

(t+t~) 312 U ( ..jt:to , log(t+to))

We(x, t) = (

1

t+to

)S/:1 V (

,

~, log(t+to))

(4.4)

vt+to

Then U({, -r), V({, -r) satisfy the system u~

{

3

- 2ue - 2u

=

v,

,.,e-~(v~- ~Ve- ~V) + (1-li'Y({e~I2))V- 2ve-~12\t( 2

Uee

= 2 +e~t 2 ...,({e~I 2 )Ue +e-~12 h'({e~I 2 )U2 N({,-r),

(4.5)

where N({, -r) = N(h({e~l2 ),e-3~1 2 h'({e~I2 )U) . We now introduce function spaces for the rescaled perturbations U, V. For T ;:::: 0, we denote by XTt Y~ the Hilbert spaces of measurable functions on R defined by the norms

We denote by

z~

the product space z~ =

II(U, V)ll~~

x~ X Y~

= IIUII~~

equipped with the standard norm

+ IIVII~~ .

143

Stability of Propagating Fronts in Damped Hyperbolic Equations

As is easily verified, if the functions (U, V) and (W, We) are related through (4.3) or (4.4), then (U(·,T),V(·,T)) E ZT if and only if the actual perturbation w(x, t) = h'(x)W(x, t/(1+e~)) satisfies

J

O

-oo

(w 2 +w!+w:)(x,t)dx+

r Jo

0

(1+x4 )eC•%(w 2 +w!+w:l(x,t)dx
o ,

2 + 'l

({

2

Ue

=o

if~$

o.

(4.7)

Therefore, it is reasonable to expect that the long-time behavior of the solutions to (4.5) will be determined by the spectral properties of the operator £ 00 on R+, with Neumann boundary condition at { = 0. Now, as is easily verified, this limiting operator is just the image under the scaling (4.4) of the radially symmetric Laplacian operator in three dimensions. Indeed, if U and W are related through (4.4), the equation UT = £ 00 U is equivalent to Wt = Wn + (2/x)Wz, x > 0. This crucial observation allows to compute exactly the spectrum of £ 00 in various function spaces; see (15]. For instance, in the space L2 (R+, (1+e6 )d(), the spectrum of £ 00 consists of a simple, isolated eigenvalue at A = 0, and of "continuous" spectrum filling the half-plane {A E C I Re A $ -1/4}. The eigenfunction corresponding to A = 0 is the 2 gaussian e-( / 4 • Therefore, we expect that the solution U(e, T) of (4.5) converges as +oo to a 0 and 6o > 0 such that, for all (Uo, Vo) E ZTo satisfying II(Uo, Vo)llzTD $ 6o, the system (4.5) has a unique solution (U, V) E C0 ([To, +oo), ZT) with (U(To), V(To)) = (Uo, Vo). In addition, there exists a• E R such that

IIU(T)- a* 0 and m, n, k E lN0 • Then we write 11 · 11, for the norm on V'(r~or2 ), 1 $ p $ oo, II·IIH• for the norm on H"(r~o r2 ), ll·llm,n for the norm on wm·""([r~or2];Hn(s~os2 )), II·IIHm·" for the norm on Hm([r~or2];Hn(s~os2 )). The preceding norms will always be understood with respect to (w.r.t .) the entire domain of the particular functions, so that the numbers r~o r 2 , s 1 and s 2 are clear from the context. We shall also use the notation 11·11,,[1] for the seminorms, defined for 0 $ t $ t1 by llfll,.,[l] ~ llfl(o,IJII,., II·IIH•[I] for the seminorms, defined for 0 $ t $ t1 by llfiiH•[1J~ llfl(o,IJIIH•• ll·llm,n,(l] for the seminorms, defined for 0 $ t $ t1 by llfllm,n,[l] ~ llf ho,IJIIm,no

II ·IIH"'·"[I] for the seminorms, defined for 0 $ t $ t1 by llfiiH"'·"[I] ~ llfl[o,tJIIH"'•"· 2.1. Statement of the Main Result. Definition 2.1. We shall call a function u a solution of {2.16}-{2.18} if and only if u E W 1•""{[0, t 0 ); L2 {a, b)) n L00 ([0, t 0 ); H 1(a, b)), u satisfies Eq. {2.16), u satisfies Eqs. (2.17} and {2.18} pointwise.

{2.19)

{2.20) {2.21)

Definition 2.2. We shall call a function h on [0, t 0 ) x [a, b) boundary-regular if and only if hE W 1·""([0, t 0); H 1 (a, b)) n L""([O, t 0); H 2 (a, b)), 1

h.,(·, a), h.,(·, b) E H (0, t 0 ) .

{2.22) (2.23)

T. Hagen, M. Renardy

150

Theorem 2.3. Given functioru f and p on [0, t 0] x [a, b], u0 on [a, b] and ub on (0, t0 ] such that p and f are boundary-regular,

> 0 on [0, to] x [a, b], u0 E H 2 (a, b), ub E H 2 (0, to), u0 (b) = ub(O), u:(o) = p(O, b) u~(b) + f(O, b).

p

(2.24) (2.25) (2.26) (2.27) (2.28) (2.29)

Then the boundary-initial value problem {2.16)-(2.18) has a boundary-regular solution such that

u

u E C 1 ([0,to];H 1(a,b)) nC([O,to];H2 (a,b)), u is unique in W 1•00 ([0, to]; L 2 (a, b)) n L00 ([0, t 0]; H 1 (a, b)).

(2.30) (2.31)

The proof of Theorem 2.3 will be outlined in the following sections. The uniqueness claim (2.31) is readily seen with the help of Gronwall's inequality.

2.2. The Compatibility Conditions. Definition 2.4. Given functions f and p on [0, to] x [a, b], u0 on [a, b] and ub on [0, to] such that

j,p E C 00 ((0,to) x (a,b]), p> 0,

u0 E C00 ([a, b]), ub E COO((O, to]).

(2.32) (2.33) (2.34) (2.35)

Then we shall say that, for N E IN0 , the functions u 0 and u 11 are compatible of order N w.r.t. f and p if and only if the sequences (V") and (W"), defined for n 2::: 0 by

(2.36) and

W0 (t) ~ u11 (t),

(2.37) 1

W"+l(t) ~ (p(t, b))- (W,"(t) - [D;J](t, b)) -

(p(t,b))- 1

(2.38)

(~ (:)w*+1 (t)[n;- *p](t,b)),

satisfy the compatibility conditions

V"(b)

= W"(O)

for 0 $ n $ N .

(2.39)

On the Equations of Melt-Spinning in Viscous Flow

151

Theorem 2.5. Suppose that the functions f, p, u0 and uli satisfy hypotheses (2.32)(2.35). Then, for N E IN0 , the boundary-initial value problem (2.16}-(2.18} has a unique solution u, satisfying uE

n CA:([O, to]; HN+l-A:(a, b)),

N+I

{2.40)

A:=O

if and only if u 0 and uli are compatible of order N w.r.t. f and p. The proof of Theorem 2.5 is straightforward, but long and tedious, and shall therefore be omitted. See [1] for details. 2.3. Approximating Sequences.

Lemma 2.6. Given a boundary-regular function h on [0, to] x [a, b]. Let M be a constant such that M ~

llhlh,l + llhllo,2·

(2.41)

Then there exists a sequence (hra)n e:N of functions on [0, to] x [a, b] such that, for

n E IN,

hra E 0 00 ((0, to] X (a, b]), hn -+ h in C([O, t 0 ]; H 1(a, b)) as n -+ oo, llhraiiH•·• $ y'2Ta"(M + 1), llhrallo,2 $ M + 1. Moreover, for each t E (0, to], there exists N E IN such that llhraiiH•·•[tJ $ Proof. Define

v'2t (M + 1)

for n ~ N.

(2.42) {2.43) (2.44) (2.45) (2.46)

h by •

h(t)

={

def

h(to) for t > to, h(t) h(O)

for 0 $ t $ to, for t < 0,

(2.47)

let H be given by def

H(t,x) =

hs(t, b) (x- b) + h(t, b)

for t E JR., x > b, • h(t,x) for t E R, a $ x $ b, { h:r(t, a) (x- a)+ h(t, a) for t E R, x < a,

(2.48)

and, by using a sequence of Friedrichs mollifiers (Jn), define the sequence {Hn)ner-1 by

Hn(t, x)

~

JEJR Jn(t - s) Jn(x - y) H(s, y) dyds

fort, x E R (2.49)

A straightforward calculation yields that, for sufficiently large K E IN, the sequence (Hn+K )ne:N, restricted to [0, to] x [a, b], has the required properties (2.42)- (2.46).

0

T. Hagen, M. Renardy

152 2.4. Energy Estimates.

Lemma 2.7. Given ub E C2([0, t•]) for some t• > 0. Let t 0 E (0, t•] and define ub d£f ubl(o,to]· Suppose the boundary-initial value problem (2.16}-{2.18} has a solution u in C3 ([0, t 0] x [a, b]) with J, p E C2([0, t 0] x [a, b]) and u0 E C2([a, b]). Let p" d£! p(·, a) and pb ~ p( ·, b). Then there exists a positive constant C, depending only on t•, such that, for 0 ~ t ~ t 0 , the solution u obeys the estimates:

llu(t)ll~1 < (llu0 11~1 + C IIPIIo,l (llubllt(c] + llu:r(·,b)ll~.[tJ) + t 11111~.1) · exp{(C 11PIIo,2 + 1) t},

(2.50)

0

llu(t)ll~• ~ (llu ll~• + C IIPIIo,l (llubll~.[t] + llu:r(·,b)ll~.(c)+

lluu(·, b)ll~.[cJ) + t ll/ll~,2) exp{(C 11PIIo,2 + 1) t},

(2.51)

llulltl,(t] ~ (llu0 11~1 + llp(O) u~ + /(0)11~1 + C llullo,2IIPII~1,1(t)+

where

ll/11~~.1[t) + C IIPIIo,l (llubll~1(t) + llu:r(·,b)ll~1[t))) · exp{(C IIPIIo.2 + C llullo,2 + 1) t},

(2.52)

1 llu:r(·,b)ll2,[tJ < ll(pb)- llao (llubiiH1[t] + C v'i 11/llo,l), 1 llu:r(·, b)IIH1[tJ ~ C ll(p•)- llao (ilu•IIH•[tJ + llfiiHt,1[tJ + 1 ll(p•)- llao (llu•IIH• + llfllo,l) IIPIIHl,l[cJ), 1 llu:r:rh b>ll2,[t] ~ ll(p6)- llao (IIU:r(·, b)IIHl(tJ + C 11PIIo,2llu:r(-, b)ll2,[t) + C v'i llfllo,2)·

(2.53) (2.54) (2.55)

Moreover

llu:r(-, a)ll~~ < (llu0 11~~ + llp(O) u~ + /(0)11~1 + C llullo.2IIPII~~.~+

ll/ll~1,1 + C IIPIIo,l (llu6 11~~ + llu:r(·, b)i1~1)) · ll(p")- 1 llao exp{(C IIPIIo.2 + C llullo,2 + 1) to}.

(2.56)

Proof. The constant C of the theorem is a generic constant that absorbs the imbedding constants of various Sobolev estimates of functions over [a, b] and [0, t•]. The stated estimates follow readily from elementary calculations involving Gronwall's inequality.

0

2.5. Existence and Regularity of Solutions. Theorem 2.8. Suppose that the functions J, p, u0 and ub satisfy hypotheses (2.2.4)(2.29). Then the boundary-initial value problem {2.16)-(2.18) has a unique boundaryregular solution u such that u E W1•00 ([0,t0];H1(a,b))nL00 ([0,t0];H2(a,b)). (2.57)

On the Equations of Melt-Spinning in Viscous Flow

153

Proof. Let M ~ IIPIIt,l + 11PIIo,2 + llllh.1 + llfllo,2· By Lemma 2.6, for p and I there are C 00-sequences and resp., with properties (2.42}-(2.45}. There exist

(It),

(pk)

sequences

(ut) and (u~) such that, fork E IN,

u: E C

((0, to)}, u~ E C 00 ((a, b)}, u~ and ut are compatible of order 3 w.r.t. It,~. u: ----+ ub in H 2 (0, t 0 ) as k --+ oo, u~ ----+ u0 in H 2 (a, b) as k --+ oo. 00

(2.58} (2.59} (2.60} (2.61}

The problems

u1A:I(t, x) = pk(t, x) u~A:I(t, x) + lk(t, x), uiA:I(o, x) = u2(x}, uiA:I(t, b) = u:(t)

(2.62} (2.63}

3

have solutions uiA:J in C ([0, t 0] x [a, b)} by Theorem 2.5. For n, mE IN define wn,m ~ uln] _ ulml.

(2.64}

By Lemma 2.7, wn,m obeys the estimate

llwn•m(t)ll~• :5 (llu~- u~ll~•

+ C (1 + IIPnllo,l +

IIPnllo,tll(pn(·,b)}- 1 11~) (llu~- u~IIH•

+

..ffo (IIPn- Pmllo,l llulmlllo,2 + 11r- lmllo.d) 2) · exp{ (C 11Pnllo,2 + 1} to}.

(2.65}

The quantities i1Pnllo,2, llrllo,2, IIPnllt,l and llrlh.t are bounded according to (2.44}(2.45}. Hence llulmlll 0 ,2 is bounded according to (2.51}. It follows that wn,m----+ 0 in C([O, t 0 ]; H 1(a, b)} as n, m--+ oo.

(2.66}

Hence (ulnl) is Cauchy in C([O,t0 ];H 1(a,b)} n C 1 ([0,t0 J;L2(a,b)} with limit u. On the other hand, ( ulnl) is bounded in W 1•00 ((0, t 0 ]; H 1 (a, b)} n L 00 ([0, t 0 ]; H 2 (a, b)} by estimates (2.51 )- (2.52}. By weak* compactness, the sequence ( ulnl) contains a weak* convergent subsequence, say (vn}, in W 1•00 ([0, t0 ]; H 1 (a, b)} n L 00 ([0, t0 ]; H 2 (a, b)} with limit v. See [3] for weak and weak* topologies. By boundedness due to (2.54}, (2.56}, we may assume that the sequences (v;(·, a}}, (v;(·, b)} converge weakly in H 1(0, t 0 ). Note that the sequence (vn) converges strongly in L 2 ([0, t 0] x [a, b]} to its unique weak* limit v. Since the sequence ( ulnl) also converges strongly in L2([0, t0 ] x [a, b]}, we have ·u = v. Suppose v.. is the weak limit of (v;(·, a)) in H 1 (0, t 0 ). We remark that (v;(·, a)) converges strongly in L2 (0, t 0 ) to its weak limit v11 • However, (v;) is weakly convergent to Uz in L2 {[0, t 0 ]; H 1 (a, b)}. Since the linear map h 1-+ h(a) from L2 ([0, t 0 ]; H 1 (a, b)} to L 2 (0, t 0 ) is bounded, the sequence (v;(·, a)) converges weakly to Uz(·, a) in L 2 (0, t 0 ) . Hence necessarily Uz(·,a) = v... Since the same argumentation works for Uz(·,b}, the claim follows.

0

154

T. Hagen, M. Renardy

Theorem 2.9. Suppose that the conditions of Theorem 2.8 hold. Then the solution u of the boundary-initial value problem (~.16)-(~.18) has the regularity property (~. 90). Proof. The solution u has to satisfy the energy estimates of Lemma 2. 7. Therefore we can conclude that

{2.67) Hence u is strongly right-continuous on [0, to). We define v{t,x) ~ u(to -t,a+b-x). v also solves a problem of the form {2.16)-{2.18). Then the boundary-regularity of u yields the right-continuity of v according to {2.67).

0 3. SOLVABILITY OF THE EQUATIONS OF MELT-SPINNING In this section we shall prove existence, uniqueness and regularity of solutions for the equations of melt-spinning, Eqs. {1.7)-{1.15). Our solution strategy is to prove the existence of a fixed-point for an appropriate solution operator. The main difficulty arises in the correct choice for the underlying metric space. Similar developments can be found elsewhere (see [4], [5], [8]).

3.1. Statement of the Main Result. Definition 3.10. We shall call a vector field (A, z, v), defined on [0, to] x (Ts, TE], a solution of (1. 7}-(1.15} if and only if

A, z, v E W 1•00 ((0, to]; H 1 (Ts, TE)) n L00 {(0, to]i H2 (Ts, TE)), A, z, v satisfy Eqs. (1. 7}-(1.9}, A satisfies Eqs. (1.10}, (1.1,4} pointwise, z satisfies Eqs. (1.11}, (1.15} pointwise, v satisfies Eqs. (1.12}, (1.13} pointwise.

(3.68) {3.69) {3.70) {3.71) {3.72)

Theorem 3.11. Given initial values A0 , z0 on (Ts,TE] and boundary values AE, VE, v5 on [0, t*] for some t• > 0 such that

A0 , z0 E H 2 (Ts, TE), AE E H 2 {0, t*), 1 00 VE Vs E W • {0, t*), A 0 > 0 on (Ts, TE], 4 < 0 on [Ts, TE]· 1

{3.73) {3.74) {3.75) (3.76) {3.77)

On the Equations of Melt-Spinning in Viscous Flow

155

Also suppose that the compatibility conditions A 0 (TE) = AE(O), ·

z0 (TE) = 0,

aTE

(3.78)

AE(O) = .jAO(TE) AT(TE) + (vs(O)- VE(O)) 0

~ 4(TE) + VE(O) =

(

1

fT, z¥-(T) )AO(T) dT ,

hs

(3.79)

0

(3.80)

are satisfied. Then there exists a t0 E (0, t•] such that the boundary-initial value problem {1. 7}-(1.15} has a unique solution (A, z, v) on [0, t 0 ] x [Ts, TE]· This solution (A, z, v) has the properties: A,z E C 1 ([0, to]; H 1 (Ts, TE)) n C([O, to]; H 2 (Ts, TE)), A, z E W 2•CX>([O, to]; L 2 (Ts, TE)), v E W 1•CX>([O, to]; H 1 (Ts, TE)) n C([O, to]; H 2 (Ts, TE)), A, z, v are boundary-regular.

(3.81) (3.82) (3.83) (3.84)

Moreover, if VE, Vs

E

C 1 ([0, to]),

(3.85)

then the solution (A, z, v) has the additional properties: A, z E

n cA:([O, to]; H -k(Ts, TE)), 2

2

(3.86)

k=O v E C 1 ([0, to]i H 1 (Ts, TE)) n C([O, to]; H 2 (Ts, TE)).

(3.87)

Remark 3.12 Solving the momentum balance (1.8) for v yields

{TZT(t,r) (fT•zT(t,T) )v(t, T) = v5 (t) + (vE(t)- vs(t)) hs A(t, r) dr hs A(t, T) dT

1

(3.88)

and

VT(t,T) ZT(t, T) A(t, T)

([T•ZT(t,T)

= (vE(t)- vs(t)) hs A(t, T)

dT

)-

1

(3.89)

Hence we can eliminate Eq. (1.8) by (3.88)-(3.89) in Eqs. (1.7), (1.9). From now on we shall assume that t•, A0 , z 0 , AE, VE and v5 are given once and for all such that hypotheses (3. 73)-(3.80) hold.

3.2. The Solution Operator. Definition 3.13. Let h = h(t, T) be a boundary-regular function with domain [0, t'] x [Ts, TE]· Then define the energy functional£ by

£(h) ~ (11hll~.2 + l1hllt1 ++II~(-, Ts)ll~~ +II~(-, TE)II~~) l.

(3.90)

156

T. Hagen, M. Renardy

Definition 3.14. For L > 0 and t' E (0, t•], let S(t', L) be the set of vector fields

(B,€)T on [O,t'] x [Ts,TE] such that

B and { are boundary-regular on [0, t'] x [Ts, TE], &(B)2 + &({)2 $ L2,

B(O, T) = A 0 (T) and B(t, TE) = AE(t), ~(0, T) = z0 (T) and {(t, TE) = 0.

(3.91) (3.92) (3.93) (3.94)

Definition 3.15. We shall say that t' E (0, t•] is admissible if and only if the set S(t', L) is nonempty and all pairs (B, {)T E S(t', L) have the properties B(t, T)

> 0 and

[T• }Ts

~BT((t, T)) dr =f. 0

fort E [0, t'), T E [Ts, TE)·

t, T

(3.95)

Lemma 3.16. Suppose S(t•, L) is nonempty. Then there exists t 0 E (0, t•] such that every t E (0, t 0 ] is admissible. This lemma follows from an easy calculation.

Definition 3.17. Lett' E (0, t•] be admissible. Define the solution operator I:t',L on S(t', L) by I:t' ,L

: (

~ ~ ~ )

(

(3.96)

) ,

where Y = Y(t, T) and ( = ((t, T) are the solutions (in the sense of Definition 2.1) of the boundary-initial value problems, stated on [0, t'] x [Ts, TE],

oT ([T•~T Yi = VB YT + (vs- VE) lTs B dT Y(O, T) = A 0 (T),

)-l

,

Y(t, TE) = AE(t),

(3.97)

(3.98)

and

(3.99)

(3.100) 3.3. Application of the Contraction Mapping Principle. In this final section we apply the Contraction Mapping Principle to the solution operator. To this end, it is necessary that S(t, L) be a complete metric space. An appropriate metric is given in the following lemma. Lemma 3.18. Suppose S(t,L) is nonemptfl. (B,{)T, (B,i_)T E S(t,L) by d

((B,~)T, (B,{)T) ~

renders S(t, L) a complete metric space.

(11 8 -

Then the metric d(·,·), defined for

iJII:.l +II{-i.II:J!'

(3·101)

On the Equations of Melt-Spinning in Viscous Flow

157

dis clearly well-defined on S(t,L). Suppose a sequence (v") = (B",{") is Cauchy in this metric. Then v" converges strongly, as n -+ oo, to a point v0 in 2 (£2 ([0, t] x [Ts, TE])) • Since £(B") 2 + £({") 2 $ £ 2 for all n, a subsequence of (v") is weak* convergent in (W 1•00 ([0, t]; H 1 (Ts, TE)) n £ 00 ([0, t]; H 2 (Ts, TE))) 2 with weak* limit w 0 E S(t, L). The convergence is strong in (£2 ([0, t] x [Ts, TE])) 2 • Hence we deduce that v0 = Wo.

Proof.

0 Theorem 3.19. Suppose S(t•, L) is nonempty. Then there exists an admissible t 0 E (0, t•] such that the solution operator Eto,L is a contraction map on S(t0 , L) w.r.t., the metric d given in {3.101}. The proof of Theorem 3.19 is not complicated, but tedious (see [1]). The main conclusion we can draw is that the fixed-point of the mapping Eto,L is the sought-for solution for the equations of melt-spinning. REFERENCES [1) HAGEN T . & M. RENARDY, On the equations of fiber spinning in nonisothermal viscous flow, to appear. [2) KASE S. & T. MATSUO, Studies on melt spinning. I. FUndamental equations on the dynamics of melt spinning, J. Pol1f171. Sci. A, 3, 2541 (1965). [3) KATO T., Perturbation Thto"ll for Linear Operotor5, 2nd corr. ed., Springer, Berlin (1995). [4) LIONS J . L. & E. MAGENES, Non-Homogeneow Bounda"ll Value Problem5 and Applicatiom I, Springer, New York (1972). [5) MAJDA A., Compru5ible Fluid Flow and Sy5tenu of Comervation Laws in Several Space Variables, Springer, New York (1984). [6) PEARSON J. R. A., Mechanics of Pol1f171er Procusing, Elsevier, London (1985). [7) PETRIE C. J. S., Elongational Flow5, Pitman, London (1979). [8) RENARDY M., W. J. HRUSA & J. A. NOHEL, Mathematical Problem5 in Viscoelasticity, Longman & Wiley, New York (1987). [9) ZIABICKI A., FUndamenta/5 of Fibre Formation, Wiley, London (1976).

T. Hagen, M. Renardy, Department of Mathematics, Virginia Polytechnic Institute and State University, Interdisciplinary Center for Applied Mathematics, Research Group on Modeling and Analysis of Fluids, Interfaces and Amorphous Polymers, Blacksburg, Virginia 24061-0123, USA Email: thagenGmath. vt. edu, renardymGmath. vt. edu

On Energy Estimates for Electro-Diffusion Equations Arising in Semiconductor Technology R. HUNLICH, A. GLITZKY Abstract. The design of modern semiconductor devices requires the numerical simulation of basic fabrication steps. We investigate some electro-reactiondiffusion equations which describe the redistribution of charged dopants and point defects in semiconductor structures and which the simulations should be based on. Especially, we are interested in pair diffusion models. We present new results concerned with the existence of steady states and with the asymptotic behaviour of solutions which are obtained by estimates of the corresponding free energy and dissipation functionals. Keywords: reaction-diffusion systems, drift-diffusion processes, pair diffusion models, steady states, asymptotic behaviour Classification: 35840, 35K45, 35K57, 78A35

1. INTRODUCTION

In the design of modern semiconductor devices and in the development of their technology, device and process simulation programmes turned out to be very important tools. The simulation of modern technologies requires the continuous improvement of the underlying physical models and their analytical and numerical investigation. One of the main steps in the preparing of semiconductor devices is the redistribution of dopants connected with or followed after the doping. In order to explain this process, different models have been developed. Of special interest are pair diffusion models (see e. g. (2,4,10)). They consist of a set of reaction-diffusion equations for charged dopants, point defects and dopant-defect pairs coupled with a Poisson equation for the electrostatic potential of the inner electric field. Besides the mentioned species, electrons and holes have to be taken into account. But we assume that their kinetics is very fast, such that the Poisson equation is replaced by a nonlinear Poisson equation for the chemical potential of the electrons (see e. g. (7,10)). Motivated-by these considerations we investigate a rather general electro-reactiondiffusion system for m species Xi. We denote by ,P the chemical potential of the electrons, by Pi('r/J) suitably chosen reference concentrations depending on ,P, and by Ui, lli = Ui/Pi('r/J}, (i = lntli the concentration, the electrochemical activity and the electrochemical potential of the i-th species where all variables are suitably scaled. We assume that the set {1, ... , m} is split into two parts {1, ... , m} = J U J', and

On Energy Estimates for Electro-Diffusion Equations Arising in...

159

formulate the initial boundary value problem which we are interested in as follows:

at+ ~ V · ii +

"

L.., (at- /Ji)Ilop II •

~ at+

=

0

on {O,oo} x fl,

0

on {O,oo} x

an,

i E J;

0

on {0, oo)

X

fl,

i E J';

= I = 0

on {0, oo)

X

fl,

on {0, oo)

X

an;

(o,IJ)E'Il

ji

E (a; - /J;) Ra11

=

(o,fl)E'Il

-V · (Vt/1) + e(t/J)-

{1}

m

L Q;(t/J)U; i=l

II·

Vt/J

U;{O}

=

Ui on fl, i

= l, . . . ,m.

Here denote I a fixed charge density, -e(t/J) the charge density of electrons and holes, and Qi(tP) = -r/;(t/J)/Pi(tP) the charge of the i-th species depending on t/J, too. Only for species i E J there is a diffusive and convective transport given by the mass flux ji = -Di(tP)

[vUi + Qi(tP) U;Vt/J], i E J.

But in all continuity equations source terms occur, generated by a lot of mass action type reactions of the form

a1X1

+ · · · + amXm 0,

u(O) = U, u E L~oc(R+, X) n Li::(~, L""(O, IR.m)), u' E L~oc(IR+, X•),

(P)

t/J E L~oc(~, H 1 (0)) n Li::(~, L""(O)). 3. THE NONLINEAR POISSON EQUATION Here we summarize some results concerned with the nonlinear Poisson equation.

Lemma 1. For any u E Y+ = L~(n, IRm) there exists a unique solution t/J to E(tp, u) = 0. Moreover, there are a positive constant c and a monotonously increasing function d: m.+ ..._. ~ such that

!IV! - ~lin•

~ c llu- ully 'Vu, u E Y+, E(.,P, u)

= E(tjj, u) = 0,

162

R. Hiinlich, A. Glitzky m

111/JIILoo ~ c { 1 + L IIUi ln UiiiLt + d(/11/JIIHt)}

'VuE Y+, E(t/J, u) = 0.

t=l

Proof. Since for u E Y+ the operator E(·, u) is strongly monotone uniformly with respect to u as well as hemicontinuous, and since for t/J e H 1 (0) the operator E(t/J, ·) is Lipschitz continuous uniformly with respect to t/J, the first and second assertions are obvious. The third assertion is a consequence of Greger's regularity result [9] and of Trudinger's imbedding theorem [14]. 0

Later on we are interested in a modified version of the Poisson equation which is obtained by setting u = ap(t/J), a E IR.m. We define E : H 1(0) x IR.m-+ (H1 (0))- by

L

{E{t/J,a),~) = {v.p · v~ + (e(t/J) + t.aap~(t/J)- f] ~} dx, ~ e H 1(0). Lemma 2. For any a E IR~ there exists a unique solution t/J to E(t/J, a) for every R > 0 there is a positive constant c(R) such that

= 0.

Moreover,

llt/JIIHt, llt/JIILoo $ c(R) 'Va E ~. llallll"' $ R, E(tjJ,a) = 0, 111/.> - ~II HI $ c(R)

!Ia - allll"' 'Va, a E IR~, llallll"', llallll"' $ R, E(t/J, a)

= E(~, a) = 0.

Proof. Again for a E JR.~ the operator E(·,a) is strongly monotone uniformly with respect to a as well as hemicontinuous. From this the existence and uniqueness result follows. The estimate

cllt/JII~~ $ (E(t/J,a)-E(O,a),t/J) $1

L

(e{O}+

t.aapao>-f]¢dxj $ c{l+llallll"')llt/JIIL'

yields the assertion on the H 1- norm of t/J. The assertion on the £""-norm of t/J is obtained from Lemma 1, for instance. Then the last assertion is obvious. 0

4.

THE ENERGY AND DISSIPATION FUNCTIONALS

In this section we investigate basic properties of the free energy functional and of other related functionals. First, we define Flo F2 : Y+ -+ 1R. by

(4) where t/J E H 1(0) n L""{O) is the unique solution to the Poisson equation E(t/J, u) = 0,

F2{u)= and set

Lt.{u.(lnp,%)-1]+p,(O)}dx, ueY+,

(5)

On Energy Estimates for Electro-Diffusion Equations Arising in...

163

The value F(u) can be interpreted as the free energy of the state u. Because of (3) it holds F(u)

2:: c { llt/JII~• +

m

L IIUi In UiiiL• - 1}. u E Y+. i=l

(6)

For u, u E Y+ and corresponding t/J, 1jj E H 1 (rl) with E(t/J, u) = E(:;jj, u) = 0 we obtain

+

1"

(s -1jj) e'(s) ds-

"

t 1"

(s -1jj)

ui

i=l

"

;::: (P(1jj), u- u)y + c llt/J -1iill~·

Q~(s) ds} dx

;::: (P(1jj), u- u)y.

(7)

From this relation it follows that F 1 is convex and continuous on the convex set Y+. We extend F 1 to Y by setting F 1 (u) = +oo for u E Y \ Y+. Then the extended functional F 1 : Y-+ 1R is proper, convex, lower semicontinuous, and subdifferentiable in each point u E Y+ where P(t/J) E oF1 (u). Because of properties of its integrand the functional F2 is convex (see [3]) and continuous (see [7]) on Y+· Again the extended functional F 2 : Y -+ lR, F2 (u) = +oo for u E Y \ Y+, is proper, convex and lower semicontinuous. For u, u E Y+ with u ;::: 6 > 0 we obtain

F2 (u)-F2 (u) =

f{f:In

Jn

i=l

~ (ui-~)+f:1"~ln~

Pi(O)

i=l u,

ui

ds}dx

;::: (In p~), u- u)y + llv'ii- viill~ ;::: (In p~), u- u)y.

(8)

Thus, F 2 is subdifferentiable in points u E Y+ with u;::: 6 > 0 and Jn(ufp(O)) E 8F2 (u). Finally, we have to extend the introduced functionals to the space X*. We define

i',. =

(F;Ix)*: x·-+ lR, k = 1,2,

where the star denotes the conjugation (see [3]) F;(v) =sup {(u, v)y- F,.(u)}, v E Y, uEY

(F;Ixt(u) =sup { 0.

F,.(u) = (9)

Omitting the tilde we can summarize the properties of the free energy functional as follows.

Lemma 3. The functional F = F 1 + F2 : X* -+ IR is proper, convex and lower semicontinuous. For u E Y+ it can be evaluated according to (4), (5). The restriction

R. Hiinlich, A. Glitzky

164

Fly+ is continuous. If u E X and u

~

6 > 0 then

u

u

( = P(,P) +In p(O) =In p(,P) EX,

( E 8F(u) .

Next we study properties of the dual functional F* : X -+

F*(()

m.,

= sup

{'t

(14) Finally, let i E J. Since t/J E V""(S, L"") and since the sequence H 6 is bounded because of (13) there exists a constant c > 0 such that IIV ..Jiifii£3(S,£3) $ c. From this V ..fiii E L 2(S, L2 ),

V

fat__.. V ..fiii in L (S, L 2

2

),

i E J,

On Energy Estimates for Electro-Diffusion Equations Arising in . ..

167

follows (see [13, Proposition 2.4]), and because of the weak lower semicontinuity of continuous quadratic functionals we arrive at 12

1 l1

f e~1 L 4 Di(?/l)Pi(?/J)IV v'iiil 2 dxdt ln iEJ 12

::; lipJJlf1 l1

Hence

f e~1 L

ln

4 Di(?/J)Pi(?/l)IV

iEJ

.;a E L {S,X}, and from {13}- {15} we obtain 2

{atl

1

1,

2

dxdt.

e~1 D(u(t))dt::; H.

{15} 0

h

Corollary 1. Along any solution {u, ?/;) to (P) the free energy F(u) remains bounded from above by its initial value F(U) and decreases monotonously,

F(u(t2)) ::; F{u(t1)) ::; F{U},

0::; t 1 ::; t 2.

Moreover, there exists a constant c depending only on the data such that m

i=l

II?/JII£00(J4,HI(rJ)} ::;

c,

II?/JIILOO(J4,LOO(rJ)} ::;

c

for any solution to (P). Proof. The first assertion follows from Theorem 1 by setting .X = 0. The remaining estimates are a consequence of {6) and Lemma 1. 0

6. INVARIANTS AND STEADY STATES We introduce the stoichiometric subspace S belonging to all reactions,

S =span{ a- {J: {o:,{J) E R} C By integrating the continuity equations over {0, t) x invariance property.

Lemma 6. If {u, t/J) is a solution to (P) then

m.m.

n one easily verifies the following

l{

u(t)- U} dx E S for all t E 1R+·

Now we ask for steady states belonging to the evolution problem (P) which satisfy such an invariance property, too. Thus we have to solve the following problem. A{u, '1/J) = 0,

E{'l/1, u) = 0,

u

~

0,

fo {u- U} dx E S, u E

X n L""{fl, 1Rm),

1/J E H 1{fl) n L""(fl).

Lemma 7. If (u, ?/;) is a solution to (S) then u E Mo and D(u) = 0.

(S)

R. Hiinlich, A. Glitzky

168

Proof. The proof is similar to that of Theorem 1. Let (u, t/J) be a solution to (S),

define a= ufp('I/J) and set u6 = u + 6, a6 = u6fp('I/J), ( 6 = 1na6 where 6 > 0. Then 6 ( ,

Vaiex, {A(u6 ,t/J),(6 }=h6

where h6 = {A(u6 , t/J)- A(u, t/J), ( 6} ~ 0 if 8 ~ 0. Furthermore, because of the estimate

1{L 0

2

4Di(tP)Pi(t/J)IV{atl +

iEJ

L

2ka,B(t/J)IVai

0 -

Vai.81

2

H }

~

.fi in Y

and

dx :5 (A(u6 ,tjJ),(6 )

(o,P)ER

we find that V.j(i.f __... V.fiii in £ 2 , i E J. Thus ..ja EX, u E Mv and 6

0:5 D(u) :5lif-}Jif(A(u ,tjJ),(6)

= 0.

0

Next, we show that there is a correspondence between the set of steady states, i. e. the set of solutions to (S), and the set A C IR.m defined by

A =

{aeiR.~ :

a 0 = a.8V(a,{J)E'R-,

L

{u-U}dxeS,

where u = ap(t/J) and t/J is the solution to E(tjJ,a) = 0 } ·

Lemma 8. If (u, t/J) is a solution to {S) then a= u/p(t/J) E A. Vice versa, if a E A and u, t/J are chosen as in the definition of A then (u, t/J) is a solution to {S).

Proof. The first assertion follows from Lemma 7 and Lemma 5. If a E A then obviously t/J E H 1 (fl) n L00 (fl), u = ap(t/J) E X n L00 (fl, IR.m), u ~ 0, Vui = Giri;(t/J)Vt/J = -'UiQi('I/J)Vt/J, i E J, A(u, t/J) = 0, and E('l/1, u) = E(t/J, a) = 0. 0 Finally, we show that the set A n int JR.~ corresponds to the set of minimizers of the function Go : IR.m ~ JR. defined by

Go(()=G(()+I5.~.(()- LU·(dx,

(EIR.m.

This function is proper, convex and lower semicontinuous, dom Go = S.l.. Because of the continuity of G (see Lemma 4) we obtain by the Moreau-Rockafellar theorem (see [3]) that 8Go(()

= oG(() +oisJ.(()-

L

Udx, ( E IR.m.

(16)

Lemma 9. If a E An intiR.~ then ( = Ina is a minimizer of G 0 • Vice versa, if ( E JR.m is a minimizer of Go then a = ~ E A n int JR.~.

Proof. Because of (16) and Lemma 4, ( is a minimizer of G0 if and only if ( E

s.l.,

oG(() - L

u dx = L { e' p(t/J)- u} dx E s

On Energy Estimates for Electro-Diffusion Equations Arising in...

169

where '1/J is the solution to E(tjJ,eC) = 0. The relation ( E Sl. is equivalent to (o - {3) · ( = 0 'V(o, f3) E 'R., or to (eC)a = (eC)P 'V(o, {3) E 'R.. Since the map ( t-+ eC is a bijection from IRm onto int IR~ all assertions of the lemma are obtained. 0

Lemma 10. The set An intiR.~ contains at most one element. Furthennore, A n int IR~ =I 0 if and only if the following condition is fulfilled:

Lu .(

dx > 0 'V( E

Sl., ( 2: 0, ( =I 0.

(II)

Proof. 1. The first assertion follows from Lemma 9 since the functions G and Gols.J.. are strictly convex. 2. If a E An intiR~ then for any ( E Sl., ( 2: 0, (=f. 0 we find

ru. (dx = lnrtaap;('I/J}(;dx > 0.

Jn

i=l

3. Now let (II) be fulfilled. According to Lemma 9 we have to show that there is a minimizer of G 0 • It is sufficient to verify the property G0 ((} -+ +oo if II(IIRm -+ +oo. Suppose this to be false. Then there exist R E 114 and a sequence (n E SJ. such that ll(niiRm -+ 00,

Go((n} = G((n) -

L

U • (n dx

~ R.

Using (11) this implies C1 {

We set ;pn

11'1/Jnll~t + ll[(n- P('I/JnWII~} -

(U, (n}Y

~ R + C2.

(17)

= '1/Jn/ll(nll, Cn = (n/ll 0, (* = Ina• E Sl., u• 2: c > 0 a. e. on 0. If (III) is fulfilled, too, then there is no other solution to (S).

Corollary 2. Assume (II) and let (u•, '1/J*) be the solution to (S) as in Theorem 2. Then for any solution (u, '1/J) to (P) it holds F{u*)

~ F{u{t))

'Vt E

114,

1oo D{u{t)) dt ~ F(U)- F(u").

R. Hiinlich, A. Glitzky

170

Proof. This follows from (7), (8) with u = u•, tjj = ,p• and from Theorem 1. 7.

0

EXPONENTIAL DECAY OF THE FREE ENERGY

In this section we shall prove that for any solution to the evolution problem (P) (with the initial value U) the free energy F(u) decays exponentially to its equilibrium value F(u•) (where u• belongs to that compatibility class which is generated by U) if the additional assumptions (II) and (III) are fulfilled. This will be a consequence of the following estimate of the free energy by the dissipation functional.

Theorem 3. Let (II) and (III) be satisfied. Then for every R > 0 there exists a constant cR > 0 such that F(u)- F(u") ~ cRD(u) VuE MR

where MR

= {u E MD : F(u)- F(u") ~ R,

k

(u- U)dx E S }·

Proof. 1. For u E MR let 1/J, a be defined by E(,P, u) = 0, a = u/p('I/J). First let us note that there is a c(R) > 0 such that 111/JIIH•• 111/JIIL"" ~ c(R) Vu E MR. Setting F(u) = F(u)- F(u") and using (7), (8) and Theorem 2, we obtain the estimates c1(R){ II~ -111~ + 111/J -1/J"II~~}

~ F(u),

F(u) ~ 0 and sequences c,. e JR.,

0 < c,. D(u,.) ~ F(u,.) ~ R. (20) Let 1/Jn, a,. be correspondingly defined. (18), (20) imply that 111/JniiHI, lly'ii;;lly ~ c(R), D(u,.) ~ 0. Therefore 1/Jn __.. ;j in H 1, 1/Jn ~ ;j in £ 2 • For i E J we find Ci; E 14 with .;a;;i ~ y'(i;" in H 1 and in each lJ'. For i E J' we have at least Bni ~ Ci; E 14 in £ 2 since for such i there is a special reaction for which

rIIT Janj/aj ln jeJ

OJ -

Bn;/a;l

2

dx

~ 0.

Fatou's lemma ensures that a0 = (itf V(a, {J) e 'R.. Setting u = ap(;j) we get u,. ~ u in Y, and thus J0 (u- U) dx E S. The estimate 111/Jn+p -1/JniiHI ~ c IIUn+p- UniiY shows that 1/Jn ~ ;j in H 1• Using properties of E we conclude that E('f/Jn, u) __.. E(;j, u) in (H 1)• ' E(1/Jn. u) ~ 0 in (H1)• Thus E( ;j, u) = E( ;j, a) = 0 is obtained. 0

On Energy Estimates for Electro-Diffusion Equations Arising in . . .

a

Summarizing, we have found that E A. Now assumption (III) ensures that and correspondingly u = u•, = ,p•. From {19) we conclude that F{u,.)-+ 0. 3. We set

;p

Wn

= ../anfa•-1,

An= JF(u,.), bn

= Wn/An,

171

a = a•,

Yn = (un-u•)/An 1 Zn = (1/Jn-1/1•)/An.

In the formula for the lower bound of the dissipation rate

D(u,.)

=

1{L 0

iEJ

2 1Vwn;l +

L

1{1 + Wn)

;

iw(wn)l :5 c

0

2 (1 + Wn)11 i } dx

-

(o,ti)E~

we use the binomial expansion

II(1 + Wn;)

0

i

iEI

= 1 + L a;Wni + w(wn), iEI

=

n

LiEI {Anlbn; l2 + A~o-ilbn;IPo}

max{2, L;er a;}. (18), (20) imply that llz..IIH•.IIbniiY :5 c(R), D(un)/A~ -+ 0. Therefore z,. -" in Hi, z,. -+ in L 2 • Now, fori E J we find bn; E 114 with bn; -+ b, in Hi and in each V, moreover, Anllb!,IILz -+ 0. Fori E J' from 2 IJ(1 +wn;)0 j -1}- 2bni- Anb!,J dx-+ 0 ln n ;eJ where Po

z

z

r1: {

it follows that 2bn; +An~; -+ 2b; E

114 in L 2 , and because of the estimate 2

~

~

llbni- b;IILz :5 c{R) {ll2bni + Anbni- 2b;IILz + AnllbniiiLz) we get bn; -+ b; in L 2 as well as Anllb~;IILz -+ 0. Fatou's lemma yields (a- (1) · b = 0 'v'(a, (1) E 'R, or in other words, bE S.J.. Setting fj = a"p(,p•)(2b- Q(,p•)Z) we get m

llYn- filly :5 c(R){ llbn- blly +liz..- ZliLz +

An(L llb!;IILz + 1) }. i=i

and thus Yn-+ fj in Y. Consequently, 0

and using

J0 fjdx E Sand (b, fj)y = 0.

Finally, we obtain

~ ;~ (E(,p•, u,.)- E(1/Jn, u,.), tPn -,p•) = -In~ Q;(t/J.)ZnYni dx, (b, fj)y = 0, in the limit we arrive at

Therefore fj = 0, and (19) gives the contradiction 1 :5 c2(R)i1Ynll~-+ 0.

D

Corollary 3. Let the assumptions (II), (III) be fulfilled. Then along any solution (u, t/J) to (P) the free energy F(u) decays exponentially to its equilibrium value F(u•), F(u(t)) - F(u•) :5 e-.u (F(U)- F(u•))

'v't ~ 0

172

R. Hiinlich, A. Glitzky

where>. depends only on the data. Moreover, there is a constant c such that

iiu(t)- u•IIL1 (0,R"'l• 111/J(t) -1/J•IIH'(OJ $ c e->.t/2 'tit~ 0 for any solution to (P). Proof. Because of Corollary 1 and Lemma 6 for R = max{1, F(U) - F(u•)} > 0 it holds u(t) E MR 'tit > 0. We set >. = 1/cR and with Theorem 3 and Theorem 1 the first assertion is obtained. In Theorem 2 have we stated that(" E Sl.. Then the estimates follow from (7), (8) by setting there u = u•, ijj = ,p•. D

8.

REMARKS

Remark 1. We consider a special version of the pair diffusion models in [2,4]. The meaning of the species and the used reactions are outlined in Figure 1. Here m = 5, i = 1, ... ,5, or in the notation of Figure 1, i = A,/, V, AI, AV, J = {2, 3,4,5}, J' = {1 }. The functions p;, Q; are given by (2), D;, k 0 p are similar averaged quantities and e(1/J) = c sinh 1/J. These functions have all properties which are required in (I).

tti···~tp

.

+++~·- ~ A~I+AV

AV~A+V

A I

A~V+AI

AI~A+I

..-·········--·-m··--.\ • ·....);m··~-------• ....t 0

, -(



• • •

v

0

AI

@

AV

@

eo

m O~V+I

host atom on lattice site dopant atom on lattice site host atom on interstice vacancy dopantinterstitial pair dopantvacancy pair

AV+AI~2A

Figure 1: Species and reactions in the pair diffusion model (see [2,4]). corresponding reaction rates are given by

= kap(V;) (aAv -

Rap= kap(?/J) (aA - Of CAV),

Rap

Rap= kap(V;) (aA- av aAI),

Rap= kap(1/J) (aAI- CA Cf) 1

The

a A av),

= kap(V;) (aAv OAf- a~) = (1, 0, 0, 0, 0), (3 = (0, 1, 0, 0, 1) and for the last

Rap= kap(V;) (1- av a1),

where e. g. for the first reaction a one a = (0, 0, 0, 1,1), (3 = (2, 0, 0, 0, 0).

Rap

On Energy Estimates for Electro-Diffusion Equations Arising in . . .

173

Next we find that dimS= 3, dimSJ.. = 2, SJ.. =span {(1, 0, 0, 1, 1), (0, 1, -1, 1, -1)} such that there are two invariants the value of which is fixed by the initial state, namely

fo[u.-~.(t) +uAI(t) +u.-~.v(t)]dx =

lt(t) =

L

l2(t) =

[ui(t) - uv(t) + UAJ(t) -

/ 1 (0),

u.-~.v(t)] dx =

/ 2(0)

'Vt E IR+-

Condition (II) means that It (0) > 0, and we easily verify that then assumption (III) is fulfilled, too. Thus our results can be applied to this special model.

Remark 2. In (1) the boundary conditions for the first set of continuity equations can be replaced by the following ones: II·

L

ii =

(a;- /3;) R~/1 on (O,oo)

X

an,

i E J,

(a,/1)E7lr

R~11 (x,u,,P) = k~11 (x,,P)

[g a~'- ga~'], m

m

x E an, u E IRm, t/J E IR, a;=

Pi~~)'

where (a, /3} E 'Rr and 'Rr C {(a, /3) E Z~ x Z~: a; = {3; = 0 Vi E J'} describes a set of additional boundary reactions. We assume that for each (a, /3) E 'Rr the function k~/1 : an X lR --+ lR satisfies the Caratheodory conditions and

c1,R(x) ~ k~p(x, t/J) ~ c2,R(x) f.a.a. x E an, 't/tjJ E IR, lt/JI ~ R, c,,R, c2,R E L~(an), llct,RIIL•(ao)

> 0.

All boundary reactions must be included into the definition of the sets S, A, and in the definition of A, D boundary integrals have to be added. Then all assertions of the paper, especially the assertions of Corollary 1 - Corollary 3, remain valid.

Remark 3. Testing the Poisson equation E(,P, u) = 0 by 1jj = 1 the global electroneutrality condition

1 n

[e(tjJ(t))-

t

Q;(,P(t)) U;(t)-

i=l

f] dx = 0 't/t E 114

(21)

is obtained. If we want to use other boundary conditions for the Poisson equation in (1) then condition (21) must be taken onto account, too. Therefore, as in [7] we consider mixed boundary conditions of the form

t/J =

on (0, oo) x fo,

(o

11·

Vt/J + r,P =

r(0

on (0, oo) x

rN

where the new unknown quantity ( 0 : IR+ --+ lR (the electrochemical potential of electrons, i. e. the Fermi level) has to be determined by means of the nonlocal constraint (21). We assume that fo,

f

D

fN

are disjoint Open subsets of

an, an= fo U fN,

n f N consists of finitely many points,

T E L~(rN),

mesrv + llriiLI(fN)

> 0.

174

R. Hiinlich, A. Glitzky

The definition of E must be changed (see [7]) and F, F" contain an additional boundary

integral. Again, all assertions of Corollary 1 - Corollary 3 remain valid.

Remark 4. As in [6] analogous energy estimates and asymptotic properties can be derived for a discrete-time version of (1) using an implicit scheme of first order. Acknowledgments. The authors are indebted toW. Ropke and N. Strecker for helpful discussion concerning the model equations. The second author is supported by the German Research Foundation (DFG). REFERENCES

[1] H. Brezis, Operateur& mazimaux monotones et semi-group& de contractions dans le1 espace& de Hilbert, North-Holland Math. Studies, no. 5, Amsterdam, 1973. [2] S. T. Dunham, A quantitati11e model for the coupled dif:Ttuion of pho1phorw and point defect~ in silicon, J. Electrochem. Soc. 139 {1992), 2628-2636. [3) I. Ekeland and R. Temam, Con11ex analy&u and 11ariational problems, Studies in Mathematics and its Applications, vol. 1, Amsterdam, 1976. [4) K. Ghaderi and G. Hohler, Simulation of phosphorw diffwion in silicon wing a pair dif!wion model with a reduced number of parameter&, J. Electrochem. Soc. 142 {1995), 1654-1658.

(5) A. Glitzky, K. Groger, and R. Hiinlich, Free energy and du&ipation rate for reaction diffwion processes of electrically charged lpet:ies, Appl. Anal. 60 {1996), 201-217.

(6] A. Glitzky and R. Hiinlich, Energetic e&timates and GBymptotics for (7)

[8] [9) (10)

[11) (12) (13) [14)

electro-m~ction­

dif!wion systems, Z. Angew. Math. Mech. 77 {1997), 823-832. A. Glitzky and R. Hiinlich, Global e&timates and GBymptoticafor electro-m~ction-diifu­ sion systems in heterostructures, Appl. Anal. 66 {1997), 205-226. A. Glitzky and R. Hiinlich, Electro-reaction-dif!wion &y&tems including clwter reactions of higher order, Math. Nachr., to appear. K. Groger, Roundedness and continuity of solutions to linear elliptic boundary 11alue problems in two dimensions, Math. Ann. 298 {1994), 719-728. A. Hofler, De11elopment and application of a model hierarchy for silicon process simulation, Series in Microelectronics, vol. 69, Konstanz, 1997. A. Kufner, 0. John, and S. Fu~fk , Function spaces, Prague, 1977. W. Merz, Analy&u und numerilche Berechnung der Dif:Ttuion 11on Fremdatomen in homogenen Strukturen, Habilitationsschrift, Technische Universitii.t Miinchen, 1998. J. Neeas, Les methodes directes en theorie des equations elliptiques, Prague, 1967. N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 {1967), 473-483.

R. Hiinlich, A. Glitzky, Weierstrass Institute for Applied Analysis and Stochastics, D-10117 Berlin, Germany Email: huenlichGwias-berlin. de, gli tzkyOwias-berlin. de

On the Boundary Conditions at the Contact Interface between Two Porous Media W.

JAGER,

A.

MIKELIC

Abstract. We consider an incompressible creeping flow through a 2D porous medium containing two different types of pores. The two parts are separated by the interface (O,L) x {0}. It is well-known that the effective flow in both parts is described by Darcy's law, with different permeabilities. For the periodic porous medium we prove that at the interface, continuity of the effective pressure and the effective normal velocity hold true. The proof uses the boundary layers for the Stokes operator in a heterogeneous periodic porous medium. Keywords: operator

homogenization, boundary layers, porous medium, interface, Stokes

Classification:

76005, 35827, 76Mxx, 35Qxx

1. INTRODUCTION

Flow through porous media, containing two or several subdomains with different micro-structures, arises in many applications. It is generally accepted that the effective conditions at interfaces, between subdomains, are the pressure continuity and the continuity of the normal components of Darcy's velocity (see e.g. Dagan[2]). However, one should not forget that Darcy's law is obtained after averaging the momentum equation and the orders of differential operators are changed. For an incompressible flow, we conclude immediately the continuity of the normal components of Darcy's velocity, at the interfaces. From the other side, the stress tensor completely changes in the weak limit and there are no a priori reasons to have the pressure continuity at the interfaces. Furthermore, it is known that, for contact between a porous medium and a channel flow, we don't have in general the pressure continuity and some additional terms appear. We refer to Jager and Mikelic [3], Jager and Mikelic [4] and Jager, Mikelic and Neuss[5] for more details. In that situation engineers use the law of Beavers and Joseph (see Beavers and Joseph [1] and, for mathematical justification, Jager and Mikelic [4]). Here the difference between the geometrical structure of the subdomains is not so big and we don't expect that the averaging procedure leads to drastic changes. Finally, there are reasons to believe in good understanding of the problem by engineers. The goal of this paper is to confirm rigorously the pressure continuity and in order to fix the ideas we introduce a model problem. We consider a slow viscous two-dimensional incompressible flow in a domain n•

consisting of the porous media 11 1

= (0, L) x R+

and S12

= (0, L)

x R_, and the

W. Jliger, A. Mikelic

176

interface E = (0, L) x {0} between them. We assume that the structure of the porous media is periodic. 0 1 is generated by translations of a cell £Z, where Z is the standard cell, Z = {0, 1)2 , consisting of an open set z•, az• E 0 00 , being strictly included in Z. Let Z F = Z \ Z• be connected and let Xt be the characteristic function of ZF extended by periodicity to R 2 • We set xl(x) = x 1 (;), x E R 2 , and define S11 by S11 = {xlx E S"lt.xHx) = 1}. is also generated by translations of a cell z•, but this time we suppose that z strictly includes the open set y• I ay· E and that YF = Z\ y•. Let x2 be the characteristic function of YF. We set x;(x) = x2 (;) on R 2

zt =

n2

and

n; =

coo'

{xix E S12,x;(x) = 1}. Then -

n• = S1~ U E U s-12.

and I E C (S1) ' suppf is compact, I is £-periodic in Xt The flow through is described by the Stokes system 00

2

n•

-~u·

+ Vp' = 1

div u• = 0 u• = 0

in

It is supposed

that~ EN



n•

inn·

{u',p'}

is £-periodic in x1 Vu' E L 2 (S1') 4 and Vp' E L 2 (S1') 2 •

f

(1.1) (1.2) (1.3) (1.4) (1.5)

The choice of periodic boundary conditions in x 1 and unboundedness with respect to x 2 eliminates the effects of outer boundaries, which are of no importance for justifying the pressure continuity. Before stating our results, we briefly discuss problem (1.1)-(1.5). We introduce the functional space W, by

W,

= {z E H 1(S1')2 : z = 0 on

div z

= 0 a.e.

in

n•,

an· \ a(n. u S12) and z is H 1-periodic in Xt}.

(1.6)

Then, using Poincare's inequality, we easily get Lemma 1.1. Problem {1.1}-{1.5} has a unique solution u• E W,. Furthermore, there exists p• E L1oc(S1f), 'Vp' E L 2 (S1") 2 such that {1.1} holds in the sense of distributions. Finally, {u•,p•} E C 00 (S1f) 2 x C00 (S1"). Section 2 contains the detailed study of the homogenized problem. In addition we give all auxiliary problems used in this paper. Their solvability and properties of solutions were established in Jager and Mikelic [3]. Our main results are stated in Section 3. Proofs are given in Sections 4 and 5. 2.

STUDY OF THE HOMOGENIZED PROBLEM AND AUXILIARY RESULTS

We start with 2 classes of auxiliary problems defining the 2 permeabilities: We are looking for {wi, 11';}, j = 1, 2, satisfying -~11 w' + V 1 11';

{

= e; in ZF;

= 0 in ZF, 111'i(y)dy = 0, wi = 0 on az·' {wi~;.;} is !-periodic. div11w'

(2.1)

On the Boundary Conditions at the Contact Interface between Two Porous Media 177

Problem (2.1) admits a unique solution {wi,1ri} E C 00 {Zp)3 and the matrix

Kij

= lzF f w;(y)dy,

is symmetric and positive definite with Ki'(, K~ Analogously, we consider the problem: Find {v',w'} E H 1 (YF) 2 X L 2 (YF) such that

{

= 1, 2

i,j

> 0.

-A11v' + V ,w' = e; in Yp; div11v' = 0 in Yp, ~(y)dy v'

= 0 on ay·,

(2.2)

1

.Yr.

= 0;

(2.3)

{v',w'} is !-periodic.

The corresponding matrix is K 11 , given by

Kij

= }y, f v}(y)dy,

i,j

= 1,2.

(2.4)

>0 0. Then. p0 E X ( -oo, 0]) and p 0 (xl, -0) = p 0 (xl, +0) on E.

[0, +oo)) n C 00 ([0, L]

W. Jliger, A. Mikelic

178

Proof: Let us fix x 1 E (0, L). It will play the role of a dummy variable. We write (2.6) in the form 0

0

dx2

OXt

oh)

d ( K'l2-+2K2t-+(K/}2+Kt2dp Op -= dx2

OX)

lJ2p0

oft

UXt

tiXt

Ku ~ + Ku ~ E C(O) dpo

dpo

/!!

((.K" - K"')/) 2 (Xt. 0) + (K2i - K21 p0 {x1 , -0) = p0 (xt. +0)

{2.7)

=

K~dx (Xt, -0)- Iq2dx (xt,+O) 2

2

in flt U 02

0

VX)

(Xt. 0)

on E

on E.

(2.8) (2.9)

After simple change of coordinates x 2 -+ ; : we write {2.7)-{2.9) as

-Z. {

= g; E L7oc(O;) in 0;, i = 1, 2, Vq

< +oo {2.10)

[ ::] E

0

= h E C(E), [p

]E

= 0.

At this stage, we redefine p 0 by setting 11'0

Then

~~

0

E

=

p0

+ X2e-s2 h{xt)

{ p0

L7oc(O), Vq < +oo and 1r0

W,~(R+uR_)) and

::IE

tn

nl

in 02. E

W1~(0).

Consequently, p0 E

{2.11)

c;:,{[o, L];

E wt-l/M(E).

Now we differentiate the equation {2.7) with respect to x 2 and arguing as above obtain

c;;o E c:,_{(O, L]; W1~(R+ U R_)).

uX2

Repeating the argument, we obtain the regularity from the statement of the proposition. 0

Corollary 2.3. For every o E N 2 there exists cSo(o) > 0 such that !l)Gp0 (x)l ~ Ce-6o(o)ls2 1 for X2

> x•(o).

Next we need auxiliary problems correcting the compressibility effects caused by corrections. They are exactly the same as in Jli.ger and Mikelic [3]:

On the Boundary Conditions at the Contact Interface between Two Porous Media 179 We are looking for -yiJ, i, j = 1, 2, satisfying

1 Kw . Z • iJ - . _; d IV"/' - 'Wi - IZFI ij m F {

'YiJ

= 0 on oz•;

{2.12)

'YiJ is 1-periodic.

For existence of a function -yiJ E C (ZF)2 satisfying {2.12) we refer to Jager and Mikelic [3]. The corresponding problem in YF reads 00

. no] _ _ _; 1 vv · v dlV"o -Vi - IYF(•ij m ~ F {

rJ = 0 on oY•; rJ is !-periodic.

{2.13)

The final class of auxiliary problems are ones corresponding to the boundary layers due to the difference between wi and vi on S = {0, 1) x {0}.

z- = u {YF- {0, k)}' z+ = u {ZF + {0, k)}' s = {0, 1) X {0} and t=l t=O = z- us u z+. Let [a]s(·, 0) = a(·, +0) -a(·, -0). We consider the following +oo

+oo

Let

ZBL problem for j = 1, 2

v 111r;,N = o in z+ u zdiv11wi·N = 0 in z+ U z[wi,bi]S =vi(-. -0)- wi(·, +0) + ez -!111wi·bl +

(2.14)

t (~(Yb

{2.15)

+0)- v4(Yll -0)) dy1 on S {2.16)

[(V11wi•bl - 1rj.NI)e2) (·, 0) = (Vvi - wil)(y11 -O)e2 - (Vwi - 1ril)(y11 +0)e2 on S 5 (2.17)

UJi·bl = 0 on oZBL \ ({{0} u {1}) {wi•bl, 1r;,w} is 1-periodic.

X

R)

{2.18)

{2.19) {1 . Remark: Since K;j = Jo ~(Yb Y2)dy11 'Vy2 E {0, 1) and /q; = Jo ~(Yt. Y2)dyz, 'Vy2 E {0, 1), the condition (2.16) can be written as

{1

[wi.bl]s =Vi(-. -0)- w'(-. +0) + (K;j- Jq;)e2 on S.

{2.20) D

Now arguing as in the proof of Theorem 3.15 from Jll.ger and Mikelic [3] we obtain the following result.

Proposition 2.4. The problem {2.14)-{2. 19) has a unique solution wi•bl E W = {z E L 2 (ZBL) : z E H 1 (Z+), z E H 1 {Z-), z = 0 on oZBL \ {{0} U {1}) x R,div11z = 0 on z+ U z- and z is H1 -periodic in y 1 }. Moreover, wi•w E Ct:,(z+ U z-) 2 and there exists 'Yo > 0 such that

IV"wi,bl(y)l ~ ce-111 1~1 and lwi•61 (y)l ~ ce-111 1~1.

{2.21)

W. Jliger, A. Mikelic

180

Finally, there exists 1r;,lll E Ct:,(z+ U z-) such that V 111r;,lll E L 2 (Z+ U z-), {2.14} holds and there are constants 'Yo > 0, C!,.,, Ct such that

IV~~~· At S

111

+ ITrj,bl(y)- H('Y2)C!- H( -'Y2)C~I

(y)J

= (0, 1) x {0}, wi•lll E

W 2•9(S)2

and 1r;,w E

W 1"(S),

~

ce-1011121

'Vq E (1, +oo).

Next we need the boundary layers corresponding to the functions

-!111-yi·',bl +

(2.22)

v 111r;,a,bl = o in z+ u z-

-yi•' and Oi•': (2.23)

div,-yi·'·bl = 1{.J•' { ell2 H( -'Y2) + e-112 H(Y2) } in z+ u zfz- e112 dy2dY1 fz+ e-112dy ['yj,&,bl]. = -"f•'(Y1t +0) + Di·'(Y1t -0) on S

[(V11-yi·',bl - 7r;,i,bll)e2Js

=0

on S

(2.27) (2.28}

{-yi·'·bl, 1r;,a,lll} is !-periodic in Yt

l (6~·'(Y1t

(2.25) (2.26)

-yi,i,bl = 0 on 8Z8 L \ ({0} U {1}) x R where 11.;,; =

(2.24)

-0}- ~·'(Y1t +O})dYt·

Analogously to Proposition 2.3 we obtain existence of the unique solution -yi·',bl which exponentially tends to zero for large I'Y21· For 7rj,i,bl we obtain an analogue to

(2.22). 3.

STATEMENT OF THE MAIN RESULTS

In this section we give two results. The first result is obtained under the hypothesis:

{

E Ctf(0) 2 , f is £-periodic in Xt and f = 0 on [0, L] x (-6, 6) for some 6 > 0

f

(3 .l)

and the second under the much weaker assumption:

f

E Ctf(0) 2 ,

f

(3.2)

is £-periodic in x 1

Theorem 3.1 Let us suppose {3.1}. Then we have

u; - t II f

k=l

(H(x2 )wk(~) + H( -x2 }vk(~ )) (!t(x)- ?

0

f

f

VXk

(x)) II

£2(0)2

~ C.;i.

(3.3}

and (3.4) weakly in £ 2 (0) 2 • Furthermore, there exists an extension fl of the pressure such that

1

r.,.. _

II 1 + Jx2J"'

po)ll

< C.;i..

I£2(0) -

The next result corresponds to the assumption (3.2}:

(3.5)

On the Boundary Conditions at the Contact Interface between Two Porous Media 181

Theorem 3.2 Let us suppose {3.2). Then we have

u; -

II f.

t

(H(x2)wk(::.)

k=l

f.

+ H( -x2)vA:(:: >)

(tt(x) -

f.

c;t (x)) II

cJXt

~ Ct 1

1 8

L2(Q)2

{3.6)

and there exists an extension fl of the pressure such that

Ill +llx21 {P'- po)t'(n)

~ Ctl/B.

{3.7)

Finally, the convergence {3.,l) holds true.

Corollary 3.3

(3.8)

weakly in L2 (E). 4. PROOF OF THEOREM 3.1

In the proof which follows we will frequently use the space V,_(O')

= {z E H 1{0'? : z = 0 on an·\ 00 and z is £-periodic in Xt}•

(4.1}

Then we start with the approximation corresponding to Darcy's law. We define u 0·• and p0 •' by

L2

k=l

t t

k=l

k=l

8p0 ) wk•'(x) ( !t(x)- -;:;-(x) ' XE VXk

?t (x)) ,

X E 02

(~t(x) - c;;o (x)) ,

X E 01

vk•'(x) (ft(x)11'k''(x)

VXk

VXt

t wk•'(x) (ft(x) - VXk c;;o (x)) ,

k=l

where

wk·•(x) { 11'k'•(x)

nl

= wk(;); = 11't(;)

vk·•(x)

(4.2}

{4.3} X E 02

= vk(;)

and wk·•(x)

= wk(!-).

{4.4)

Then we set

{4.5)

W. Jager, A. Mikelic

182

Our first estimate is given by the following proposition Proposition 4.1. Lett/> E V,..,.(O•). Then we have

{ (VU~Vt/>- 0 Ilo;un;

P div t/>)1 :5

C { IIV{f- Vp0 )11£2(11)4

+ tll~(f- Vp0 )11£2(11)2} IIVt/>11£2(0(!;- {)po)l ax, · i=l

(4.6)

Proof: We start with the weak formulation corresponding to (1.1}-(1.5}:

f

ln;u~

Vu; Vt/>- {

t-2p• div t/> = { f.- 2 /t/>,

'v't/> E V,..,.(n•).

(4.7}

lo;uo; inThe introduction of U0 and P0 corresponds to the elimination of the forcing term. t

We have:

f VU~Vt/>- f PO lo;un; ln;u~

ht {Bj +(vwi·•-

div t/> =

t- 11ri.• -

E j=l

Vvi·•

+t-lwi.ti) (1;- {)po)} e2t/> +t 1 t/> t A~ ax, 0: i=l

j=l

(4.8}

where

.

. (! ·-{)po) . 1. ) - + (2Vw'••--1rul

A'=w'··~



'

ax;

l

·V ( !-'

-{)po) ax;

in01

(4.9)

and

(4.10} Following Jager and Mikelic [3] we obtain

It 1 t/> t A~~ :5 CIIVt/>IIL2(n: = E-2p~- E-2p0- E-lpO,c-

(/;-

j=l

apo) .

(4.16}

ax,

Then we have

, (vu:v¢J _ P: div ¢J)I :5 IJlli~

C {IIV(J- Vp0 )IIL2(o>• + v'fii~(J- Vp0 )11£2(0l2} ·IIV¢JII£2(o>•·

(4.17)

Proof: The variational equation for {Uf, J>:} reads

{

Jlli~

(VUiV¢J-

P; div ¢J) =

kf;(K2;- Iq;)e2 2

(

® V /;-

where

A;·111 •

= (2vwi·111· · - !11';,!11,•1) v f

apo) e ¢J + ~ /0: ¢J ]; 0 is considered as a parameter in the above system. In many situations the displacement current E8tE• is small in comparison with the charge currents and is often neglected. Therefore, it is the aim of this paper to

F. Jochmann

188

investigate the singular limit e --+ 0. By letting e --+ 0 one obtains formally from 1.1-1.4 the quasi-stationary Maxwell system curl h

= S(t, x, E)+ jo

(1.6)

curl E = -iWth

(1.7}

divE= 0 on [O,oo} x no

(1.8}

supplemented by the initial-boundary conditions

nA E = 0 on [0, 00) X 8n and h(O) =

(1.9}

ho on n.

(1.10}

This problem has been investigated in [8] in the case that the domain n is bounded and S is linear with respect to E and uniformly positive. See also [3] and [4], where a temperature-dependent electrical conductivity is considered. However, in this paper n is an exterior domain and the electrical conductivity vanishes on a subset no = n \G. Note that (E, h) is not uniquely determined by 1.6-1.10, since (E + F, h) also solves 1.6-1.10 provided that curl F = 0 on n, iiAF = 0 on an, F = 0 on G and div F = 0 on In order to also determine the electric field E uniquely, the following boundary condition is imposed.

no.

( iiE(t, x)dS = Q~c for all t E [0, oo}, k E {1, .. , N}. )8C•

(1.11}

\no.

Here ck ,k = 1I • • , N denote the connected components of c ~ JR.3 the complement of the "vacuum-region" The physical meaning of 1.11 is that the total electric charge Qk on each C~c is prescribed. (In the case that C is not connected it is not sufficient to prescribe only fBC iiEdS in order to achieve uniqueness.) The initial data must obey the compatibility conditions

no.

div Eo

= 0,

curl

ho = jo(O} on no

=n \ G

(1.12}

and ( iiE0 (x)dS = Q~c for all k E {1, .. , N}. (1.13} )8C• Note that 1.6-1.11 can be considered as a problem of mixed type. It is elliptic on (with respect to the space variable x for fixed time) and parabolic on (0, oo) x G. It is shown in this paper that 1.6-1.11 admits for given ho, j 0 and Q~c a unique global weak solution (E, h). Moreover, it is shown that (E£, ~) ~ (E, h) in a suitable weak topology. Finally, the asymptotic behavior of the solution to 1.6-1.11 fort--+ oo is investigated in the case j 0 = 0. It is shown that llh(t)ll£2 decays exponentially provided that

no

k

J.Lhogdx = 0 for all g E L2 (n) with curl g = 0.

This condition includes div (J.Lho) = 0 on n and iih = 0 on weakly. The aim of this article is to summarize the results in [6] without presenting detailed proofs.

an

On the Semistatic Limit for Maxwell's Equations

2.

189

NOTATION, DEFINITIONS AND ASSUMPTIONS

For an arbitrary open set K C /R3 the space of all infinitely differentiable functions with compact support contained in K is denoted by C(f(K). Moreover, 'D'(K) is the space of distributions on K. For p E [1, oo) we define ~r1 (K) as the space of all E E V'(K) with curl E E V'(K). 0 HP curl (K) denotes the set of all E E H:.0r1(K), such that

he E curl h- h curl Edx = 0 for all hE ~r1 (K),

which includes a weak formulation of the boundary condition ii /\ E = 0 on oK. Here ~ = 1. W 1.P(K) is the usual first-order Sobolev space consisting of all functions in

+.;.

0

V'(K) whose distributional gradient belongs to V'(K). W 1.P(K) denotes the closure of C(f(K) in W 1.P(K). In the sequel n c /R3 is an open set with bounded complement, G c nand no~ n\G. It is assumed that C ~ /R3 \no = Ct U .. U CN is a bounded Lipschitz-domain, where C~c are its connected components with Ci n C~c = 0 if i =f k. p. E L""(JR3 ) is an uniformly positive function. Let ~ > 0, such that C C BRo ~ {lxl < ~}. Next, let PoE (2,6), such that for all p E (2,Po) the following holds: 0

tp tl.tp

E

EW1•2 (no n B2Ro) and

w-t,p(no n B2Ro) ~ ( W1,p' (non B2Ro)) • ===> "Vtp

E V'(flo

(2.14)

n B2Ro)·

Since no n B 2Ro = B 2Ro \ C is a Lipschitz-domain, it follows from the result in (2) that there exist such Po > 2. The following assumptions are imposed on s : [0, 00) X n X JR3 -+ JR3 • S{t,x,y)

=0 if

X

E

no= n \ G,

{2.15)

S(·,·,y) E Ll:([O,oo),L""(n)) for all fixed y E JRl.

(2.16)

Next, S is assumed to be Lipschitz-continuous, i.e. there exists L E (0, oo), such that IS{t,x,y)- S{t,x,y)l ~ Lly- :YI for all t ~ O,y,y E

ma and

X

En. (2.17)

Moreover, IS(t,x,y)l ~ Co-r(x)IYI and yS(t,x,y) ~ -y(x)IYI 2

(2.18)

for ally E /R3 , t ~ 0 and x E G with some 'Y E L""(G), 'Y > 0 and Co E {0, oo). Finally, S is assumed to be monotone, i.e. (y- y)(S{t,x,y)- S{t,x,y)) > 0 for ally

=f y

E

JRl,t ~ 0 and x E G. (2.19)

F.Jochmann

190

The following compatibility conditions will be imposed on Po and 'Y· -y- 1 E Lro/(2 -rol(G) with ro

dJf Po·

{2.20)

All these assumptions are fulfilled in the frequently occurring linear case S(t,x,y) = u(t,x)y for ally E lff, t;::: 0 and x E G

{2.21)

with some positive u E Ll:,([O, oo), L (G)) satisfying u(t, x) ;::: -y(x) with 'Y as above. Let g and g• be the weighted £ 2-space consisting of all functions f : G -+ fL8 and F : G -+ afl with 00

llfll~ ~

k

lfl 2 -rdx < oo and

IIFII~· ~

k

2

1

IFI -y- dx < oo, respectively.

It follows from 2.18 and Holder's inequality that

Q c Lr0 (G)

(2.22)

S(t,x,f) E g• C L2 (G)

(2.23)

and for f E Q and t ;::: 0 one has with IIS(t,x,f)llv• = independent of t, f.

IIS(t,x,f)'Y- 1/ 2 11£2(G)

$

Kllfllv with some K E (O,oo}

Next, some function spaces are introduced. Let ~ IP E L6 1V~P E L 2 (JR3 )}. It is a Hilbert-space endowed with the scalar

z

{

def

-

product < !p, 1/J > z = fRs cV!pVt/Jdx. Let Z 0 be the closed subspace consisting of all1p E Z with V1p = 0 on C, i.e. which are constant on each component C,. For E E LJ.,,:('ri) with div E = 0 on no define q;(E)

~

fo EVx;dx =/no EVx;dx,

(2.24}

where Xi E Cg"(Jl3) obey x;lc~ = t5;,t.· Due to the condition divE= 0 this defintion is independent of the choice of the functions X;. Now, the following function spaces are defined. Let for q E [1, 2] Y09 be the space of all measurable functions E: n-+ fL8, such that E E L9 (n n BRo} and E E L 6 (Jl3 \ BRo)· Recall that G c BRo and .IR3 \ n c BRo. Y19 is defined as the space of all E E Yo9 with the property that 0

div E = 0 on no, curl E E L2 (n) and 1pE eHvcurl (n) for all1p E Cg"(Jl3). The latter condition means 1\ E = 0 on 00. The following norms are introduced:

n

IIEIIv: ~ IIEIIL•(R3 \BAol + IIEIIL•(nnBRoh IEI11 ~II curl Ell£2(0) + IIEIIL•(G) +

N

L

k=l

and IIEIIY:'I ~ IEiy;tI + IIEIIY.!· D

lqt.(E)I

On the Semistatic Limit for Maxwell's Equations

Next, let X be the space of all h E .H!r~(O) with curl h E g• and curl h Moreover, let X be the space of all E E Y}0 n g, such that

k

h curl Edx =

k

E curl h for all h E

X.

191

= 0 on no. (2.25)

Let w be the space of all T E 1J ((0, oo) X n) n L1oc((O, oo), Q), such that T = 8tA with some A E Ll:,((O, oo), X) n Wt!:([o, oo), Q). For A E Ll:,((O, oo), X) n W,~([O, oo), 9) and T ~ 8tA E W we define q~c(T) E 1J1 ((0, oo)) by q~c(T) ~ 1tq~c (A(t)). Next, the notion of weak solutions to 1.6- 1.11 is given precisely. 1

Definition 1. Let j 0 E L1oc([O, oo), L2 (0)), ho E L2 (0) and Q~c E JR. Then (E,h) E W x C((O,oo),L2 (0)) is called a solution to 1.6- 1.11, if curl h

= S(t,x,E) +jo,

q~c(E)

curl E

= -iWth,

(2.26)

= Q~c for all k E {1, .., N},

(2.27)

h(O) = ho.

(2.28)

and

Note that by the definition of Wit follows Elco,ao)xG) E L1oc((O, oo), Q). Therefore S(E) E L1oc([O, oo), L2 (0) n g•) .

3. A

BASIC ESTIMATE

The following estimate will be used frequently.

Theorem 1. For r E (ro, 2), i.e. r• ~Po, there exists a constant Kr E JR+, such that for all E E Y{ the estimate IIEIIy; = IIEIIL•(Ili\BJto) + IIEIILP(flnB~~o)

~ KriEIY{ = Kr (11 curl Ell£2(0) + IIEIILP(G) + ~ lq~c(E) I) holds. In particular I · IY{ and II · IIY{ are equivalent norms on Y{. By 2.22 and the previous theorem the space X is complete with respect to the norm IIEIIx ~II curl EIIL2(0J + IIEiallc; +

N

L

lq~c(E)I forE EX.

lc=l

The estimate IIEIIL•({Izi>Ro}) + IIEIILPo(nnB~~ol ~ KIIEIIx for all E EX

(3.29)

holds. The proof of this estimate involves the following extension theorem for the spaces

H!,r,(O):

F. Jochmann

192

Lemma 1. Let 0 1 c JR3 be a bounded Lipschitz-domain. Then there exists a bounded operator T: L 1(0 1, , b< 1l) and (E(t))- q(E(t)) = 0.

(4.36)

Since A E Ll:,(lR, Yi"0 ), it follows from 4.34 - 4.36 and lemma 1 that A is constant with respect to t and hence also E = 81A = 0. o Now, Maxwell's equations including the displacement current

co EE = 1

curl hE- j 0

-

S(t, x, EE),

(4.37)

(4.38) are considered, where c boundary conditions

> 0 is a

parameter. 4.37, 4.38 is supplemented by the initial-

= 0 on (O,oo) X 00,

(4.39)

= E 0 (x),ht(O,x) = ho(x).

(4.40)

iii\Et EE(O,x)

For the weak formulation of 4.37 - 4.40 the operator B defined by B(E, h) ~ ( curl h, -J.'-l curl E)

F. Jochmann

194 0

for (E, b) E D(B) ~lfl curl (0) x ll!r1(0) is introduced. It turns out that B is a densely defined skew self-adjoint operator in the Hilbert-space X 0 ~ £ 2 (0, (Lii) endowed with the scalar product < (E, h), (F, g) > Xo~ J0 (EF + ~-&hl)dx . Setting w~ ~ (t 1 13 E~, b~- g 0 ) 4.37-4.40 reads as 8tw~ = t- 113 Bw~

+ t-l/:Z~(wc)(t) + fo

and w~(O) = w~.o ~

(t1/ 2

Eo, ho -

(4.41)

go(O)),

where f0 ~ -(0, 8tgo). Here~: L1oe([O,oo),Xo)-+ Lloc([O,oo),Xo) is defined by (~(E,h))(t,x) ~- (S(t,x,E(t,x)),O) for (E,h) E L~oc([O,oo),Xo).

The unique weak solution w~ E C([O, oo), X 0 ) to 4.41 is given by the variation of constant formula wc(t) = exp (t-112 tB)wc,o (4.42)

+ fo' exp (t-113 (t- s)B)[t- 1/2~(wc)(s) + fo(s)]ds, where exp (tB), t E JR. is the unitary group generated by B. Since R. is Lipschitzcontinuous as an operator from £ 2 ((0, T), £ 2 (0)) to £ 1 ((0, T), £ 2 (0)), it follows from a standard result that this integral equation has a unique solution w~ = (e 112 Ec, b~) E C([O, oo), X 0 ); see [9], chapter 6. In particular the energy balance 1d 2 dt llw~(t)ll~o =< t- 112~(w~)(t) + fo(t), Wc(t) > Xo (4.43} =

-fa E~(t)S(t, x, E~(t))dx + k~-&(go(t)

-

~(t))8tgo(t)dx

holds, since exp (tB) is unitary in Xo. The aim of the following considerations is the investigation of the limit e -+ 0. Let A~ e C(JR., £ 2 (0)) be defined by def

A~(t) =

lor E~(s)ds fort~ 0.

(4.44}

By using 4.43 and theorem 1 the following lemma is obtained.

Lemma 4. The family (w~ = (e112Ec, b~- g0 ))~>0 is bounded in L00 ((0, T), Xo), (E~)oo = (8tA~)~>D is bounded in L 2 ((0, T), 9)

and (A~)oo is bounded in £ 00 ((0, T), X) C £ 00 ((0, T), Y{0 for every T

(4.45) (4.46)

n 9),

(4.47)

> 0. Moreover, div E~(t) = 0 on

Oo,

qt(E~(t)) =Qt.

(4.48)

On the Semistatic Limit for Maxwell's Equations

195 (4.49)

By the previous lemma 4 there exist A E L'foc(JR, X)nw,:; ([0, oo ), 9) and a sequence E:n, n E IN with limn-+oa E:n = 0, such that A ... ~ A in Loa((O,T),L6 ({jxj > Ro})) weak - •,

(4.50)

Loa((O, T), Lr (0o n BRo)) weak -*and in weakly 2 with curl A ... ~ curl A in Loa((O, T), £ (0)) weak - •, Since S(·, EcJneiV is bounded in £ 2 ((0, T), g•) by 2.18 and 4.46, it can also be assumed W 1•2 ((0, T),Q)

0

that

S(·, E ... )~ J in £ 2 ((0, T), g•) weakly. From 4.44, 4.48 and 4.50 it follows that

(4.51)

qt(A(t)) = tQk for all k E {1, .. , N} and

(4.52)

Ee,.l(o,oa)xG

= 8cAe,.l(o,oa)xG ~ e ~ 8cAico,oa)xG

2

in £ ((0, T), 9) weakly, in particular A(t)lc Moreover, it follows from 4.49 and 4.50 that

h...

=

(4.53)

= f~ e(s)ds E Q.

ho- JJ- 1 curl A ... ~ h ~ ho- JJ- 1 curl A

(4.54)

2

in Loa((O,T),£ (0)) weak -•. Since lle:nEc,.IIL.. ((O,T),£'(0)) ~ 0 by 4.45, it follows easily from 4.37 that curl h = J +joE L~oe([O, oo), L2 (0)), in particular h(t) - g 0 (t) E

(4.55)

X with

curl (h- go) = J E L~oe([O, oo), g•). Next, let E ~ 8cA E W C 1Y((O, oo) x 0). It follows from 4.52 and 4.54 that 8c(JJh) = - curl E and tic(E) = Q,..

Theorem 2. (E, h) E W x C([O, oo), L2 (0)) is the unique solution of problem 2.262.28 in the sense of definition 1. Moreover,

A. ~A in Loa((O, T), J1"0 ) weak and in

W 1•2 ( (0, T), 9}

*

(4.56)

weakly, in particular

E. ~ E = 8cA in V'((O, oo) x 0) and in £ 2 ((0, T), 9) weakly (4.57)

and h. ~ h =

ho- JJ- 1 curl A in Loa((o, T), £ 2 (0)) weak - *· (4.58)

Finally,

S(-,E.) ~ S(·,E) in £ 1.2((0,T),g•) weakly

For the detailed proof see [6].

(4.59)

196

F. Jochmann

5. ASYMPTOTIC BEHAVIOR: DECAY OF THE MAGNETIC FIELD

In this section the asymptotic behavior for t -+ oo of solutions to 2.26-2.28 in the case (5.60) is investigated. By the assumptions on S there exists a constant K 3 E {0, oo), such that

IIS(t, ·, E)lli2(G)

~ K3

k

S(t, x, E(t, x)) · E(t, x)dx

(5.61)

for all E E L~oc([O,oo),Q). In the sequel let Hcurl,o(O) be the space of all hE H~r1 (0) with curl h = 0 on n. It is a closed subspace of £ 2 (0). Its orthogonal complement with respect to the scalar product < h, g >,.~In J.lhgdx is denoted by X,.. Let P be the orthogonal projection on X,. C £ 2 {0) with respect to < ·, · >,. . Since V'l/1 E Hcurl,o(O) for all '1/J E H 1 (0), it follows that each hE X,. obeys div (J.lh)

= 0 on n and rih = 0 on an weakly

(5.62)

in the sense that- fnJ.lh'V'I/ldx = 0 for all cp E H 1 (0). Next, let (E, h) E W x C([O, oo), L2 {n}) be the solution to 2.26-2.28. In order to apply a compactness theorem, it is assumed that JR3 \ 0 is also a Lipschitz-domain. The solution (E, h) satisfies the energy balance

~dd lly'jm(t)/li2 =- f S(t,x,E(t,x))E(t,x)dx. 2 t la

{5.63)

{This is obtained from a regularization with respect to the time variable as in the proof of uniqueness.) Now it follows from 5.61 and 5.63 that there exists some 6 > 0 (depending only on S), such that

d 1 2 dt 21/y'jm(t)ll£2 ~

2 -611 curl h(t)ll£2(n)·

(5.64)

The aim of the following considerations is to estimate llhll£2 by II curl hilL• for h E X,..

Lemma 5. Let h E L2 {F) with curl h E L6 15 (F) and div (J.lh) E L 615 (F). Then

lihl/£2(.ftS) ~ K(/1 curl hi/L•I•(RS) + II div {J.lh)IIL•I•(Rs)) with some K E {0, oo) independent of h.

A consequence of the result in [5] is

Lemma 6. Let q E (6/5, 2] and (hn)neN be a sequence in X,., such that ( curl hn)neN is bounded in £9(0), supp ( curl hn) C BR for some R E (Ro, oo) independent of n. Then (hn)neN is precompact in L 2 (0). With the aid of the previous lemmata the following estimate can be proved.

On the Semistatic Limit for Maxwell's Equations

197

Theorem 3. Let R E (0, oo). Then there exists a constant K 9 ,n E (0, oo), such that for all h E X~a n H~r1 (0) with curl h(x) = 0 if lxl > R the estimate

llhll£2(0) ~ K9 ,nll curl hiiL•(nnBa} holds. Now the asymptotic behavior of the solution (E, h) is investigated. Let h{ll(t) ~ Ph(t) E X~a.

Theorem 4. There exists some C, d > 0, such that

llh(ll(t)ll£2(0) ~ Cexp(-dt). Proof: Suppose that E = 8tA with some A E Ll::([O, oo), X). Then J.Lh + curl A E Ll::([O, oo), £ 2 (0)) satisfies 8t [J.Lh + curl A]= 0 and hence

h(t) + p.- 1A(t) = h 2 for all t E (0, oo)

(5.65)

2

with some h 2 E £ (0). Since Hcurl,o(O) C X, it follows from 2.25 that p.- 1 curl A(t) E X~a. Hence 5.65 yields

h(t) = h< 1>(t) + (1- P) (h2 - p.- 1A(t)) = h 0 if T $ Tc. (Jv is Jacobian of ii}. In our model (1} the transport part of the equation is 8tb(u) + divF(t, x, u) = 0 and its (corresponding) velocity field is v = F~·'c:t> since formally we have

V _ a,u+ F~(t,x,u) b'(u) . u-

divsF(t, x, u)

b'(u)

Even if the solution of (1} would be smooth we cannot guarantee IV F~·'c:f'> loo < c < 00. In Section 1 we present our approximation scheme and discuss the convergence of the method. In Section 2 we discuss the approximation scheme and the convergence for Navier-Stokes system. In Section 3 we present some numerical experiments. 1. APPROXIMATION SCHEME

Let us denote wh * z the convolution of the mollifier Wh with z where and for lxl > 1 and

2

) w1(x) = xexp ( lxllxl 2 _ 1 ,

forlxl $ 1,

w1(x)

=0

J w1(x)dx = 1. Denote by

i t.p.,(x) := x- TWh •

(F~(ti,X,Ui-1}) v(x) ,0 < v

(n}

E L 00 u ,

h=r"',O= II In(b(u) - b(Uo))8tV Loo(QT), v(T) = 0.

'r/v E L2(J, V)

n Loo(QT), 8tV

E

The existence of a variational solution u is proved in [AL]; see also [K 1 ]. The uniqueness is proved by F. Otto in [OJ under some additional restrictions on .F, f, g. Also ui E V in (4) is considered as a variational solution of (4). 'r/v E V.

(10)

If Ui-1 o rp~, E L2(!l) then the existence of ui in (10) is given by the Lax-Milgram theorem. The proposed method is based on the following lemma.

Lemma 11. If 1J.i = Gi(v) for any 0 < v E L 00 (!l) and w 0 < c1 < det(Jrp~,) < c2 forT$ To> 0 uniformly fori= 1, ... , n.

+

d

< 1 then

For the proof see [K3]. As a consequence rp~, (x) possess inverse and both rp~ and ( rp~.) -l are Lipschitz continuous and

204

J. Kocur

(12) Then if u, E L2(0), then also Ui-1 o tp1 E L2(0). Under some more special structure of data, e.g., i) J,g,divzF(t,x,s)

=0;

ii} g:: O,Ai ~ q > 0 (for some q E R),L E L 00 (QT) in H3); iii} f(t, x, s)s ~ 0, g(t, x, s)s ~ 0, H(t, x, s).s ~ 0

it can be proven (see [K3])

i = l, ... ,n.

(13}

In this section we shall assume (13) without additional restriction on data. The convergence of iterations (8)1 can be guaranteed in the following theorem (see

[K3]). Theorem 14. Suppose {13} and IIVUillo ~ c Vn, i = 1, ... , n. If p + d + /fw < 1 then the iterations (8)1 are converyent, provided T ~ To (To > 0 being sufficiently small). Remark 15. The regularization bn(s) of b(s) in H 1 (see i-v) is not so much restrictive with respect to asymptotic behavior of b'n, b~. For example, let us consider b(s) = jsjP sgn s,p E (0, 1). We can take for bn(s)

for s1

~

for - s1

s

~

0

d. Let /Ji in {8} be of the form /Ji = Gi(/Ji-t,t0 ) (see {8),). If w + d < 1, 'Y + 2d + p < w then uft ~ u in L.(QT) Vs > 1 and uft ...... u in L2(I, V), where un is from (4}-(6}, {22} and u is a variational solution of (1). If the variational solution u is unique then the original sequence {un} is converyent. For the proof see [K3 ]. A stronger convergence result can be obtained under the assumption

Application of Relaxation Schemes and Method of Characteristics to Degenerate . . .

0

< e :5 b'(s) :5 M < oo a .e. in R (d = 'Y = 0 in Hl) and assumptions (5), (6) are satisfied with JI.JI 00 instead of

205

(17)

II.Jio.

Theorem 18. Let the assumptions of Theorem 16 and the assumptions (17} be satisfied. Then uft -+ u in L2(I, V) where u is a variational solution of {1}-{9} and {un} are from {10}. Similar convergence results can be obtained when elliptic problem (10) is projected to a finite dimensional space V~ (e.g., by FEM) provided V~ -+ V for .>. -+ 0 in canonical sense. Then in the place of un we obtain Ua with o = (T, .>.), o-+ 0 (see

[K3]).

2.

CONVERGENCE OF THE METHOD FOR NAVIER-STOKES PROBLEM

Let us consider the Navier-Stokes problem

Otu+ (uV).u- -y~u = -Vp+f(t,x, u) inn u = 0 on an X I, u(x, 0) = Uo and

X

I (19)

div u = 0

where uV and V =

= (ut8., ,u28.,,, ... ,Un8.,,.). Denote by V = {v E C~(n) : div v = 0} V where the closure is in the norm of the space [ W~(O)] N (Cartesian 1

N

o

product of WHn)), N = 2, 3. We denote by L2(n) = [L2(n)] , (tp, '1/J) = tpt/Jdx, (u, v) = u.vidx, ((u, v)) = 1 (Vut, Vvi) and b(u, v, w) = 1 Ef.;=l u;8.,;viwidx which is well defined for u, v, wE V since W:f(O) C. L 4 (0) (continuous imbedding, N = 2, 3). The variational solution to (19) is defined as u E L 00 (l, V) n L 2 (/, L 2 ) satisfying

In

E! In

In

d

E!

dt (u, v) + ((u, v))

+ b(u, u, v) = (f(t, u), v)

with

\fv

eV

(20)

u(O) = Uo·

We assume that f(t, x, s) is continuous in its variables and

lf(t,x, s)l :5 c(1 +lsi).

(21)

The existence of the variational solution can be found, e.g., in [F]. The uniqueness is proved for N = 2 in [LP] and for N = 3 it is an open problem. If u E L 8 (/, L 4 ) then the variational solution is also unique for N = 3. The method of characteristics has been applied to (19) in [P], [S] etc. and its convergence has been proved under

rather strong regularity assumptions.

206

J. Katur

We apply our concept of the method of characteristics from Section 1 and prove its convergence without the assumption llulloo ~ c {norm in L00 (11 X I)). In that CB.'le we can truncate the convective term and apply our concept of approximation to (see (19)) a,u + (uKV}.u- 'Y~U = -Vp+ f(t,x, u) u = 0 on

an X I, div u

u(x, 0} =

inn

X

I

Uo

=0

for lcp(x)l < K, ( ) r I ( )I sgn cp x 10r cp x ~ K. Remark 22. Suppose that f(t, x, s) is Lipschitz continuous in s. It can be proved that (18}K possess the unique solution in the cB.'le N = 2. In the case N = 3 it holds true under the additional assumption that a variational solution of (19) or (18)K is an element of L 00 (1, V). The unique variational solution of (18}K coincides with the bounded variational solution of (19}. Now our modified method of characteristics applied to (18}K reads where u

K

K

K

.

= (u 1 , ... ,uN) With cp

(Us- ~- 1 ocpi,

K

:=

v)

{ cp(x)

K

+-y((Ut, v)) = (fi,v)

lr/v E V

(23}

fori= 1, ... , n (ui:::::: u(x, ti}, ti = ir) where cpi(x) := x- Wh * ~~ 1 , h = ~ for any w E (0, 1}. Problem (23} is the stationary Stokes problem for determination of ui E V. The existence is guaranteed by Lax-Millgram argument provided Ui-1 o cpi E L2(0). In this case (since I~Kioo ~ K) we verify that cpi(x) with its inverse are Lipschitz continuous for h = ~ for any w E (0, 1} and hence Ut- 1 o cpi E L 2 (0) for Ui- 1 E L2(0). In our CB.'le we cannot prove the boundedness IIUtlloo $ c < oo as in Section 1 since (23} represents a system coupled through V. Thus we shall not assume (13} in this section. We can prove the a priori estimates Lemma 24. The estimates

(25}

i=1

hold true uniformly fori= 1, ... , n. Proof. We put v

= UiT into (23} and sum it up for i =

j

~)ui- Ui-t. Us) +'Y i=l

j

L

2 llt1tll r $

i=1 +

L I(Ut-1- Ui-1 o cpi, Ut)l+ i=1

j

L lt1ti2T +c. i=1

1, ... , j. Then we obtain

;

(26}

Application of Relaxation Schemes and Method of Characteristics to Degenerate • . •

207

The first term on R.H.S. we estimate in the following way

Ui-1- Ui-1 o vi= -r(w" *

~~ 1 V)

1 1

1~-1(x + +r ([~~ 1 - w" * ~~ 1 ] V) 1~-l(x + =

(~~ 1 V)

-T

Ui-1(x + s(rp'(x)- x))ds =

1

1

s(rp'(x)- x))ds+ srp'(x)- x))ds = -rJl + rJl

n

where ~-1 is a prolongation of Ui-1 on n· :::> (llullwl(O•) can estimate the first term on R.H.S. in (26) by

t

t, fo T

1Jll

:5 t

2

dx + c(t)

(27)

~ IIUill~r +

t fo

j

/c

i=l

i=1

T

:5 llullwHO)>· Then we

1Jll 2 dx :5

L IIVu,ll~r + c(t, K) L IIUill~r.

Since Ef= 1(u;- Ui-1• U;) ~ !llu;ll~- !IIUollg. Then the Gronwall argument in (26) yields the a priori estimate (25).

Lemma 28. The sequence {u,} is compact in L2(QT) i.e. 3u E £2(/, V)such that u"-+ u in L2(QT)· Proof. We sum up (23) fori= j + 1, ... ,j + k and then we put v = u;+lc- u; and again we sum it up for j = 1, ... , n- k. The first term in (23) we split in the form

(29) Then using the a priori estimates of Lemma 24 we successively obtain n-/c

~)u;+lc- u;, UJ+Ic- u;)r :5 ckT

j=1 from which we conclude

for any 0

< z :5 zo

(30)

and for all n (un(t) is the corresponding Rothe's function). From the estimate (see (24)2)

208

J. Katur

L

we deduce

lun(t,x + y)- un(t,x)l 2dy :5 clyl

which together with (29) gives

Then Kolmogorov's argument implies the compactness of {un}. consequence of Lemma 24.

The rest is a

Theorem 31. Let {ut}~ 1 be the solution of the Stokes problem (29). Then un--. u in L 2 (I, V) where u is a variational solution of (18)K· Proof. Let v E 0 1 (/, V) be a smooth test function, v(x, t) = 0 fort= T. We put it into (23) and integrate it over (0, T). In the first term J 1 = Jl + Jf in {23) we use (29) and obtain (vi = v(x, ti))

Easily we obtain that the second term converges to 0 with n --. oo. Using Abel's summation in the first term we obtain

We can prove

n

Jf:: ~)ui-1- lli-1

o

qi, v) __. ((uKV)u, v)

i=l

since un --. u a.e. in QT, Vun _.. u in L2(/, V) and v E 0 1 {/, V). Similarly we obtain J2 ((un, v))dt ((u, v))dt

=

Ja

=

1

1

--.1 --.1

(f(t, un(t- r)), v)dt

{f(t, u), v.)dt

and hence we deduce that u is a variational solution of (18)K· To prove the strong convergence of {un} we shall need 8tftn _.. 8tU in L2(/, v•). We obtain from (23) (using {29)) by duality argument

Application of Relaxation Schemes and Method of Characteristics to Degenerate . . •

209

where (26) has been used. Hence we deduce

and 8[un _...Btu in L 2 (/, V"). Moreover, we use integration by parts formula

1' (8eu, u)ds =

~llu(t)JI~- ~Jiuall~

and liminf 1t(8[u.n, u.n)

(32)

~ ~Jiu(t)JI~- ~JiuaJI~.

(33)

We put v = u- u.n into (23) and integrate it over (0, t). The first term we split in the form (29). Using (32), (33) in parabolic terms we obtain liminf1t (8[u.n , u.n- u)ds n-+oo

0

~ 0.

(34)

We can estimate

11' (lin - u.n o VJ, u.n - u)dsl-+ 0 for n -+ oo.

(35)

Since u.n -+ u in L2(/, L2). Similarly we have

11' (f(t, u.n(t- r)), u.n - u)dsl-+ 0 for n-+ oo.

(36)

The elliptic term we estimate as follows:

1' ((lin, u.n - u))ds = 1t ((u.n - u, u.n- u))dt + 1' ((u, u.n- u))dt

1'

~ c Jlun - uJI 2 dt- 0(1) (0(1) denotes any term c,. with c,. -+ 0 for n-+ oo). Finally we obtain

from which we obtain the required result.

~

210

J. Katur

3.

SOME NUMERICAL EXPERIMENTS

We present a numerical solution of a mathematical model 8t(8u + pS) + div (iiu- DVu) = 0 p8tS = k(w(u)- S), llf(C) = ~tC",p E (0, 1) describing contaminant transport with sorption governed by Freundlich isotherm. In the equilibrium mode, i.e. k -+ oo, we obtain s = w(u) = ~tuP. Then b(u) = 8u + p~tuP (here b'(O) = oo) where 8 is the porosity of the porous media, ti is the velocity field of the water and u is the concetration of the contaminant. This problem has been studied in [Kn], [DDG], [DKn], etc... We use the following data: 8t (

~u + 1.5uP) + 8z(3u- 0.058zu) = 0

with p = 1.2; 1; 0.8; 0.6; 0.4. We consider !l = {0, 100), T = 30 and the Dirichlet boundary conditions u(t, 100) = 0 and i) u(t, 0) = 1, ii) u(t, 0) = 0 with the corresponding initial conditions i) u(O, x) = uA(x), ii) u(O,x) = ~(x) where U:,(x) (i = 1, 2) are piecewise linear functions of the following form: uA(x) = 1 for x E (0, 0.1), uA(0.2) = 0, uA(x) = 0 for x > 0.2; ug(x) = 0 for x E (0, 100)\(0.1, 0.5), ug(0.2) = ug(0.4) = 1. The solutions are drawn in three time moments for various p (p = 1.2- dash-dash-dotted line, p = 1.0- full line, p = 0.8dash-dotted line, p = 0.6- dashed line, p = 0.4- dotted line) in Figure 1 for t = 2, in Figure 2 for t = 4 and in Figure 3 for t = 6 in the case i. The case ii is drawn in Figure 4 for t = 1, in Figure 5 for t = 2 and in Figure 6 for t = 6.

0

2

4

FIGURE 1

6

8

10

Application of Relaxation Schemes and Method of Characteristics to Degenerate . ..

1

211

\i --~ \ .. .

0.8

It

\

~

I

0.6

..

I

..'.

0.4

\

!\ I!~ i!

0.2



\

..·~ j!

0

2

4

6

' ' ....______ '·, 8

10

12

14

FIGURE 2

0.8 0.6 0.4 0.2

6

10

8

12

FIGURE

0

1

14

16

18

20

3

2

FIGURE 4

3

4

5

212

J. Katur

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

2

4

6

FIGURE

0

5

10

FIGURE

8

10

5

15

20

6

REFERENCES (AL) H. W. Alt, S. Luckhaus: QuMilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), pp. 311- 341. (B) E. Bii.nsch: Numerical ezperiment.t with adaptillity for the porous medium equation. Acta Math. Univ. Comenianae. 84(2) (1995) , pp. 157-172. (DOG) C. N. Dawson, C. J. Van Duijn, R. E . Grundy: Lorge time Mymptotie& in contaminant transport in porous media. SIAM J. Appl. Math. 58(4) (1996) , pp. 965- 993. (DR) J . Douglas, T . F. Russel: Numerical methods for convection dominated diffusion problem& bMed on combining the method of the charoctemtie& with finite element.t or finite difference&. SIAM J. Numer. Anal. 19 (1982), pp. 871- 885. (DKn) C. J. Van Duijn, P . Knabner: Solute tromport in porous media with equilibrium and nonequilibrium multiple-site adsorption: 7hweling waves. J. Reine Angew. Math. 415 (1991), pp. 1-49. (F) M. Feistauer: Mathematical methods in fluid dynamiC&. Pitman Monographs and Survey in

Application of Relaxation Schemes and Method of Characteristics to Degenerate . . .

213

Pure and Applied Mathematics, Longman, Scientific & Technical, 1993. (JKl] W . Jager, J. Kaeur: Solution of porou, medium systems by linear approximation scheme. Numer. Math. 60 (1991), pp. 407-427. (JK2] W . Jager, J . Kaeur: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO Model. Math. Anal. Numer., 29(5) (1995), pp. 605--627. (Kl] J . Kaeur: Solution of some fru boundary problems by relaxation schemes. Preprint No. M394, Faculty of Mathematics and Physics, Comenius University (1994), pp. 1- 25, SIAM J . Numer. Anal., to appear. [K2] J. Kaeur: Solution to strongly nonlinear parabolic problems by a linear approximation scheme, Mathematics Preprint No. IV-M1-96, Comenius University Faculty of Mathematics and Physics, (1996), pp. 1-26, IMA J . Numeric, to appear. (K3] J . Kaeur: Solution to degenerate convection-diffwion problems by the method of charo.cteristics, to appear. [KHK] J. Kaeur, A. Handlovitova, M. Kaeurova: Solution of nonlinear diffwion problems by linear approximation schemes, SIAM J . Numer. Anal. 80 (1993) , pp. 1703- 1722. [Kn] P. Knabner: Finite-Element-Approximation of Solute 1hmsport in Porous Media with Genero.l Adsorption Processes. "Flow and Transport in Porous Media" Ed. Xiao Shutie, Summer school, Beijing, 8-26 August 1988, World Scientific (1992), pp. 223-292. [KJF] A. Kufner, 0. John, S. Futik: fUnction spaces. Noordhoff, Leiden, 1977. (LP) J. L. Lions, G . Prodi: Un theoreme d 'e:ristence et d'unicite dans les equations de N avier-Stokes en dimension 2. C. R. Acad. Sci. Paris 248 (1959), pp. 3519-3521. [M1) K. Mikula: Numerical solution of nonlinear diffwion with finite extinction phenomena. Acta Math. Univ. Comenianae, 64(2) (1995), pp. 223-292. [M2) K. Mikula: Solution of nonlinear curvature driven evolution of plane convex curves. Appl. Numer. Math. 28 (1997), pp. 347-360. [OJ F. Otto: L 1 -contraction principle and uniqueness for quasilinear elliptic-parabolic equations, C . R . Acad. Sci. Paris T 321, Ser. I (1995), pp. 105-110. [P) 0 . Pironneau: On the tro.nsport-diffwion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38 (1982), pp. 309-332. [S) E . Siili: Convergence and Nonlinear stability of the Lagro.nge-Galerkin Method for the NavierStokes Equation.Numer. Math., 53 (1988), pp. 459-483.

J. Kacur, Department of Numerical Analysis, Faculty of Mathematics and Physics, Comenius University, 842 15 Bratislava, Slovakia

Symmetrization - or How to Prove Symmetry of Solutions to a PD E B. KAWOHL

Abstract. Many partial differential equations are Euler equations of variational problems, and this circumstance can be used to derive symmetry or monotonicity properties of their solutions. I survey some recent results on symmetrization methods, i.e. on procedures in which a supposedly nonsymmetric minimizer is deformed into a symmetric one with lower energy. I also compare this variational approach with arguments involving the maximum principle. There are specific advantages and disadvantages of both methods. Another symmetry approach uses the unique continuation principle. Finally I address the recently emerging subject of isoparametric surfaces. Keywords: symmetrization, symmetry, rearrangement, unique continuation, isoparametric surfaces, Ginzburg-Landau Classification:

35805, 49Kl0, 26010, 35J20

"Nihil est sine ratione" (Nothing is without a reason) was- according to G.Polya [46]- a fundamental idea of W.G.Leibniz. He phrased it in Latin, the lingua franca of the time when Charles University was founded 650 years ago. With my lecture in today's Latin I follow a suggestion of the organizing committee to talk about symmetry. Symmetry of solutions is often taken for granted in symmetric problems, but by no means automatically true (36, 37]. There are many discussions of this topic in the mathematical literature, e.g. (10, 45, 46, 54]. In this lecture I focus on symmetry of elliptic equations, and here in particular on symmetrization methods. For parabolic equations I refer to [31] and the lecture of P.Polacik. Other methods to prove symmetry, for instance via stability arguments, are listed in [36].

I. What is symmetrization? Let me present the geometric idea of symmetrization in one dimension by means of a simple example. Symmetrization maps the nonsymmetric function u4 (z) into a symmetric function u*(z) = uo(z).

u4(z) --

~in

{ ~ .m 1+4 1-4

[-1 ' a] [ a, 11

..-+

u*(z) = 1 -lzl in [-1, I]

Symmetrization- or How to Prove Symmetry of Solutions to a POE

u.

-I

4

1

I



215

u•(x)

-1

Figure 1: The graphs of U 4 and u• Remark 1: One immediately observes the following two properties, which seem to be predestined for use in variational problems. J~ 1 F(uo) dx = J~ 1 F(u•) dx (Cavalieri's principle) for any Borel measurable function F. • g(a) := J~ 1 lduo/dxl" dx = (1 + a)l-p + (1 - a) 1-" ;?: 1 ldu• /dxl" dx = g(O), because g is even and (even strictly) convex.



t

Definitions: For u(x) E W~·"(O, JR+) consider the level sets Oc(u) := { x E 1R I u(x) ;?: c }. Then their rearrangements are defined as

O~(u) := { [-IOc(u)l/2 , IOc(u)l/2] if Oc(u) ::/: 0, 0

otherwise,

and u•(x) := sup { c E 1R I x E n;(u) }, which is defined on n• = 0 0(u), is called the symmetrically decreasing rearrangement of u. Note that the level sets Oc(u•) of u• coincide with O~(u) by construction.

u(x)

u•(x)

Figure 2: The graphs of u and its symmetrically decreasing rearrangement u• To symmetrize a function of several variables, we have to symmetrize its level sets in a suitable way. One of them is Steiner symmetrization with respect to y=O. For u(:r:) E W~'"(O,JR+) with{} C JRn set x = (x',y) with x' E 1Rn-l. Now let us consider the one-dimensional level sets Oc(u(x',·)) := { y E IR I u(x',y);?: c },

216

B. Kawohl

and define

n•( (

He U X

I

•)) · - { 1

·-

n~(u) :=

1

[-lflc(U(X , ·}}l/2, lflc(u(z', ·}}l/2] if flc(u(z', ·)) ::/:0, 0 ' otherwlSe,

u

{x'}

X

u•(x) := sup { c E lR

n~(u(x', ·}},

IX

the Steiner symm. of nc(u}, the Steiner symmetrization

En~ },

or symmetrically decreasing rearrangement of u in direction y.

y

(j(n. ......_____....)

y

x'

~! -

y

x'

Figure 3: A level set Oc and its Steiner symmetrization with respect to y Another suitable way to symmetrize level sets and functions is known as Schwarz symmetrization (not Schwartz symmetrization please!}, named after Hermann Amandus Schwarz, in which for u(x) E wJ•"(O, .JR+) with n c /Rn the level sets nc(u) := { X E mn I u(x) ~ c} are replaced by concentric balls centered at zero. Therefore Schwarz symmetrization amounts to defining

n~(u) := {ball of same volume

0 nc centered at 0

if nc ::/:0, otherwise,

u•(x) := sup { c E lR I X E n~(u) }. Let me remark in passing that Schwarz symmetrization can be obtained as a limit of consecutive Steiner symmetrizations (similar to making dumplings or snowballs}. Remark 2: The following inequalities are common to all these rearrangements; see for instance (33].

• In F(u) dx =In· F(u•) dx • In IVul" dx 2: In· IVu•IP dx • In· u• v• dx ~ In u v dx

(Cavalieri's principle} (Polya-Szego inequality) for p E (0, oo) (Hardy-Littlewood inequality)

Application 1: The Krahn Faber inequality At (0) ~ At (n•) is a typical example for the usefulness of Remark 2. Recall that the first Dirichlet eigenvalue for the Laplace

Symmetrization- or How to Prove Symmetry of Solutions to a PDE

217

operator on any domain n is characterized as A1(f2) = min {R(v; f2) 0 :1= v E 2 H~' (f2)} and R(v; n) = IIVvll2/llvll2, where II · ll2 denotes the norm in L 2{f2). Therefore

and equality holds only if n is a ball. This last observation about equality was (as I learned in 1993) first made by F.Schulz in his diploma thesis [49]. It rests on the observation that sometimes one can discuss the equality sign in the second inequality of Remark 2. In fact Schulz did this 1977 for the first eigenfunction and for the Dirichlet integral In1Vul 2 dx, while I discussed it in 1984 [32] for integrals of the type F(x', u)G(IVul) dx with positive F, strictly convex and strictly increasing G and e.g. analytic functions u. Moreover, I showed that for F = 1 and G(t) = tP and for a dense subset of W 1•P(f2) one could discuss equality and conclude that it implies symmetry of u. But be careful. There are also functions, even functions, with large sets of critical points for which this conclusion is wrong. As a counterexample, imagine a radially symmetric cutoff function 1]1 , centered at zero, with a flat top B 1 {0), and add to it another radially symmetric cutoff function, centered at x 0 "=/= 0, such that supp 1J2 C B 1 {0). Then the Dirichlet integrals of TJ = 111 + 712 and TJ* coinincide, but 7J :1= TJ*. As for Schwarz symmetrization, these results were later refined by J .E.Brothers and W.T.Ziemer [12, 13] and by R.Tahraoui [52]. Incidentally, the result of Brothers and Ziemer was first presented at a winter school 1987 in Srni, organized by Charles University. Brothers and Ziemer considered integrals of type A(IVul) dx under Schwarz symmetrization, with A{O) = 0, A increasing and A 1/P convex for some p E [1, oo). They discussed the equality sign in A{IVul) dx ~ A(IVu*l) dx under the assumptions p > 1 and 1 that {x En· i IVu*(x)l = 0} n u•- (0, M) is a nullset. HereM =ess sup u• :5 oo. Fortunately, solutions of elliptic equations are often almost nowhere flat, so that these results are applicable for our purposes. Tahraoui considered more general integrals of type F(u, IVul) dx under Schwarz symmetrization, where F satisfies natural growth assumptions and is convex in its second argument [52]. Finally, Brock considered integrals of type F(x, u, Vu) dx under Steiner symmetrization with structural assumptions on F that recover all previous (weak) inequalities and that are too detailed to be reproduced here; see [7] and also [33, p.83].

In

coo

In

In

In

In

In

Application 2: Nonlinear ground states on JRn are in a simple setting characterized as minimal energy solutions of - ~u

= >.f(u)

u(x)

4

0

and

u

>0

in mn, as lxl4 oo.

218

B. Kawohl

Modulo suitable assumptions on f, there is a metatheorem which states that any ground state u is radial and radially decreasing. This approach was used for instance in [4) or [25). Of course there are other ways to get such results via the moving plane method; see [29) and [39). To be a little more precise, in the recent paper [25) of M.Flucher and S.Miiller the ground states are characterized as maximizing F(v) dx over all functions v with Dirichlet integral not exceeding 1. Here F' = J, 0 :5 F(t) :5 clt1 2" with 2• = 2n/(n- 2) and F is upper semicontinuous and F '1. 0 in £ 1-sense. This includes the case that /(0) = /'(0) = 0, which was not covered in [29). In fact, in [29) the assumption /(0) = 0 and /'(0) < 0 was used, and only later removed in the paper of Y.Li and W.M.Ni. I refer to the lecture of H.Berestycki during this meeting for further details.

JR..

Application 3: Oddness and monotonicity, even one-dimensionality in cylinders can be proved via Steiner symmetrization as well. To this end let n = w X JR. with w c JR.n- 1 and consider minimizers u of the Ginzburg-Landau energy J(v) = 0 21Vvl 2+(v 2-1) 2 dx among those functions v which go to ±1 as Xn -+ ±oo.

J

Claim 1: The nodal surface of u is a plane; without loss of generality it is the plane {xn = 0}, so that u(x',O) = 0, and u is odd in Xn, i.e. u(x',xn) = -u(x',-xn) in

n.

Proof: Substitute w := (1- u 2 ), then w ~ 0 and w -+ 0 as lxnl -+ oo. For every x' E w the function w(x', ·) has a maximum of height 1 somewhere. Under this tranformation the energy is converted into a convex functional

-

·- f

1Vwl2

2

J(w) .- Jn 2(1- w) + w dx. Incidentally, the convexity of this transformed functional j might explain the somewhat unexpected uniqueness-, symmetry- and Liouville-theorems for GinzburgLandau type problems. To prove the oddness of u, Steiner-symmetrize w and set u(x', xn) := sign(xn)v'1 - w•. Then it is easy to see that J( u) :5 J( u) and equality holds only if u is odd and monotone. More can be shown, however.

Claim 2: Not only the nodal surface, but every level surface of u is a plane, and therefore u depends only on Xn. Proof: For each c E ( -1, 1) the level surface {(x', Xn); u(x) = c} can be parametrized by x' as Xn = fc(x'). Replace it by its average Xn = fw /c(x') dx', i.e. replace the level sets of w = 1 - u 2 by cylinders with cross section w and with plane ends. It can be shown that this decreases energy.

Symmetrization- or How to Prove Symmetry of Solutions to a PDE

219

This approach can be found in a nice paper of G.Carbou (17]. It was recently generalized by F.Brock (8] to apply to travelling waves in cylinders, generalizing previous results of H.Berestycki & L.Nirenberg (3]. What about boundary conditions other than Dirichlet? We just saw an example with zero Neumann data on the lateral boundary x JR. Here are two more.

aw

Application 4:

Steiner symmetry for nonlinear boundary conditions can sometimes be achieved without symmetrization arguments. The following theorem is new, and its assumptions are far from optimal. Rather than specialize in generalizations, I just want to expose the idea of proof. Theorem 1: Let n c /Rn be Steiner symmetric in Xn and consider minimizers u of the functional J(v) = fn{1Vvl 2 - f(x)v} dx + f 80 j(v) do with f E C 1 (n) nonnegative and Steiner symmetric and j E C 1 ( JR) convex and even. Then u is Steiner symmetric, i.e. symmetrically decreasing in Xn· Note that u satisfies the nonlinear boundary condition -aujov = 8j(u) on an. Proof of Theorem 1: By strict convexity a minimizer u is unique, thus symmetric. Moreover it is nonnegative, because otherwise we can replace it by lui. It remains to show that u is symmetrically decreasing in Xn or equivalently, that u decreases in Xn on n> := n n {:en > 0}. Clearly U:.:,. = 0 on the hyperplane n n {Xn = 0} because of the symmetry, and U 20,. $ 0 On n {:en > 0} due to the boundary condition, thus w = U:.:,. $ 0 on an>. But now we can apply the maximum principle to w. In fact u satisfies the Euler equation -du = f in n. Therefore differentiation with respect to Xn yields -dw = !:.: .. $ 0 in n>. Thus w < 0 inn> and u is Steiner symmetric.

on

Remark 3: It seems to be difficult to prove Theorem 1 via Steiner symmetrization. It can be shown that the boundary integral in J decreases under symmetrization. The L 2 -product of u and f increases under symmetrization due to the Hardy-Littlewood inequality in Remark 2. Thus the second term in J decreases. What about the Dirichlet integral? It can· in fact increase under Steiner symmetrization, unless we know already that for a. e. x' the minimum of u(x', Xn) is attained on the boundary points. Otherwise the obstruction from (33, p.56] causes problems. An example (due to my student Miroslaw Michaelowski} for which the Dirichlet integral increases under Steiner symmetrization is provided by the following function. Let n := (0, 1] X (-2, 2} u {(x, y)l X E [1, 2), IYI < 3- x} and u(x, y) := 0 if IYI < 1 and u(x, y) := IYI- 1 otherwise. Then a simple calculation shows that Jn 1Vul2dx = 3, while Jn IV'u'l2dx = 4.

220

B. Kawohl

I wish to point out, however, that Steiner symmetrization can be sucessfully applied to functions which are periodic in Xn. This observation is used for instance in circular symmetrization, where a disk in JR.2 is interpreted as a rectangle in polar coordinates, by identifying x' =rand Xn = t/>, see [33, Ch.II.9) or [21). Second proof of Theorem 1: If we are willing to assume C 1-smoothness off and an, we can argue that the function w = u.,,. satisfies the equation -aw- f., .. (x)w = 0 in n and Xn w $ 0 on an. Hence w $ 0 in n n { Xn > 0} by the maximum principle. Application 5: Optimal insulation of bodies against heat loss was recently studied in [21). Let n be the unit disk, and let '11. be a stationary temperature in n, subject to uniform heating, i.e. -au= 1 inn. On the boundary an we have Newton's law of cooling -(aujav) = hu, where h represents conductivity. We want to maximize insulation in the sense that the average temperature should be maximized among admissible variations of h. Depending on the material which bounds an, the conductivity constant h is allowed to attain two values h 1 and h2 with 0 < h 1 < h 2 < oo, but the proportion of an on which h = h 1 is prescribed. Its location is free. Mathematically this leads to the problem of minimizing

J(v, h):=

f {1Vvl 2 ln

2v} dx +

f

hv 2 ds

lao over v E W 1 •2 (n); h: an-+ {hl, h2} and Ian h ds = c E (21Th}, 211'h2}·

One can think of the subset an 1 of an on which h = h1 as a wall, and of an 2 = an\ an 1 as windows in a room. Then one of the the results in [21) states that having a single large window is preferable to having many small ones. This result was obtained by means of circular symmetrization. A similar problem was studied in (24), where the first eigenvalue was minimized under mixed DirichletNeumann boundary conditions. These occur as limits h2 -+ oo and h1 -+ 0 in our formulation. II. Continuous symmetrization

Until now my message has been: Global minimizers of variational problems are symmetric or monotone. An apparent advantage of symmetrization lies in the fact that often it requires less regularity than for the moving plane method. Disadvantages of symmetrization until now seemed to be a) that the method did not apply to nonvariational problems and b) that it did not yield results for local minimizers or critical points. At least the second one of these difficulties has recently been overcome. F .Brock has been able to apply continuous symmetrization methods to local minimizers of variational problems and to critical points. For functions of a single variable this

Symmetrization- or How to Prove Symmetry of Solutions to a POE

221

concept had been developed by H.Matano and myself over ten years ago [42, 34), and for functions of several variables with convex level sets it goes back about 50 years to G.Polya and G.Szego [47, pp.201-204]. At the beginning of my lecture I presented an example of continuous symmetrization. Remember the function Ua from Figure 1? Consider a as a parameter for a family {Ua} of equimeasurable functions and send a to zero. Then Ua tends to uo = u•. It is easy to extend this approach to functions with convex level sets, and this was done in [47]. In continuous symmetrization a more general function u is continuously deformed into a family of equimeasurable functions ut and, as t -+ oo, into u•. To this end the level sets of u are shifted toward their final location with a speed that depends on their shape and their distance to the final location. The details of the definition are too long to be reproduced here; see [5, 6]. However, for Ua from Figure 1 above, Brock's ut corresponds to our ub with b = a e-t. To give you a flavor of Brock's results I list one of them.

Theorem 2: (Brock [6, Theorem 9.1]) Let n be Steiner-symmetric in Xn, p E (1, oo), u E wJ·P(O)nC(O) a nonnegative solution of -~pu = -div(IVuiP- 2 Vu) = l(x',xn,u) in 0, and let

I be symmetrically strictly decreasing in Xn

and continuous. Then u

= u• .

The idea of the proof proceeds as follows. Test the differential equation with ut - u. Then by the convexity of the function s ~ sP and since Ut is close to u for small t

L

IVuiP- 2 VuV(ut- u)dx =

$ Thus

L

l(x, u)(ut- u)dx

!

r(IVutlp - IVuiP)

Plo

( = o(t) ) dx $ 0

L

(IVutlp - IVuiP) dx = o(t)

and this implies via a delicate quantitative analysis that u is symmetric. To get a feeling for this qualitative analysis consider the function g(a) defined in Remark 1 and note that g(a + t)- g(a) = o(t) if and only if a= 0. F.Brock has many other results, for instance on mean curvature type equations, on equations in /Rn et cetera. Further applications of continuous symmetrization have recently emerged, namely, • Applications to overdetermined boundary value problems. More precisely, if u solves -~pu = l(u) in D, u = 0 and 8uf8v = c on 8D, where D C /Rn is simply connected and convex, and where I is continuous and nonincreasing, then D is a ball. This result in [9] is a considerable generalization of [26], where I 1.

=

222

B. Kawohl

Another variant is given in (19, Sect. 4], this time for solutions of -~u

= ..\1u under

the same boundary conditions, and again assuming convexity of n. Here continuous

Steiner symmetrization leads to the well-known conclusion that n is a ball. • Results on the attainability of eigenvalues by D.Bucur, G.Buttazzo, !.Figueiredo (13]. The Krahn Faber inequality states that given the volume of n the range of ..\1 (0) is a half-infinite interval. In (13] the authors look for the range of the vector (A 1(0}, A2(0}}, given the volume of n, and prove via continuous rearrangement that it is a closed set E in JR2 which is convex in both cartesian directions. This shows, in particular, that given A1(0) one can maximize or minimize A2(0} and vice versa. • More general shape optimization problems require the convergence W~·"(nt) -+ W~·"(O•) as t -+ s- in a suitable sense (of Mosco) in order to prove existence results. Such a convergence has been derived by D.Bucur and A.Henrot via continuous symmetrization in (14].

III. Symmetry without symmetrization Let me now come to a somewhat different subject. There are domains with reflection symmetries which are not Steiner symmetric, for instance spherical annuli in JR.", i.e. 0 = BR,(O} \ BR, (O) with 0 < Rt < R2 < oo. How about symmetry of solutions on those domains?

Figure 4: Domains without Steiner symmetry One successful approach to proving symmetry is based on the unique continuation principle. Theorem 3: (Lopez (41]} Let u minimize J(v) = fo{IVvl 2 + F(v)} dx and let

Then u is symmetric in x 1 •

For the proof cut n in half, into Or = the energy J(u) is equidistributed,

n be symmetric in Xt.

n n {x1 > 0} and its reflection 0 1• Then

Symmetrization - or How to Prove Symmetry of Solutions to a PDE

223

Otherwise, if say the left-hand side is smaller than the right-hand side, we could define a function -( ) _ { u(x1, ... xn) if x E 0,, u x1, .. . xn ( ) .f n U -:Z:l ···:Z:n 1 :z:Eur, 1

< J(u); contradicting the minimality of u. Therefore we see that if u minimizes J, so does u. But u = u in O,. Consequently u = u on all of 0, provided the solutions of the associated Euler equation have the unique continuation property, i.e. if I is for instance Lipschitz continuous.

such that J(u)

This trick was used by O.Lopez, and it works even for some elliptic systems such as the Lame system with variable Lame constants (see [1]}, but it is not entirely new. M.E.Gurtin and H.Matano used it for instance in 1988 to prove monotonicity results in [30, p.308]. On annuli there can be symmetry breaking, but then the solutions still show some symmetry under certain group actions. This effect was shown in [18, Sec.3], [22], [25, p.148 and 163], [33, pp.95-99], [38) and [40) for positive solutions of

= l(u) u= O

-~u

in 0, on an,

and for suitable I with subcritical growth. In these papers prototypes for I are l(u) =uP- u or l(u) =uP.

IV. Isoparametric surfaces are level sets of solutions to the overdetermined system -~u

= l(u) in 0

and

IVul = g(u) > 0 in 0.

The following result is well known among differential geometers, but maybe not so well known in the PDE community. It was found around 1920 in attempts to describe the propagation of wave fronts in geometrical optics.

Theorem 4: If u solves system (*) above in a connected domain 0 C JR.", if I is CO and g is C 1 , then the level surfaces of u are either planes, spheres or cylinders, i.e. cartesian products of spheres and planes. For a recent proof in the three-dimensional case, see [48); for the general history of Theorem 4 and its generalizations to noneuclidean geometry I refer to [53). Before

giving a generalization of this result, let me demonstrate its usefulness.

224

B. Kawohl

Application 6: In 1971 H. Weinberger published an elegant approach to a special overdetermined problem in DC !Rn -~u= 1 u= 0

and

au = c -:/:; 0 ov

-

onoD

for a bounded domain D, and showed that D is a ball. This way he recovered a special case of J.Serrin's seminal results [50). His strategy of proof went through first showing

2 1Vul2 + -u = c2 n

in D

Jc2-

and to derive radial symmetry from this [55). If we set g(u) ~u2, and identify n with D \ {set of critical points of u}, we can apply Theorem 4 to see that D must be a ball. It seems only natural that Weinberger's trick might extend to more general semilinear equations -~u = f(u), and that then P(x) = 1Vul2 + ~F(u) should be constant. Let me dismiss any hope about this by pointing out that P is not even constant for the linear function /(u) = ~ 1 (D)u and if Dis a ball.

Application 7: An open problem of DeGiorgi, that was already presented in H.Berestycki's lecture, reads precisely as follows: "Consider a solution u E C 2 (JRn) of ~u = u 3 - u in !Rn such that lui ~ 1, ~ > 0 in the whole !Rn. Is it true that for every ~ E lR the sets {u = ~} are hyperplanes, at least if n ~ 8 ?" In [36) I gave an explanation why the dimension n ~ 8 makes things easier. L.Caffarelli, N.Garofalo and F.Segala (16) and later X.Chen (20) showed that the auxiliary function P(u, x) := IVu(x)Fl - 2F(u(x)) is nonpositive in !Rn. Here F(u) = (u2 - 1) 2 /4. This tells us that the "kinetic energy" or Dirichlet integral in the corresponding functional is dominated by the "potential energy" term, except that both terms are infinite. Moreover, under the further assumption, that there is an x 0 E JRn such that P( u, x 0 ) = 0 they were able to derive a Liouville-type result, namely P = 0. Once this result is known one can see that P,_, the partial derivative of u in direction - Vu must vanish. This leads to the observation, see [36), that the (n - 1) dimensional level surfaces of u are minimal surfaces, and then a famous theorem of Bernstein applies, provided n- 1 ~ 7 or n ~ 8. Both in [16) and [20) the one dimensionality of u is derived for any n, provided P is constant. But in this case we can cut their proof short by observing that one can identify g(u) with ..j2F(u) and apply Theorem 4 to reach the conclusion. Incidentally, N.Ghoussoub and C.Gui have recently found another proof of DeGiorgi's conjecture without assuming anything about P, but unfortunately only under the restriction n = 2 or 3; see [27).

Symmetrization - or How to Prove Symmetry of Solutions to a POE

225

Application 8: O.Mendez and W.Reichel looked recently at the overdetermined exterior boundary value problem in !Rn \ D

-Au=O

u

=c> 0

au - = -1

and

811 with D a bounded convex domain D C !Rn and n

[43].

on8D 3, and showed that D is a ball

~

They showed first that the auxiliary function P(x) = 1Vu(x)l2 u(x)C2-2n}/(n-2}

equals a constant C in JR" \D. If we set g(u) = v'cu(x)0

inn,

(1)

inn.

(2)

Here a and g are assumed to be of class and f E CO. For a = 1, i.e. if (1) has the special structure -Au = f(u), we recover Theorem 4. However, not even the ellipticity of the operator in (1) is needed to derive its conclusion.

C1

Theorem 5: Suppose that a> 0, then the level surfaces S1c~(u) := { x E S1 I u(x) = d} of solutions u to (1), (2) are subsets of planes, cylinders or spheres. Proof: Without loss of generality we may assume that g

r g(t)1 dt

= 1, otherwise we substitute

u(x) := lo

for u. If we denote differentiation with respect to x; by subscript;, and let p stand for 1Vul 2 , we can rewrite (1) using Einstein's summation convention as

-a(u,p)Au - 2a,.(u,p)1J.iu;1J.i;- au(u,p)p = f(u,p). Consequently, observing that p = g

(3)

=1

-Au= 2a,(u,1)1J.iUj1J.ij + au(u,l) + f(u, 1). a(u,l)

(4)

226

B. Kawohl

But the right-hand side of (4) looks almost as if it depends only on u. Let v(z) = -Vu(z)/IVu(z)l be the direction of steepest descent of u in z. Then the term UiU;t.Li; can be expressed as u~u.,.,, but g., = (1Vul 2)., = 2u.,u.,., = 0. Therefore (4) degenerates to -du = j(u) = [a,.(u, 1) + f(u, 1))/a(u, 1),

(5)

a variant of (1). But now the proof of Theorem 5 is reduced to Theorem 4.

Remark 3: Let H(z) denote the mean curvature of the level surface through z. One can then rewrite du as u.,., + (n- 1)H(z)u., and (3) as (see for example [35]) -(n -1) a(u,p) H(z) u.,- [a(u,p) + 2pa,(u,p))u.,., = a,.(u,p)p+ f(u,p), and, using u.,.,

=0, arrive at

H(z) = a,.(u, 1) + f(u, 1) =: h(u) (n- 1)a(u, 1)

inn,

i.e. at the fact that the mean curvature of each level surface is constant and the constant depends only on the level u. Other equations that u satisfies, once (1) (respectively, (5)) and u., = -1 hold, are dqu = div(IVulq- 2 Vu) = IVulq- 2 [du + (q- 2)u.,.,] = du = i(u) for any

q E

(1, oo), or div (

Vu ) = j(u) ../1 + 1Vul 2 .f2

inn,

i.e., the mean curvature of the graph of u depends only on its level u. (Incidentally, if we know in addition that the level surfaces of u are compact, a famous result of Alexandrov implies that they are spheres.) This shows how overdetermined the system (1), (2) is.

Remark 4: Let me conclude with an open problem. In [2) the authors look for a variant of Serrin's result, i.e. for special symmetries of solutions to overdetermined boundary value problems on unbounded domains, and state: "It is tempting to conjecture that a much more general result is true. Assuming that n is a smooth domain with nc connected and that there exists a positive bounded solution of"

=0

and

u=O

and

du+ f(u)

u>0

ou on = c "' 0

inn, on an,

Symmetrization- or How to Prove Symmetry of Solutions to a PDE

227

n

"for some Lipschitz function f, then is either a half-space, a ball, the complement of a ball or a circular-cylinder type of domain: JR.i x B with B a ball." But such a problem seems to be amenable via a P-function, and maybe one can solve it by a combination of results in (51] with Theorem 4 above. This concludes my little survey. I hope to have demonstrated that there is more than one way to prove symmetry. I learned Liouville theorems in Prague during my time in 1979 as a postdoctoral student of J.Necas. Therefore I feel entitled to say: Universitas Carolina vivat crescat et fioreat! REFERENCES

[1] Ang,D.D., M.Ikehata, D.D.Trong & M.Yamamoto, Unique continuation for a stationary isotropic Lame system with variable coefficients. Comm. Partial Differential Equations, 23 (1998) pp.371-385. [2] Berestycki,H., L.Caffarelli & L.Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains. Comm. Pure Appl. Math 50 (1997) pp. 1089-1111. [3] Berestycki,H. & L.Nirenberg, Travelling fronts in cylinders. Ann. Inst. H. Poincare, Anal. Non Lineaire 9 (1992) pp.497-572. [4] Berestycki,H. & P.L.Lions, Nonlinear scalar field equations, I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983) pp.313-345. [5] Brock,F., Continuous Steiner symmetrization, Math. Nachr. 1722 (1995) pp. 25-48. (6] Brock,F., Continuous Rearrangement and Symmetry of Solutions of Elliptic Problems. Habilitationsschrift, Leipzig (1998) pp.l-124. [7] Brock,F., Weighted Dirichlet-type inequalities for Steiner-symmetrization, Calc. Var. and PDE, to appear. [8] Brock,F., An elementary proof for one-dimensionality of travelling waves in cylinders, manuscripts pp. 14 (1998), Journ. Inequalities & Appl., to appear. [9] Brock,F. & A.Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, manuscript (1998). (10] Brothers,J.E., Invariance of solutions to invariant nonparametric problems. Trans. Amer. Math. Soc. 264 (1981) pp. 91-111. [11] Brothers,J.E. & W.P.Ziemer, Minimal rearrangements of Sobolev functions. 15th winter school in abstract Analysis, Srni, Acta Univ. Carol. Math. Phys. 28 (1987) pp.l3- 24. [12] Brothers,J.E. & W.P.Ziemer, Minimal rearrangements of Sobolev functions. J. reine angew. Math. 384 (1988) pp.153-179. [13] Bucur,D., Buttazzo,G. ·& !.Figueiredo, On the attainable eigenvalues of the Laplace operator. Preprint 9, Besancon, SIAM J. Math. Anal., to appear. (14] Bucur,D. & A.Henrot, Continuous Steiner symmetrization and stability for the Dirichlet problem. Preprint 16, Besancon, Potential Analysis, to appear. [15] Byeon,J., Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli. J. Diff. Equations 136 {1997) pp. 136165. [16] Caffarelli,L., N.Garofalo & F.Segala, A gradient bound for entire solutions of quasilinear equations and its consequences. Comm. Pure Appl. Math. 47 (1994) pp. 1457-1473. (17] Carbou,G., Unicite et minimalite des solutions d'une equation de Ginzburg-Landau. Ann. Inst. H. Poincare, Anal. Non Lineaire 12 (1995) pp.305-318.

228

B. Kawohl

[18] Catrina,F. & Z.Q.Wang, Nonlinear elliptic equations on expanding symmetric domains. manuscript (1998) [19] Chatelain,T., M.Choulli & A.Henrot, Some new ideas for Schiffer's conjecture, in: Modelling and Optimization of distributed parameter systems, Eds: K.Malanowski, Z.Nahorski & M.Peszinska, Chapman & Hall, London (1996) pp. 90-97. [20] Chen,X., Global asymptotic limit of solutions to the Cahn Hilliard equation, J. Differential Geom. 44 (1996) pp. 262-311. [21] Cox,S. & B.Kawohl, Circular symmetrization and extremal Robin conditions, 50 (1999). Zeitschrift Angew. Math. Phys. (ZAMP), to appear. [22] Coffman,C.V., A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984) pp.429-437. [23] DeGiorgi,E., Convergence problems for functions and operators. in:Recent Methods in Nonlinear Analysis, Eds: E.De Giorgi, E.Magenes & U.Mosco; Pitagora Editrice, Bologna (1979) pp. 131-188. [24] Denzler,J., Windows of given area with minimal heat diffusion. Trans. Amer. Math. Soc., to appear. [25] Flucher,M. & S.Miiller, Radial symmetry and decay rate of variational ground states in the zero mass case. SIAM J. Math. Anal. 29 (1997) pp. 712-719. [26] Garofalo,N. & J .L.Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math. 111 (1989) pp. 9-33. (27] Ghoussoub,N. & C.Gui, On a conjecture of De Giorgi and some related problems. Math. Ann. 311 (1998) pp. 481-491. [28] Gidas,B., W.M.Ni & L.Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979) pp. 209-243. [29] Gidas,B., W.M.Ni & L.Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Math. Anal. and Applications Part A, Ed. L.Nachbin, Advances in Math. Suppl. Studies 7A, Academic Press {1981) pp. 369-402. (30] Gurtin,M.E. & H.Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math. 46 (1988) pp. 301-317. (31] Hess,P. & P.Polacik, Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems. Proc. Roy. Soc. Edinburgh Sec A 124 (1994) pp. 573-587. [32] Kawohl,B., On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems, Arch. Rational Mech. Anal. 94 (1986) pp. 227-243. [33] Kawohl,B., Rearrangements and convexity of level sets in PDE. Springer Lecture Notes in Math. 1150 Springer Verlag, Heidelberg (1985), pp.1-136. [34] Kawohl,B., Continuous symmetrization and related problems, in: Differential Equations, Eds: C. Dafermos, G.Ladas, G.C.Papanicolaou. Marcel Dekker, New York {1989) pp. 353- 360. (35] Kawohl,B., Comparison results between linear and quasilinear elliptic boundary value problems, J. Math. Anal. Appl. 142 (1989) pp. 162-169. (36] Kawohl,B., Symmetry or not? Mathematical Intelligencer 20 (1998) pp. 16-22. (37] Kawohl,B., Was ist eine ronde Sache? GAMM Mitteilungen 21 (1998) pp. 43-56. (38] Li,Y.Y., Existence of many positive solutions of semilinear equations on annulus. J. Diff. Equations 83 (1990) pp. 348- 367.

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229

(39] Li,Y. & W.M.Ni, Radial symmetry of positive solutions of nonlinear equations in Rn, Comm. Partial Differential Equations 18 {1993) pp. 1043-1054. (40] Lin,S.S., Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus, J. Diff. Equations 103 (1993) pp. 338-349. (41] Lopez,O., Radial and nonradial minimizers for some radially symmetric functionals. Electronic Journal of Differential Equations 1996 (1996) No. 3, 1-14. On WWW under URL:httpI I ejde.math.swt.edu (42] Matano,H. & B.Kawohl, Continuous equimeasurable rearrangement and applications to variational problems, unpublished manuscript (1988). (43] Mendez,O. & W.Reichel, Electrostatic characterization of spheres, manuscript (1998). (44] Modica,L. & S.Mortola, Some entire solutions in the plane of nonlinear Poisson equations. Boll. Unione Mat. Ital. 17 B (1980) pp. 614-622. (45] Palais,R.S., The principle of symmetric criticality, Comm. Math. Phys. 69 (1979) pp. 19-30. (46] Polya,G., Inequalities and the principle of nonsufficient reason. in: Inequalities Ed: O.Shisha, Academic Press, New York (1967) pp. 1-15. (47] Polya,G. & G.Szego, lsoperimetric inequalities in mathematical physics. Ann. Math. Studies 27 Princeton Univ. Press, Princeton (1952) (48] Pucci,P., An overdetermined system, Quart. Appl. Math. 41 (1983) pp. 365- 367. (49] Schulz,F., Der Faber-Krahnsche Satz fUr Dirichlet-Gebiete in n-Dimensionen, Diplomarbeit, Gottingen {1977). (50] Serrin,J., A symmetry problem in potential theory, Arch. Rational Mech. Anal.43 (1971) pp. 304-318. (51] Sperb, R., Maximum principles and their applications, Academic Press, New York (1981). [52] Tahraoui,R., Symmetrization inequalities, Nonlinear Anal. Theory Meth. Appl. 27 (1996) pp. 933-955. (53] Thorbergsson,G., A survey on isoparametric hypersurfaces and their generalizations. In: Handbook of Differential Geometry, Eds: F.Dillen, L.Verstraelen, Elsevier Science, to appear. (54] Waterhouse,W.C., Do symmetric problems have symmetric solutions? Amer. Math. Month. 90 (1983) pp.378-388. (55] Weinberger,H., Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43 (1971) pp. 319- 320. B. Kawohl, Mathematical Institute, University of Cologne, D 50923 Cologne, Germany Email: kavohlGmi. uni-koeln.de

A Bubble-Type Stabilization of the Ql/Qt-Element for Incompressible Flows P.

KNOBLOCH,

L.

TOBISKA

Abstract. Starting with the unstable finite Ql/Qt-element a general class of bubble-type spaces V~ is constructed and used for enlarging the space Vl approximating the velocity. It is shown that the new pair of spaces satisfies the Babu§ka-Brezzi condition. Using the techniques of reduced discretizations and static condensation some new and old stabilized schemes are obtained. Keywords: Babu§ka-Brezzi condition, stabilization, Stokes problem Classification: 65N30

1. INTRODUCTION

In this paper we introduce a general class of stable finite element spaces suitable for a numerical solution of the Stokes equations - I I .6 u

+ Vp = I'

div u

=0

in 0'

u = 0 on

an.

(1)

Here, u is the velocity and p is the pressure in a linear viscous fluid contained in a bounded domain 0 c R 2 with a polygonal boundary an. The parameter II> 0 is the kinematic viscosity and I is an external body force, e.g. the gravity. Denoting

a(u, v)

=

fo Vu · Vv dx,

b(v,p)

=

the usual weak formulation of (1) reads: Given 11 u e HJ(0) 2 and p e L~(O) such that

11a(u,v) +b(v,p)- b(u,q)

= (l,v)

-In

pdivv dx,

> 0

and

I

E H- 1 (0) 2 , find

(2)

where L~(O) consists of £ 2 (0) functions having zero mean value on 0. It can be shown that this problem has a unique solution (cf. [9], p. 80, Theorem 5.1). A conforming finite element discretization of (2) reads: Find uh E Vh and Ph E ~ satisfying

(3) where Vh c and Qh c L~(O) are finite element spaces based on a triangulation 7i. of 0. In this paper, we shall consider only triangulations consisting of rectangles T

HJ(0) 2

A Bubble-Type Stabilization of the Q1 /Q1-Element for Incompressible Flows

(cf. Section 2), thus, we implicitly assume that We are interested in equal order spaces

231

n can be decomposed into rectangles.

V~ = {v E HJ(n) 2 ; vir E Q1(T) 2 V T E 7h}, Qh = {q E H 1 (n) n L~(n}; qjT E Q1(T) V T E

7h}

for approximating the velocity and the pressure, respectively. However, this pair of spaces does not satisfy the Babuska-Brezzi condition

3.8>0: which often causes that the problem (3) with Vh = V~ is not solvable or that its solution contains spurious oscillations. One way to suppress these oscillations and to assure the solvability is to add some extra terms to the discretization (3} (cf. e.g. (6], (10], (5]). Another way is to enlarge the space V~ by a space V~ so that the BabuskaBrezzi condition is satisfied. Here we start with the second possibility and construct a general class of spaces V~ assuring the fulfillment of the Babuska-Brezzi condition. Then, by using the concept of reduced discretizations [11] and static condensation, we shall show that, for suitable spaces V~, the V~-component of uh and the function Ph are solutions of the stabilized methods of (6] and [10]. In case of the mini element (1], which is defined by enriching continuous piecewise linear functions by cubic bubble functions, the close relation between stabilized methods and Galerkin methods with bubble functions was already established in [13] and (3]. In an abstract framework, this equivalence was investigated for linear problems in (2]. It was also realized that bubble functions can help to design new stabilized methods (cf. e.g. [7], (8]). There are a lot of further papers devoted to investigations of discretizations stabilized using bubble functions, but most of them are restricted to triangular elements and to linear problems. In this paper, we deal with quadrilateral elements and, in addition, we consider more general functions than the standard bubble functions used before. Apart from investigating the relations to some well-known stabilized methods, we shall also derive, eliminating a suitable space V~ from the discretization, a new type of stabilization which can also be applied to the NavierStokes equations (12]. The space V~ added to V~ to satisfy the Babuska-Brezzi condition will be defined in a general way as Vh2

= span{u>i ... t*}l!" •=1 .

(4)

where tp' E HJ(n) and t• E R 2 are some suitable functions and vectors, respectively. The proof of the Babuska-Brezzi condition for the spaces Vh :::: V~ EB V~ and Qh, which uses some ideas of (4] and a modification of the Verftirth trick (14], requires that the functions cpi have localized supports and that, for any tpi, there exists a point A' E 0

232

P. Knobloch, L. Tobiska

such that

ttl' dx = 8qh )f 8qh Ot' .,. Ot' (A') 0

f ,,, lo .,. dx

,qh,h ~cIt} ,~. 1 :~(A')r

"'v qh E Qh, '. = 1, ... , Hh, "'

(5)

7i..

(6}

"'Qh E Qh. T E

A;ET

The plan of the paper is as follows. In Section 2, we introduce some notations and summarize the assumptions on the triangulations and the functions cp' needed for proving the Babu~ka-Brezzi condition in Section 3. In Section 4, we give examples for the functions cp' defining the supplement spaces \1. In Section 5, we investigate discretizations obtained from (3} by eliminating the \1-component of uh. In particular, we show the equivalence to the stabilized methods of [6] and [10] and establish a new type of stabilization. 2. ASSUMPTIONS AND NOTATIONS We assume that we are given a family {7i.}h>o of triangulations of n consisting of closed rectangular elements T having the usual compatibility properties. Let hT diam(T) ~ h for any T E 7i. and let the ratio of the longest and the shortest edge of each T satisfy

=

~-

h!F'" ~ Ct

with some constant C 1 independent of h. Before introducing the functions cp' used for defining V~ in (4}, we define the points A' mentioned in the introduction. Thus, for any triangulation 7i., we define a set {A'}~1 of points A' E 0 different from the vertices of 7i. and we assume that card{ A';

A' E T} ~ c2

"'T E 7i.

I

where the constant C2 is independent of h. In order to be able to use the same function

cp in (4} twice (each time with another t'}, we admit A'= Ai fori :f. j. For any A', we

denote by pi the union of the elements containing A', i.e., pi consists of one or two elements. Further, for any A', we introduce a unit vector t' E R 2 and if A' belongs to an edge of some T E 7i., we require that ti coincides with the direction of this edge. We choose a fixed Cartesian coordinate system with axes parallel to the edges of the triangulation and, for any t' = (ti, t~}, we define n' = (t~, ti}. For getting the property (6}, we assume that, for any T E 7i., there exist indices i, j, k E {1, ... , Nh} such that A', Ai, A A: E T and gi~A: = (n' x ni) nA: ·AA:+ (ni x nA:) n' ·A'+ (nA: x n') ni · Ai

satisfies

(7} where the constant c3 is independent of T and h, the vector product a as a 1 ~ - b1 a2 and n' ·A' is defined as n' · (A' - 0}.

X

b is defined

A Bubble-Type Stabilization of the Q1/Q1-Element for Incompressible Flows

Remark 1. If, particularly, n;

233

= nJ .l nA:, then = lni · (Ai- A1)1,

~~~tl

which illustrates the meaning of (7). Finally, for each Ai, we introduce a function tpi E HJ(O) such that supprpi C pi,

(8)

ni ·

(9)

£

(x- Ai) tpi(x) dx = 0

lfn rpidxl ~ c.IPil,

(10) (11)

where the constants C4 , C5 > 0 are independent of h. We shall see (cf. Lemma 1 in the next section) that from (9) the property (5) can be concluded. The functions .. N {tp' t• };=\ are assumed to be linearly independent. 3. BABUSKA-BREZZI CONDITION

In this section, we prove that, under the assumptions made in Section 2, the spaces V11. = V~ EB V~ and Q~~, satisfy the Babuska-Brezzi condition with a constant independent of h. First, in Lemmas 1 and 2, we prove the validity of (5) and (6). Then, in Lemma 3, we establish a Babuska-Brezzi condition with a "wrong" norm of q~~, and, finally, in Theorem 1, we prove the desired Babuska- Brezzi condition applying the modified Verftirth trick.

Lemma 1. We have

r

r

aq~~, id = aq~(Ai) id ln at• I{J x at• ln I{J x Proof Consider any T C pi and let q~~,(x) = {o+{lxl +{2x2+6x1x2 for any x E T, where {0 , ••• , {3 are some real numbers. Denoting Ai = (a11 ~), we have for x E T

Hence

:~(x) = :~(Ai) +{ani· (x- Ai) and the lemma follows using (9).

0

Lemma 2. For any T E r,., we have

2 54 I ~ I8qh i) 12 IQhll,T ~ C~ Tl i~, ati (A A'ET

(12)

P . Knobloch, L. Tobiska

234

Proof. Consider any T E 7i. and let AI, A2 , A3 E T be points satisfying (7). Let us set for x E T and i = 1,2,3 q'(x) = {n'+l (ni-l· A'- 1)- ni-l (ni+1 · A'+l)} · (-x11 x2) + (n'+l x n'- 1) X1 X2,

=

with the convention that i -1 3 fori shows that any qh E Q" satisfies

= 1 and i+ 1

1 ~ qh(x) = {o + 8123 ~ h

i=l

=1 fori = 3. An easy calculation

oq" n/2 where ln+ denotes the positive part of the logarithm. Combining both estimates and using Gronwall's inequality one obtains existence until the vorticity blows up- a result proven by Beale, Kato and Majda (3) in Rn and by Ferrari [1) in domains with boundary. We shall reconsider well-posedness questions in all space dimensions in C 1•0 using arguments which generalize to many other function spaces. The case n = 2 is different from the case n > 2. There the proof simplifies and results are stronger: one obtains global solutions. The velocity vector fields with Dini continuous vorticity provide an attractive setting for well-posedness questions. Our approach seems to interpolate between the approach of (8) and (4]. We want to obtain qualitative information about solutions. A by-product of the existence proof in this work is the estimate for n = 2

(1)

llu(t)llc• ~ cecllu(D)IIc•tllu(O)IIc•.a.

It would be desirable to know whether this estimate is sharp, i.e. are there solutions for which sup"' IVu(x, t)l grows exponentially? On the other hand the V-norm of the vorticity is conserved and thus IIVu(t)llv is uniformly bounded in t for all p < oo. Finally we turn to stability issues: we almost completely and explicitly describe the essential spectrum for the linearized equation. We give a strong necessary condition for (linearized and nonlinear) stability which shows that, among other things, all twodimensional shear flows are nonlinearly (Ljapunov) unstable. Linear instability criteria have been found by Vishik (6] for spatial periodic problems and applied by Friedlander and Vishik (7]. Moreover Friedlander, Strauss and Vishik (5] have shown that some particular shear flows are unstable. In this paper we restrict ourselves to a study of stability in two dimensions. We use completely different methods which yield precise results in domains with boundaries. 2. PRELIMINARIES

Let n c Rn be a bounded domain with smooth boundary r. Let C"·0 (0) with a nonnegative integer k and 0 ~ a ~ 1 be the standard space of functions with Holder continuous derivatives. We denote by V(O) and W".P(O) with nonnegative integer k and 1 ~ p ~ oo the standard Lebesgue and Sobolev spaces.

H. Koch

242

2.1. The Euler equation. We study the Euler fluids: u 1 + (u · V)u + Vp = 0 V'·u = 0 (2) U•ll = 0 u(.,O) = Uo

equation for incompressible inviscid

inn X (O,T) on n X (O,T) on r X (O,T) on n.

Here u is the velocity vector field, p is the pressure which is determined up to a function = 0 at r . One easily sees that the kinetic energy 0 lul2 dx is formally conserved. The velocity vector field u defines an evolution operator ~ by

Po(t), T is a positive number and Uo is the initial datum which satisfies Uo · 11

lJ

81 ~(x, t, s)

(3)

~(x, t, t)

u(~(x, t, s), t)

=

=

x.

Clearly 1

IV~(x, t, s)l ~ exp(/. sup IV'u(x, u)loo)

(4)



%

where lvl denotes the length of the vector and IVul = supl.,l=l,lwl=tl(v · V)(u · w)l. We will use a similar notation for higher tensors. 2.2. A representation of the solution. Let t/J = ~(t, o)- 1 and define

~.u(x) := L(~,p;u;(t/J(x)). j

Let Q be the standard projection to solenoidal vector fields. Lemma 2.1. Suppose that u is sufficiently smooth and tangential at the boundary. Suppose that ~ is defined by (3). Suppose that u(t) = Q~(t, O).Uo. Then u satisfies the Euler equation in the sense that there exists p such that (u, p) satisfies (2). 2.3. Dini continuity. We define the space Co of Dini-continuous functions as the subset of functions of L 00 for which the norm

11/llco :=max {

00 sup lf(x)- f(y)l , 11/IILao} Jo lo:-wi!S" u

(

is finite. A simple check reveals that properties of the space Co:

coo is dense in C0 .

There are three important

1. The imbedding Co C L 00 is compact. 2. Suppose that the operator T: L2 (n) ~ L2 (n) has a kernel k which satisfies

(5)

lk(x, y)l ~ clx- Yl-" and T(l) E L 00 •

Then T defines a unique operator from Co to C(fi) and

liT/lloo ~ ~11/llco·

Instability for Incompressible and Inviscid Fluids

243

3. There are logarithmic interpolation estimates:

11/llco ~

ll/~oo ( ~ + 2ln+ ( ~%~))

holds whenever the right-hand side is finite. 2.4. Relation between velocity and vorticity: n ~ 2. The spatial derivative of u is a square matrix depending on x and t. Its symmetric part is the tensor D = ~(Vu + Vut) and its antisymmetric part is the vorticity~ = ~(Vu- Vut). There are two constraints for the velocity vector field u: it is divergence free and its normal component is zero at r, i.e. there is no flux through the boundary. The velocity and the vorticity ~ are related by the problem

D.ui

(6)

U•ll

t;vi(aiui - 2~i)

= = =

28i~ij

0 siiuiti

inn on r on r

for all tangential vectors t and s'' is the second fundamental form of r . Equations (6) are the Euler-Lagrange equations of critical points of

~ fn1Vul 2 + 2~iiaiuidx + ~ frsiiuiui . As a consequence the null space is finite dimensional, u is smooth if~ is smooth, and T : ~ -+ P~-+ Vu where Vu is orthogonal to the null space and P is the projection to the range, is a singular integral operator with kernel satisfying (5). 3. WELL-POSEDNESS

We shall construct a local solution as the fixed point of the map u -+ v where v is the projection of the transported initial velocity vector field. For simplicity we only formulate the result for Holder spaces. The case of Sobolev spaces is more tedious, but it does not require new arguments.

Theorem 3.1. Suppose that the bounded domain n c Rn has a smooth boundary with the exterior unit normalv. Suppose that Uo E C 1•0 (n) satisfies u 0 • 11 = 0 and V · u0 = 0. Then there exist t 0 > 0 and a unique solution u E C 1•0 (n x [0, to]) to (2) .

an

Sketch of the proof: Let ~ be the evolution operator defined by the vector field u. The solution is characterized as fixed point of the map u-+ v =

Q~(t).Uo.

At first sight this map seems to lead to a loss of derivatives. But it implies that the vorticity is again transported as a tensor by u. There the loss of derivatives disappears. The result follows by careful but elementary estimates and standard fixed

point arguments.

0

244

H. Koch

Remark 3.2. The proof implies that the first derivatives of u grow at most exponentially if n = 2. It would be interesting to understand whether this is optimal, i.e. whether there are solutions for which the derivatives of the velocity grow exponentially or not. To the knowledge of the author the estimate by e.r•• for llu(., t)llc'·o is the best which is known up to now.

4.

INSTABILITY FOR THE NONLINEAR PROBLEM

Let n = 2 in the sequel. Nonlinear instability is a consequence of the following result and the next section. Theorem 4.1. Let u E ct+2 •0 be a solution with initial velocity Uo. Let J.L(T) = IIV¢(., 0, T)lloo and let c and R be nonnegative. There exists a solution v in Clc+l,a

with llv(O) - u(O)IIcHl,o ~ R and llv(T) - u(T)IIcHl,o ;:: R(J.L(T) - c)lc+a.

If k ;:: 1, 1 < p < oo then there exist solutions such that llv(O)- u0 llwHl,p $ R and llv(T)- u(T)IIwHl... ;:: R(J.L(T)- c)/c. Proof. The proof consists of several steps.

Step 1: There exists a point x 0 and a unit vector w such that, with x 1 = ~(x0 , T, 0), (w · V)~(x1o T, 0) = J.L(T). We suppose that x 0 lies in the interior. Otherwise we approximate x 0 from the interior. We will essentially work in a neighborhood of this

point. Step 2: Let 1Jo E Ccf(B0 (1)) have mean zero. There exists a unique Ill with ~Ill= 716,p and llllr = 0. We set v6,p the solution with initial data u 0 + (O:!Ill,-8tlll), w = v - u and 71 the vorticity of w. We omit the indices and p in the notation of w and 71· The solution V6,p has the desired properties for properly chosen p and o. Up to a term of minor order 1Jo is transported by the vector field v, which is close to the vector field u. The support of 7Jo gets distorted under the flow. This causes the growth of the norm of the vorticity. 0

o

ca

5. STATIONARY SOLUTIONS There are many stationary solutions to the 2-d Euler equation. Suppose that

(7)

~'1/J

'1/J

= =

f(.,P) 0

on on

n

r

for some C 1 function f. Then u = (0:!'1/J, -8t'I/J) is a stationary solution to the Euler equation. In particular any velocity profile for a shear flow gives a stationary solution. Every stationary solution is the Hamiltonian vector field to the Hamiltonian function .,P with respect to the symplectic structure on IR2 which comes from the identification with C. Let 41 be the flow defined by this vector field. The next lemma is surprising and crucial for the sequel.

Instability for Incompressible and Inviscid Fluids

245

Lemma 5.1. Let A be the maximal real part of the eigenvalues of the linearization at stagnation points (which we define to be 0 if there is no stagnation point). Then lnsup IV~(x, t)l '" -+ A as t -+

ItI

±oo.

The proof makes heavy use of two dimensionality and the fact that u 0 is a Hamiltonian vector field .

Lemma 5.2. Let u be a sufficiently Hamiltonian vector field on 0, tangential at the boundary. Let ~ be the associated flow. If there exists c such that IV~(x, t) I

independently of t and x then there exists T ~(x,T)

$ c

> 0 such that

= x.

Proof. If there is no regular point of the Hamiltonian t/J then u(x) = 0 for all x. Then the assertion is trivial. Let x be a regular point of 1/J. The orbit through x is periodic. There is a neighborhood consisting of periodic orbits. We choose a maximal connected neighborhood U with this property. A check of V~(x, T) at multiples of the period shows that the period is locally constant since otherwise V~ grows linearly. The boundary of U consists of elliptic fixed points. Hence U is dense if 0 is connected. This proves the assertion. 0 Remark 5.3. This is all we need for Ljapunov instability for most stationary solutions of the Euler equation. For example all stationary solutions are unstable in domains with more than one hole. All shear flows are unstable. 6.

THE LINEAR EQUATION

Let (u,p) be a stationary sufficiently smooth solution to the Euler equation. In this section we study the linearized problem:

v1 + (u · V)v + (v · V)u + Vq V·v

V·V

v

=

= =

0 0 0 vo

inn X (0, T) on n X (0, T) on r X (0, T) on n.

Let Sx (t) be the semi-group defined by this equation on the function space X which we shall specify later. We denote the set where Sx(t)- J1. is not Fredholm by the Fredholm spectrum. Here we complexify space and operator in the natural way. In this section we determine the Fredholm spectrum of S(t): if A> 0 then it is an annulus with radii e±cxA. Outside the Fredholm spectrum of Sx(t) there are only isolated eigenvalues of finite multiplicity. There are two steps: a reduction to a composition operator, and a study of the spectrum of the composition operator. We begin with the reduction. Let ~be the flow generated by u and let Sx(t) be the semigroup X 3 f-+ f o ~( -t) EX for suitable function spaces.

H. Koch

246

Lemma 6.1. 1. Let X= W""(O), W~"(O) with k:::! 0 and 1 < p < oo, their dual, or C"·0 (0) with 0 < k + o. Then Sx is a (wecJ() continuous semigroup. 2. Let X = W""{O) with k ~ 0 or C"+l,a with k + o > 0. Then Sx is a (wecJ() continuous semigroup. 3. Let A be the map {0 -t Uo defined by Uo = (8-.Jw,-~w) with ~1{1 = {0 and 1{1 = 0 at the boundary. Then, if k ~ 0

IIS(t)Uo- AS(t){ollw'+'·• ~ ciiUollw•·" and, if in addition 0

0 one can look, e.g., for decaying functions or such from the smaller space H 1 {R~; A). Fork = k1 > 0 one puts the Sommerfeld radiation condition

ara u- iku = o(r-,--) l=ll

to guarantee uniqueness of these b. v.p. 's.

as

r~OO

{7)

E. Meister, A. Passow

256

Now the second fundamental situation in diffraction theory is given by Definition 2 (Transmission problem (T.p.) for R~:). For given F = (F+, F_)T E C0 (R~) x Co(R~) or E L2 (R~) x L2 (R~}, and f E CJ(~}, g E C0 (~) {classically) or f E H,I!.,2 (~;- 1 } n S'(a,:.-1 }, g E Hj~ 12 (a,:.- 1 } n S'(a.:.- 1 ) {weakly) and constants k± = kr + ikf with kr > 0 and kf ~ 0, e± similar, find a function u = (u+, u_)T E ( C2 (R~) X OZ(R~)) n ( CJ(R~) X CJ(R~)) {classically} orE (H1~(R~;~) x H1~(R~;~))

(~n

n (S'(R~) x S'(R~)) (weakly) such that

+ (k*}2 }u* = F±

in R~

(8}

in the strong or weak sense, respectively. Additionally the two transmission conditions have to hold

(9} Tr. u

==

_!_au+ {!+

I -_!_au-~

axn a•-1 ~

{!-

axn a•-1

= g.

(10}

~

Remark 2. In order to get a well-posed problem one has to add conditions concerning the behavior of the solution u for r = 1~1 -+ oo in R±. They could be similar like in Remark 1: global L 2 -integmbility if k± E c++ or fulfilling the appropriate radiation conditions like (7} with different k 's. For incident waves

ut..,(*) ""eiJ::I:, with lg_l

(11}

= 1 one arrives at the well-known Snellius law at the interface 8R~ = 8R~ =

~;-1.

Remark 3. The above-mentioned boundary and transmission conditions may be generalized to other differential equations and systems of them of elliptic type for smoothly bounded finite domains n c Rn and 0' := RD\lt There exists a fully developed theory making use of integral representations with fundamental solutions for the Lame equations of linear elastodynamics, thermo- elastodynamics, and also for Maxwell's system - which for the stationary case is not even elliptic [2, 7, 9, 10, 14]. There has been a growing interest by physicists and engineers in the behavior of structures having singularities in their geometry, like edges and vertices, under different constant or time-changing loads, like microwaves in radar technique or seismic waves in geophysics, but also the scattering and absorption of sound waves and pulses by traffic noise shielding walls. Nondestructive testing and tomography - as inverse scattering problems - rely very strongly on the well-understood behavior of diffracted fields near the corners, at far distances, shortly after a wave hit the obstacle or a long time afterward. As the main canonical obstacles there are considered those with semi-infinite boundaries, like half- and quarter- planes, cones, octants, and pyramids. They model locally the most important geometrical singularities, not including cusps and needles.

Scattering of Acoustical and Electromagnetic Waves by Some Canonical Obstacles 257

+ Tr20 (u] = 0, Tr21 [u]

FIGURE 1.

=0

Sommerfeld problem with four media

The actual physical behavior of the scattered fields also strongly depends on the types of boundary/transmission conditions like in Definitions (1) and (2), but there is now a great demand to improve the models of boundary material response to impinging waves. The combination of different materials to composites in elasticity or microwave electrical circuits lead to the mixed problems for thin layers of dielectrics and metal coatings on such substrates [20]. Of great interest also are the chiral media. One may find some information about the state of the art in (14]. In the following two figures, you will see the basic canonical boundary-transmission problems for the scalar wave fields which arise in acoustics and for specially polarized electromagnetic waves having only one component of the electric or magnetic field The goal in the scalar diffraction vectors parallel to the infinite line-like edges. theory is to solve the initial boundary- transmission problems for the d'Alembert wave equation

(~ :

2 -

~) U(~,t) = F(~,t)

(12)

in the time half- cylinder Or := 0 x (0, T), 0 < T $ oo classically, i.e. U E C2(0r) n C 1 (nT') or in Sobolev-space valued functions with special weights w.r.t. time t to allow for a Laplace transformation

(13)

258

E. Meister, A. P8880w

Tr30 [u]

= 0,

Tr31 [u]

Bt [u] = It

=0

Q3,k3

I

0'1

Tr40 [u] = 0, Tr41 [u] = 0

--+-

Q.,k.

I I 0'4

FIGURE 2. Wedge diffraction problem with three media

existing as parameter-dependent Sobolev space functions from H1~(0; ~} when there are given (compatible} initial data U(~. +0} = Uo(~}

3tU(~. +0}

= Ut(~}

n S'(O; ~}

E $(0}.

(14}

The main questions in this context are the short-time and the long-time behavior, particularly the validity of the limiting amplitude principle and the existence of the general wave operators of scattering theory.

Remark 4. In the case of finite scatterers n such that 0' = R 0 \11 is an exterior domain a lot is /mown about those questions [10], but in the case of semi-infinite obstacles, still a lot has to be done starting with the representation of the resolvents of the stationary boundary-transmission problems, particularly for the vectorial cases in electro- and elastodynamics. 3. ELECTRODYNAMICAL BOUNDARY- TRANSMISSION PROBLEMS

Let there be given a canonical domain an. Then we have

n C R 3 with a piecewise smooth boundary

Definition 3. If there are given the electric current density J. E R 3 and the electrical charge density {! E R, then we look for a quadruple of vectorfields E R 3 , the electric fieldstrength E., the magnetic fieldstrength H, the electric displacement I2. and the

Scattering of Acoustical and Electromagnetic Waves by Some Canonical Obstacles 259

magnetic induction I1 fulfilling the Maxwell equations

curl E. + 8tB.. = Q, curl H - OtD = J.,

div I1 = 0, div D = ll·

(15) (16)

Remark 5. This system of eight partial differential equations has to be solved in the classical or weak form - for (~. t) E Or = n X (0, T) with additional boundary-transmission conditions on OOr = an X [0, T], where on n initial conditions are prescribed. Here the different types of boundary-transmission conditions will be formulated after specifying the constitutive equations Definition 4. The material equations are

/2(;r, t) = 12 (E(~. t), H(;r, t)) B(;r, t) = .li (E(~, t), H(;r, t))

(17) {18)

which in general are nonlinear functions and in general depend on time history which causes hysteresis effects. Neglecting these influences, we arrive at the simplest situation of linear dependency

D.(;r, t) IJ.(;r, t) J.(;&., t)

'=•

= ,.E.(;r, t)

= =

(19) (20)

gH.(;&., t) fl. E.(;&., t)

(21)

Ohm's law

~ and g are called, respectively, the permittivity, the permeability, and the conductivity tensors. If they are multiples of the identity tensor / 3 the material is called isotropic, otherwise anisotropic.

By energy considerations with the Poynting vector it can be seen that for lossless media the relations and hold, which specialize to ~{ e}

(22)

= ~{J.L} for '= = e/3 and g = J.L/3

in the isotropic case.

Definition 5. If n. denotes the unit normal vector to a surface S C R 3 being sufficiently smooth and Il. E R 3 will be a vector field continuous on S - or having integrable traces Ills of fields E HJ:c(O) n S'(O), m E No, S C fi, then we have

n.. A E...cls = L

and

n.. A H.cls = ~

(23)

as electric and magnetic boundary conditions for the scattered fields given by

E...c := ~ - £.""'

H.c := H..toc - H.;nc

(24)

which may be augmented by conditions for the normal components w. r. t. S

< n,.,'=E. >Is= fn

and

< n,.,gH >Is= gn.

(25)

These boundary conditions hold reasonably for electrically or magnetically ideal conducting surfaces at high frequency, but have to be replaced by such who model better the absorption and reflection properties of the scatterer's surface S. In the

E. Meister, A. Passow

260

following sections we shall concentrate on the application of the so-called Leontovich

boundary conditions of different orders, the simplest being given by n. 1\ E.ls = Zn. 1\ (n. 1\ H)is

(26)

with the impedance matrix #,. E C 3 xS. Many authors (8, 19, 20) studied this electromagnetic boundary value problem for smooth surfaces in the isotropic case. Like for acoustics or elastodynamics there are scattering problems for different, not ideally conducting media in R 3 -space leading to Definition 6. Let there be given at least two domains

n and n' c

R 3 such that

n u n' = R a and n n Q' = 0 but S = nn IT ::/- 0 being an oriented piecewise smooth surface. Let ~. ~ and e_, e_' be permittivity and permeability tensors, respectively, continuously depending ~n ;: E -

0

ft,

and ~ E

and being strictly Hermitean -

here

for~(*.)

'v'{ E C 3 , then

*. E Q,

(27)

E. and H fulfill the transmission conditions at the interfaceS, if n, 1'1£,x = 'fl. I\ lL.n

< n,, ~E.tot > < n..gH..un >

nl'll:4ot,

(28)

n" 1£.,,

(29)

=

J.b)T E with Y±

= Yla~ E [Ho,t (R!) x Ht,o(R!>J2

which is a weak solution of

a -(

azyin

R~

0

(iwJ.'o)- 1 (k~~

+ D.)li

-(iweo)- 1 (k~~ + Il)l:f_ ) 0

= 1, 2;

(42)

y

(43)

with the differential matrix operator n._ ( g.-

~ tfJ 8%8!1

and the 1r/2-rotation operator

li=

i)

(0-1) 1

0

'

(44)

(45)

and fulfills the Leontotlich-boundar7J conditions

±/iy± + z.±·~= l± 10 ~

(46)

and the transmission condition

(47) for Yl~o e [H_M(R2 ) x Hkt(R2 )] 2 ; l = 1,2; the traces of the electrical and magnetic tangential fields on z = 0. These data l± have to fulfill the compatibility condition

t+ - r e [k_1 0, the map (x, 8, t) ...... u(x, 8, t) - u(x, 8, t) - at attains its max value on the boundary of [-R, R] x 1 x (0, T], and then of sending a to zero and R to infinity. Corollary 1 (uniqueness). A classical solution of the DCMA associated with a given initial datum Uo E C~ is unique. In order to ensure the existence of classical solutions of the DCMA, we now restrain the space of initial data. Definition 4. For n ;;;: 1, we write exists a movie v E cy- 1 such that

V:

u8 + vu.,

the space of movies u E

=0

on

IR x 1.

C~

for which there (3)

v is called a velocity map of u. In addition, the space V:"' is defined as elements of 1 C~"' admitting a velocity map v E ".

c:-

L. Moisan

276

Remark: Consider a movie u E v;. When u.,(x, 9) = 0, (3) implies u,(x, 9) = 0, and 2 if n ~ 2, differentiating (3) with respect to 8 and x shows that uss + 2vus:z: +v Un = 0 as soon as U:z: = 0. A consequence is that if u E DCMA, then any velocity map v of u satisfies on n

v:•

1

is a classical solution of the

(4) We now build explicit solutions of the DCMA. As we said before, the main idea is to notice that the trajectories (i.e. the curves x(9) along which u is constant) are smoothed by the monodimensional heat equation. For that purpose, we need to introduce the natural domain I* for such trajectories. If I =]9t. 92 [ then I* = I, and if I = S 1 , then I* = IR (the natural injection S 1 01 and for some a, b depending on N. This implies that we cannot leave the linear term on the right-hand side and we must involve it into the left-hand side. The weighted estimates for the transport equations are proved in [7]. Now, on the left-hand side of (13} we have an operator which is similar to the (classical) Oseen problem. We shall call such a problem modified Oseen problem. It can be verified (see [10]) that the properties of the modified Oseen problem are similar to those of the Oseen problem and sometimes we can directly apply the results for the 1 More

precisely, there appear some logarithmic factors, see [6).

286

M. Pokorny

(classical) Oseen problem (as e.g. the weighted V'-estimates); sometimes only very

small changes are needed. Let us study the linearized system (13)-(17). Our proof is based on the following version of the Banach fixed point theorem.

Theorem 1. Let X, Y be Banach spaces such that X is reflexive and X 0, C2 = Kf/2 > 0, Ca = C2()2, C4 = {)2f2 and 'Ph, r.Oh are arbitrary elements of vh. Now, let us consider the first ordinary differential equation, in its weak form; we have fort< T

On the Galerkin Method for Semilinear Parabolic-Ordinary Systems

305

Set 1/Jh = u2,h(t) - Wh E wh, where wh is an arbitrary element of wh. As before, this yields

~ !IIU2II~ $ CsiiU2II~ + ~II(U2)ell~ + ~llu2- whll~ + K:ll8dl~ + K:IIP1II~ + K:IIU311~, (21) with Cs = 2 + K?. Moreover, choosing (u2,h)e - tPh E Wh in (20), where tPh is an arbitrary element of Wh, we have for every t E]O, T[

II(U2)ell~

$ ll(u2)e-

~hll~ + C6ll81ll~ + C611Pdl~ + C6IIU2II~ + C6IIU311~,

where C6 = 10K?. The last inequality can be combined with (21) yielding fort< T

~i1tiiU2II~ $ ~ll(u2)e- tPhll~ + ~llu2- whll~ + qiiU2II~ + qll81ll~+ +qiiPdl~ + q11u311~. for suitable constants q e q. Similarly, the second ordinary differential equation yields the inequality

(22)

~1tiiU311~ $ ~ll(u3)e- 4ihll~ + ~llu2- whll~+

{23) +qiiU311~ + qll8dl~ + qiiPdl~ + C~IIU2II~, for 4>h, w arbitrary elements of Wh and for suitable constants q, q > 0. Now, we sum up (19), (22) and {23) and integrate over [0, t], t < T; using {17), for a suitable constant C we get

1

ll81(t)ll~ + IIU2{t)ll~ + IIU3(t)ll~ $1181(0)11~ + IIU2(0)II~ + IIU3(0)II~+

1

+

[ll(u2)e(s)-

tPhll~ + 5llu2(s)- whll~ + ll{u3)e{s)- 4ihll~ + 5llu3{s)- whll~+

+Ch 2 llul(s)- 0 sufficiently large as to include the initial datum Uo,h = {ul,O,ht u2,o,h, UJ,o,h) in a closed neighborhood Ec~ of E and let Ki > 0 be constants such that the Lipschitz condition (8) holds in E6 • Heuristically we might argue that, since the approximate solution uh is always going to be close to u, it belongs to E6 • In order to show that this is the case, we have to provide maximum norm estimates for the approximation error, since closeness in the sense of £ 2 or H 1 does not automatically imply that uh belongs to E6 for small h. Now we state the main theorem of this section. Theorem 2.5. Assume that 9i• i = 1, 2, 3, satisfies the Lipschitz condition {8}. Let {Vh} be of class S 1,. (JJ. ~ 2} and {Wh} of class S1,.• (J1.' ~ 2} and suppose that {25}, {26}, {27} and {28} hold for some v, v', with v < J1. and v' < J1.1• Then, if the solution u is such that u1 E H 1{0, T; H~'(n)) and u2 , u3 E H 1{0, T; H~'' (n)), there exists an ho such that, for h :5 ho, the solution uh of {12) exists fort :5 T, and for these t we have lluh - ullo :5 c hmm(,.,,.,). Proof. Let th be the largest number less than or equal to T such that uh exists and belongs to E6 fort :5th, i.e. th = sup{s :5 T : uh(t) E E6 'Vt :5 s}. Similarly to Theorem 2.2, we get the estimate (24) fort < th and, taken into account the assumptions on vh and wh and the smoothness of u, we have

ll(ul,h(t),u2,h(t),u3,h(t))- (u1(t),u2(t),u3(t))llo :S chi sup{l/'(w)l; wE /(u, u,.)}, where l(b, c) denotes the interval with bounds band c. As is well-known (an analysis of the linear stability is convincing), the uniforrr stability of an approximation as T-+ o+ is strongly related to the asymptotic stabilit~· as t -+ +oo with a fixed value of T, say T = 1. On the other hand, the semi-group;; defined by either (2) or {3) share the properties ofT. This is clear for the viscous approximation, whereas it has to be slightly adapted for the relaxation in the following way: we define the characteristic variables w, z :=au± v, which satisfy Wt

+

aw., = I (w2: z) + z~ w,

Zt-

az., = -I (w 2: z) + w; z.

The right-hand sides are monotonous increasing/decreasing with respect to w or z, within the box [A,C] x [B,D), provided a~ sup{ll'(u)I;A+B :5 2au :5 C+D}. This

D. Serre

314

box is then a positively invariant domain, so that the semi-group has the following properties (see [9]): : It is contracting for the L 1-distance

d1((u, v), {u', u')) := llw- w'lh +liz- z'lh· : It is mass-preserving, provided v0

-

v0 "vanishes at infinity":

L sup{l/'(m)l; mE [A, B]}. Then lim llu(t)- u+lll + llv(t)- /(u+)lh = 0. t-++oo

Let us make two remarks. First, we do not need to assume any integral condition on v0 , since the system (3) preserves only the mass of u (or u- u+ here). Second, the condition of zero mass is a meaningful restriction in Theorem 3 and Conjecture 2, since we cannot come back to this case by a shift of the constant state. When the integral does not vanish, we expect that a diffusion wave develops as t increases. In both cases, this wave spreads on a domain of width ../t and decays in L 00 as t- 1/2; however, it does not decay in L 1. We shall prove Conjecture 2 in the linear case. Theorem 4. Let f be affine: f(u) = bu + c with lbl < a. Let u+ be a corutant and let Uo E u+ + L\ v0 E f(u+) + L 1 be such that the disturbance has zero mass:

L 0.

1/(u(x,t))l ~ Mu(x,t) 2 ,

1.2. A nonlinear inequality. For such a data, we define A(t) := sup llu(s)ll 2s 114 , 0:99

which is a non-decreasing function. By the Duhamel's principle, u(t)

= K1 • Uo -fo'(a.,Kc-.) • f(u(s))ds,

where K is the heat kernel. Using a Young inequality, we obtain llu(t)112

~ IIK1 II2IIUolh + fo'118.,K 1- 'II2IIJ(u(s))ll1ds.

This yields 1!u(t)ll2t1/ 4

+ c2Mt1/ 4 fo' (t- s)-314 llu(s)ll~ds

~

c11!Uolh

~

cd!Uolh + C2M A(t) 2t 114 cd!Uolh

fo' (t- s)-31's-112ds

+ c3MA(t)2.

Therefore

(8) Actually, to-+ u(t) is continuous with values in £ and vanishes at t = 0. We now make the restriction

2

,

so that to-+ A(t) is continuous

(9) where o := (4c1c3M)- 1. Then the roots of the equation X = cd!Uo lh + c3MX 2 are real distinct positive numbers. By continuity, A(t) remains smaller than the least one. This root clearly is bounded by c,I!Uolh· Therefore: Lemma 1. Let uo E L 1 n L 00 be such that I!Uolloo ~ 1 and IIUolh ~ \lu(t)112

:5 c,IIUolht-

114

,

t

> 0.

o. Then

D. Serre

318

Let us point out that up to now, we did not use the assumption (6). Let us also

point out that this lemma is not the best possible. Abourjaily and Benilan [1] proved

that llu(t)ll 2 ~ {2t)-114 11Uolh, without smallness assumption on the data. Their idea consists of writing the energy equality llull:z! llull:z +

llu,.ll~ = 0,

then to write a Sobolev-Gagliardo-Nirenberg inequality !lull~ ~ 2llull~llu,.ll 2 • Last, one uses the decay of llulh· They obtain the following inequality:

d

211Uoll1 dt llull:z + !lull~ ~ 0, from which the conclusion follows easily. 1.3. The decay for small data. We still assume that IIUolloo ~ 1 and IIUolh ~ J. We use again a Young inequality: llu(t)ih

~ IIKt • Uolh + lua,.Kt-•lhlif(u(s))lltds t

ds llu(s)ll~. ~ yt- s t ds

$

liKt * Uolh + c5 lo0

$

IIKt * Uoih + Ci;IIUoll~ lo0

Jt (t - s}

= IIKt * Uolh + C711Uoll~. Because of {6), we know that

IlK' • Uolh

tends to zero as t -t

+oo.

Therefore

l := lim llu(t)lh $ c711Uoll~· t-++oo

However, since llu(s)lh and llu(s)lloo decay, u(s) satisfies all the assumptions used above, so that, replacing t = 0 by t = s, we also have l ~ c7llu(s)ll~· Passing to the limit as s -t +oo, it comes l ~ c 7l 2 , where we already know that l ~ IIUolh· Finally, we have Lemma 2. Let Uo E £ 1 n L"" satisfy {6} and IIUolloo $ 1,

IIUolh $ min{4,1/c7)·

Then lim llu(t)lh

t-++oo

= 0.

1.4. The conclusion. It works as in the proof of [3]. Let define X to be the metric space of measurable functions v E L 1 n L"" such that llvlloo $ 1 and fR vdx = 0, endowed with the £ 1-distance. It is convex, thus a connected space. It is positively invariant by the flow defined by equation (2), because of maximum principle and mass conservation. Let B be the attraction basin, in X, of the origin. On one hand, the semi-group is £~-contracting, so that B is closed. On the other hand, we proved (Lemma 2) that B is a neighbourhood of u = 0; since the semi-group is continuous, this implies that B is an open set. Finally, B is non-void. We conclude that B = X.

L 1-decay and the Stability of Shock Profiles

319

Now we recall that the solution enters X provided Uo E L 1 n L 00 • Thus Theorem 3 is valid with the condition that uo be bounded. Last, by density, this condition is removed.

QED

2.

THE CASE OF LINEAR RELAXATION

Here we consider the linear system with damping, which is the linear case of the relaxation model. Up to a change of variable, we choose c = 0 and u+ = 0: Ut

where lbl

+ v., =

0,

Vt

+ a 2 u., =

bu - v,

(10)

< a. Here we prove Theorem 4, that is ( uo, vo

eL1 and JRf u.odx = o) ~

lim llu(t)lh + 1iv(t)ll1 = 0.

t-++oo

As before, the contraction property allows us to restrict to a dense subset of initial data, namely to bounded data of compact support. Let (u.o, v0 ) be such a data, supported in [-L, L]. Then (u(t), v(t)) is supported in [-L- at, L +at]. Moreover, fR u(x, t)dx remains null. Therefore we may introduce a potential function p(t), compactly supported, so that u = p., . Then v = Pt· Let us now define 1 E(u, v,p) := a 2 u2 + v2 + -p2 - pv, Q(u, v) := a2 u2 + v2 - 2buv,

2

which are positive definite quadratic forms. The following identity holds:

pu) + Q = 0.

OtE + 8., ( 2a2 uv + ~p2 - a 2 Integration on R x (0, T) yields

kE(u,

v,p)t=rdx +

kTkQ(u, v)dxdt = kE(u.o, vo,Po)dx < +oo.

This implies that u and v belong to L!,t· On the other hand, (10) may be rewritten in characteristic form as Wt

+ aw., =

"fZ -

ow,

Zt - az.,

= ow -

"(Z,

where "(, o = 1/2 ± b/2a are positive numbers. The functional

J1(t) := llw(t)lh + llz(t)lh is a Lyapunov function (this is a particular case of Natalini's results). Let l be its limit as t -+ +oo. The Cauchy-Schwarz inequality gives

l ~ llw(t)lh + llz(t)lh ~ ..j2(L +at) (llw(t)ll2 + llz(t)ll2), so that

l2

L +at ~ 4{llw(t)ll~ + llz(t)ll~).

Since the right-hand side is integrable on (0, +oo), l must be zero.

QED

D. Serre

320

2.1. The case b =±a. Let us consider for instance the case b =a. Then

the equations

w 1 + aw., = z, can be integrated by hands. One obtains

Zt -

az.,

o= 0 and

= -z

u(t) = Uo(·- at)+ L 1Zo, where

L 1Zo(x) := e- 1Zo(x +at)- Zo(X- at)+

/o' e-•Zo(x- at+ 2as)ds.

Using the shift Tat : y t-+ y +at, one has

(raeL1Zo)(y) = e- 1Zo(Y + 2at)- Zo(Y) +

/o' e-•Zo(Y + 2as)ds.

The first term of the right-hand side converges to zero in L 1• The second one converges since it does not depend on time. The last one is the convolution by the kernel 1

m'(s) := 2a e•/24X(-2Gt,OJ· By Young's inequality, it converges in L 1 to m 00

* Zo·

Finally,

lim llu(t)- (Uo + MZo)(·- at)ih

t-++oo

where we define

MZ(y) :=

= 0,

fo+oo e- Z(y + 2at)dt- Z(y). 1

This calculus shows that the conclusion of Theorem 4 does not hold in this borderline case: u(t) does not decay to zero in general. We may give a more precise statement as follows. The condition for llu(t)lh converging to zero is M Zo + Uo = 0. This is equivalent to the differential equation ~ = U:, - Uo/2a. Since Uo is given with the condition (6), we know that there exists a unique absolutely continuous function p 0 such that p~ = Uo and Po(±oo) = 0. Then Zo = Uo- Po/2a. The hypothesis ZoE L 1 is ensured if we assume moreover that Uo E L 1 ((1 + lxl)dx). Acknowledgment: The author is very much indebted to Ph. Benilan and Kevin Zumbrun, for fruitful discussions on this topic. REFERENCES

[1) C. Abour;aillf, Ph. Blnilan, Symmetrization of quasi-linear parabolic problems. Preprint (1996), Besan~n. (2) J.-F. Collet, M. Rtude, Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography, Z. Angew. Math. Phys. 47 (1996), pp 400409.

[3) H. Freutilhler, D. Serre, L 1-stability of shock waves in scalar viscous conservation laws. Comm. Pure Appl. Math. 51 (1998), pp 291-301. [4) Shi Jin, Zhouping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), pp 235-277. [5) S. Kruzkhov, Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order. Dokl. Akad. Nauk SSSR 187 (1969), pp 29-32.

L 1-decay and the Stability of Shock Profiles

321

[6] C. LllttGJUio, D. Serre, Stability and convergence of relaxation schemes towards systems of conservation laws, in preparation. [7] T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), pp 153-175. [8) C. Ma6cia, R. Natalini, L 1-Nonlinear stability of travelling waves for a hyperbolic system with relaxation. J. Differential Equations 132 (1996), pp 275-292. (9] R. Natalini, Convergence to equilibrium for the relaxation approximation of conservation laws, Comm. Pure Appl. Math. 49 (1996), pp 795-823. [10) D. Serre, Systemes de lois de conservation. Diderot, Paris, 1996. (11] D. Serre, Relaxation semi-lineaire et cin~tic.ue des systemes de lois de conservation. Ann. Inst. H. Poincar~ Anal. Non-lineaire, to appear.

D. Serre, Unite de Mathematiques Pures et Appliquees, (CNRS UMR #128), ENS Lyon 46, Allee d'ltalie 69364 LYON Cedex 07

On Positive Solutions of the Equation .6U + f(lxi)UP = 0 in Rn, n > 2 TADIE

Abstract. Positive solutions of the problem (E) below when f is nonnegative and decreasing have been investigated thoroughly by many authors (see references herein). Here we provide some sufficient conditions for their existence even in some cases where f (or/') changes sign. Some estimates of the solutions are provided. Keywords: semilinear elliptic equations, super-sub solutions methods Classification: 35B05, 35J65

1. INTRODUCTION Consider the problem E(U) := b.U + l(lxi)CJP = 0 { U(O) > 0; u ~ 0 'Vx E Rn

in Rn; n

~

3;

(E)

where 1 E 0 1 ( (0, oo)) is positive at least in some (0, p) and satisfies everywhere for some 02 ~ 0

02 + lxla ll(lxl)l ~ (1 + lxl)';

2 + o > 0;

8 ~ 0.

(f)

This work investigates some conditions for the existence of radial solutions (i.e.

u(x) depends only on r := lxl) of (E). Such solutions will be in 0 2 ([0, oo}) if I

no

2 ((0, oo}) otherwise. The case where is continuous at r = 0 and in 0([0, oo}) ~ 0 and decreasing has been thoroughly investigated in the literature (e.g. [2] - [6]}. The novelty here is that we allow I not to be decreasing in the whole~· When I is positive in some (0, p), (E) has a solution which is positive in some [0, R) and can be extended by a solution in the whole~ (see e.g. [5]). The super- and subsolutions will be the main tools for our investigation. So we recall some results for nonnegative and continuous 1: Rl: ([6], [7]} The existence of a decreasing weak supersolution v (i.e v E 0 1 (~}; v ~ 0; v' ~ 0 and E( v) ~ 0 in ~ a.e.) is a sufficient condition for the existence of a classical (nonradial} solution and of a weak (in 0 1 (~}) radial one, each of which is bounded above by v;

l(r)

323

On Poaitive Solutions of the Equation ••.

R2: ([1] , [2]) Let I be decreasing; if a classical solution u of (E) satisfies for large lxl and some ao, k > 0; u(x) = aolxi-A: + g(x) where lg(x)l \, 0 and IVg(x)l = o(lxi-A:- 2 ) then u is radially symmetric around some x 0 E Rn;

R3: ([7])

For more general l(r, w), if a radial solution u is positive in [0, R) with u(R) = 0 the Pokhozhaev's identity reads with F(r, T) := J0T l(r, s)ds

Rnu'(R)

2=foR r"- 1HI(u)dr

where

Hl(¢)(r) := 2nF(r, ¢) + 2rFr(r, ¢)- (n- 2)¢1(r, ¢).

(Po)

(3i) If H I(T) $ 0 VT > 0, then (E) has a classical and radial positive solution; if 00 in addition / 0 r l(r, T)dr = +oo 'IT > 0, then that solution decays to 0 at oo; (3ii) Let v be a radial and decreasing weak supersolution of (E). If for some to > 0 (a) Hl(v) < 0 for r >to and HI(¢)$ 0 for r >to and any 0 $ ¢ :5 v; (b) 3Ro >to; VR > Ro f 0tor"- 1 HI(v)dr+ J::r"- 1 HI(v/2)dr < 0, then (E) has a radial and classical solution u such that 0 $ u $ v;

R4: ([9])

Let u be a positive solution of (E) with u(x) = 0( lxl- 11 ) at oo; b > 0 and l(x) ~ lxl_.,. at oo; Then if T + pb > n, b = n - 2. Main results: Theorem 1.1. {Uniqueness) Consider in lp := {0, p), (p E ]0, oo]) the "radial" problem in

l::.U + g(r, U) = 0; r := lxl { U'(O) = 0; U > 0 in lp;

T;:::

0.

an

>0 U(p) = 0

(P)

nc

1 where g E C([O, oo)) ((0, oo)). lf'lr E lp and T > 0 1) 8g(r, T)/8T < 0 or 2) "llf(r, T) := 8{ g(r, T)/T }/8T does not change sign, then (P) has at most one solution in C 2 ([0, p)).

Theorem 1.2.

Assume that I is nonnegative and ll(lxl)l ~ lxla /(1 + lxl) 6 • 1) For any p > (n +a- 8)/(n- 2), (E) has a (nonradial) solution U and a weak radial one in C 1 (~) such that ix(t~l' U(x) is bounded. 2) i) Let p;::: (n + 2 + 2a)/(n- 2) and 8 < 2 +a. Then (E) has a slowly decaying (i.e. r"- 2 U is unbounded) radial solution.

Tadie

324

ii) Let 8 < 2 +a and p >Po := (n + 2 + 2a- 29)/(n- 2); a} if n < 2( a+ 2 - 9) or otherwise b) 1 < m < 1 + 2(a- 9)/(n + 28- 2a- 2) with

p = P(m,a,9) := 1 + m(2+a- 9)/(n- 2),

then

U ~ r 2 -n

(E) has a (not necessarily radial} solution U such that Moreover, U is radial iff is decreasing, 8 ~ 1 +a and Po< p < (n + 2 + 2a)/(n- 2).

at

oo.

Theorem 1.3. a) If either Hf(J)(r):= {n + 2- p(n- 2)}f(r) + 2rf'(r) ~ 0 'r/r > 0 b) or 3R1 > 0 such that Hf(1)(r) > 0 in (0, R1), Hf(1)(r) ~ 0 for r ~ R1 and 'rJT > 0 limpl'oo J~, rn-lHf(r)(r)dr = -oo; then (E) has a radial solution. Theorem 1.4. Assume that f is nonnegative, f' changes sign and for some Ro > 0, 'rJt > 0 H f(t)(r) > 0 in (0, Ro) and H f(t)(r) < 0 for r > Ro. Then 1} Hu := 6u + f(r)u" = 0 has a radial solution uo such that 0 < u 0 ~ v in [0, Ro]; uo has an extension Uo, which is a radial solution of Hu = 0 in 14; it could be a solution of (E) or a crossing solution (i.e. 3R > Ro such that Uo(R) = 0 and U0(R) < 0}. 2} If in addition 'rJt > 0 limpl'oo rn-l H f(t)(r)dr = -oo then (E) has a solution U such that 0 ~ U ~ v .

J;,

Remarks 1) Theorem 1.3 applies also when f(r, u) := fo(r)u" + g(u) with fo satisfying (!) and specially when g(u) < 0 for u > 0. 2) If for f in Theorem 1.2, a E (-2, 0), then (E) has a radial and continuous solution which is positive in some [0, Pl)· In fact it is enough to notice that for U(r) = u(O}- dt{f:(sft) 0 f(s)u(s)"ds} and fl(s) :=s-o /(s), IU(r)- u(O}I ~ SUP[o, r]{t 0 +2fl(t)u(t)"} for small r > 0.

J;

2. PRELIMINARIES

A supersolution for (E) ([6]):

define for some b, A

v(r) := (1 + (Ar) +o )2

Simple calculations show that for

b--

r

tOr

6v + 'rip~

11

;

>0

2 +a> 0.

(2.1)

C(m) := (n~ 2 ) 2 (m- 1), n-2 ( )

m2+a C(m)A2+0 r 0 vP

(1 + Ar)B

P(m,a,9) := 1 +

and

m

> 1,

~ 0 in [0, oo) m(2+a- 9) . n- 2

(2.2)

On Positive Solutions of the Equation •..

325

For f(r, v) = r 0 (1 + Ar)-11 v",

vJ>+lra

H f(v)(r) = (p + 1)(1 + Ar)ll+l {Ar(n + 2 + 2a- 29- p(n- 2} )+ + n + 2 + 2a- p(n- 2)}

( . ) 23

whence

(•) • p ~ n+2+2a n _ => Hf( v )() r $ 0 'Vr > 0 2

P.a := n + 2 + 2a- 29 < p < n + 2 + 2a => Hf( v )(r ) < 0

(n")

'f 1

(2.4}

r>

R A :=

n- 2 n-2 n + 2 + 2a- p(n- 2} {p(n- 2} + 29- (n + 2 + 2a)}A ·

If a decreasing and bounded function

(2.5}

f satisfies for some R 1 > 0,

{n+2-p(n-2}}/(r}+2r/'(r}(R) = v.(R). For 4> = v, b.(v- 4>) ~ B(v- 4>) in JR or for a:= n- 1 (r"(v'- 4>') )' ~ Br"(v- 4>) in lRi r"(v'- 4>')1r=O = 0. If we suppose that 4> > v in [0, T), T ~ R then v' < 4>' and v "decreases" faster than 4> in JR; That conflicts with the fact that v.(R) = 4>(R). Thus 4> ~ v in JR. Similarily we obtain that w ~ 4> in JR. SoT maps C{IR) toE:= {u E C(JR); w ~ u ~ v in JR}· The continuity of g and the fact that v E C 2 imply that TC(IR) c En02{JR) by elliptic theory. Because E is closed and bounded, T has a fixed point u, say, b.u - Bu = -g(r, u) - Bu in JR) which is such a required in En C 2 (JR) {i.e. solution. For the desired extension, consider the sequence (Ui)i> 2 where for ~ := i.Ro, Ui is the solution of Gu = 0 in [0, ~]; uHO) = 0; Ui(r) = v.(r) for r ~ ~. For any k > 2, Gu~; = 0 in [0, R~;]; u~el(o, R,] = u1 ' n and R4 applies. 0

On Positive Solutions of the Equation .. .

3.

327

PROOF OF THE THEOREMS

Proof of Theorem 1.1. Let u and v be two solutions of (P) with u(O) > v(O) and a := n- 1 (note that if u(O) = v(O) then u = v in lp is a well-known result). 1) As {r11 (u' -v')}' = r 11 {g(r, v)- g(r, u)} in lp with (u' -v')lr=ca = 0, there is R > 0 such that (u'- v') > 0 in (0, R) . Thus u(r)- v(r) ~ u(O)- v(O) > 0 in [0, R] and u > v as long as v ~ 0. 2) Let W(r) := u'(r)v(r)- u(r)v'(r). From the equations of u and v, vu" + (a/r)vu' = -vg(r, u) and uv" + (afr)uv' = -ug(r, v) whence (r 0 W)' = r 0 uv{g(r, v)/v - g(r, u)/u} with W(O) = W(p) = 0. This implies that Jt r 11 uv{g(r, v)fv - g(r, u)/u}dr = 0 and that cannot hold if 'Vr > 0, t t-+ g(r, t)/t is monotone in t > o.• Proof of Theorem 1.2. 1) For p > (n +a- 8)/(n- 2), 3m > 1; p = P(m, a, 8) whence v in (2.1)-(2.2) is a supersolution for (E) and R1 applies. 2)As 8 < 2 +a, 3m > 1 such that p :=Po = P(m, a, 8). i)If p ~ (n + 2 + 2a)/(n- 2) then Hf ~ 0 'Vr > 0 and if 00 8 < 2 +a, 0 r f(r, T)dr = +oo and 3i) applies. The slow decaying of the solution follows from Lemma 2.5 of [4) . ii) From (2.4), p >Po==* H/ ~ 0 in some (R,oo). Lemma 2.4 implies that U(x) ~ O(lxl 2 -n) at oo. Finally in (2.10) we have IVt/.11 = o(lxl-n) if 8 ~ 1 +a . R2 completes the proof. • Proof of Theorem 1.3. a) It follows from 3i) of R3. b) Let v be a positive and decreasing supersolution of (E). Suppose that f > 0 in [0, R 1 ). By Lemma 2.1 and Theorem 2.2, 'VR > R 1 , (E) has a radial solution u such that 0 ~ u ~ v in [0, R] with u(R) = v.(R). Define H/(V)(r) := min{H/(v)(r) I r E [R1 , R]}. R can be chosen such that 1 R rn- 1 Hf(v)(r)dr+ J~ rn- 1 Hf(V)(r)dr = 0. For the nonincreasing extension of 0 these u and that R (see Theorem 2.2), J:rn- 1 Hf(u)(r)dr < 0R 1 rn- 1 Hf(v)(r)dr+ J~ rn- 1 Hf(V)(r)dr = 0 and from (Po), u ~ 0 'Vr > 0; in fact for r > R we cannot have u(r) = 0 and

J

J

J

u'(r) < 0. •

Proof of Theorem 1.4. Part 1) follows from Theorem 2.2 and the principle of the extension of local solutions for elliptic equations. Part 2) is an application of Theorem 1.3.• REFERENCES

1. Amick C .J . & Fraenkel L.E., The uniquene., of Hill 's spherical vortu, Arch. Rational Mech.

Anal. 92 (1986), 91-119. 2. Gidas B., Ni W .M. & Nirenberg L., SJ~mmetry of positive solutions of non linear elliptic equations in Rn, Math. Anal. Appl. A; Adv. in Math. Suppl. Stud., A (1981), 369-402. 3. Gidas B . & Spruck J., Global ond local bevavior of positive solutions of non linear elliptic equation.., Comm. Pure Appl. Math. 34 (1981), 525-598.

328

Tadie

4. Kawano N.,Yanagida E. & Yotsutani S., Structure theoreYM for positive radial solutioM to .O.u + K(l2:1)uP = 0 in Rn, F\mkcialaj Ekvacioj S8 (1993), 557-579. S. Ni W.M., On the dliptic equ11tion .O.u + K(2:)u -1, a similar result can be obtained using suitable travelling wave solutions as barriers. If on the contrary inf (To (·)) < -1 the solutions of (5)-(8) might develop singularities in a finite time, a fact that was firstly derived in (18). By singularity formation for (5)-(8) we understand that the velocity of the interface ~~ (t)l becomes unbounded in finite time. Taking into account (6), (7), this implies that the temperature function becomes very steep near the interface as one approaches the time of formation of the singularity that we will denote from now on by t• .

331

Singularity Formation for the Stefan Problem

In the paper [8] several questions were analyzed about the possible continuation of the solutions of (5}-(8} after the formation of the singularity. A complete classification of all the possible asymptotics of (T (x, t), s (t)) as t -+ (t•)- was recently derived in [12]. At the time of formation of the singularity the interface s (t) approaches to a point that we will assume to be the origin without loss of generality. It then turns out that the interface exhibits one of the following behaviours: s(t) = 2(t* -t)lllog(llog(t• -t)l)li (1 +o(1}} as t-+ (t*)-,

(9}

or: S

(t) =

c (t*- t):O (1 + 0 (1}}

as t-+ (t*)-

(10}

1

where C > 0 and m = 3, 4, ... In [12] the asymptotics of the temperature T (x, t) as t -+ (t•)- , as well as the final profile at the singularity time were also computed. More precisely, in the cases (9}, (10} above we have:

T (x, t*) T (x, t*)

1 as 2llog (I log (I xi) I) I -1 - K lxlm-:1 as X -+ o-

-1 -

X I

-+

o-

1

m = 3, 4, ...

(11} (12}

where K E R. In higher dimensions the situation is rather different. Notice that (6} as well as (11}, (12} imply that T (x, t) develops a jump near the interface of value -1 as t approaches t• in the one-dimensional case. On the contrary, it was pointed out in [15] that in higher dimensional situations another mechanisms of singularity formation should exist, for which the jump of the temperature (or undercooling) at the origin would be arbitrarily small. In [19] a mechanism of cusp formation in dimensions two and three for (1}-(4} was described using matched asymptotic expansions. Detailed asymptotics for the temperature and the inteface profile were derived there. For instance, it was obtained in [19], that in the two-dimensional case the interface behaves asymptotically at the time of formation of the singularity as: lx2l "'K

1 lx 1 llog (llog lx1ll)ll

as lx1l-+ 0, K

> 0.

(13}

where we have assumed, without loss of generality, that a singularity appears at x = 0. For this mechanism of singularity formation the undercooling at x = 0 can take any value in the interval (-1,0}. The constant Kin (13} is a function of the local undercooling at the singularity, and it turns out that it approaches zero if the undercooling approaches zero, and on the contrary K approaches +oo if the undercooling approaches -1. In other words, as the undercooling approaches -1 the cusp formation mechanism above resembles more the one-dimensional singularity previously described.

332

J.J.L. Velazquez

Formula {13) describes the asymptotics of the interface just at the time of formation of the singularity. It is interesting to remark that for times t < t• , the interface is smooth. At distances (t•- t)l llog (llog (t•- t)l)ll from the origin, the interface resembles a parabola of size

(t"-t)i

llos(llosW-tllll

i. The onset of such a parabola is due to the

fact that the tip of the cusp can be approximated, after a suitable rescaling, by an explicit travelling wave of {1)-{3) that was discovered by Ivantsov {cf. [14]). In [11] the problem of the stability of the singularity mechanism so far described was addressed. More precisely, in that paper, the evolution of small perturbations of one-dimensional solutions with the asymptotics (9) was considered. Detailed asymptotics were obtained showing that under small perturbations, a cusp formation as in {13) takes place. These results strongly suggested that cusp formation as described in [19] is the generic singularity formation mechanism. However, it was recently obtained in [20] that besides the singularity formation mechanisms previously described, other possibilities exist. More precisely, the way in which a corner formation can occur in two and three dimensions was examined. It is apparent from the arguments therein that many other singularity formation mechanisms can be obtained for {1)-{4) in the undercooled case. The reason behind such a high degree of arbitrariness in the possible singularities of the undercooled Stefan problem is related to a well-known physical instability, that is called the Mullins-Sekerka instability {cf. [17]) and that we will describe shortly. In the analysis of Mullins and Sekerka, small perturbations of a planar front solving {1)-{3), moving at a constant velocity c in the undercooled case, were considered. Let us write the position of the interface as (14) where I (x2 , t) is a small perturbation. Solving the free boundary problem in the linearized approximation we derive the following catastrophical growth for high frequencies:

j (k, t) = n (k) eiA:It

J

(15)

where (k, t) denotes the Fourier transform of I (x2 , t) with respect to the x2 -variable. The catastrophic instability (15) is usually cut for high frequencies adding to (1)-(4) the effect of surface tension. Actually {15) could raise doubts about the solvability of (1)-(4). In fact solvability results for (1)-(4) obtained in [19] were derived under the assumption of analyticity on the initial data and the interfaces. Nevertheless, although (1)-(4) can be solved (locally in time) under such stringent assumptions, the instability (15) finally shows up in the possibility of having an extremely rich set of possible singularity formation mechanisms. A problem that requires further research is the analysis of the effect on the singularities of the usual smoothing mechanisms that must be introduced in (1)-(4), namely, surface tension and kinetic effects.

Singularity Formation for the Stefan Problem

333

3. SINGULARITY FORMATION FOR THE CLASSICAL STEFAN PROBLEM The understanding of the structure of singularities that might appear for (1)(4) in the classical case (i.e. for To (x) ~ 0) is somehow more complete than for the undercooled situation. In the paper (1] a rather thought-out description of the structure of singularities has been given by using asymptotic methods. The main results of (1] will be reviewed here. There are different ways in which the interfaces of the classical Stefan problem can develop singularities, i.e. points of nonsmoothness. The simplest one is just the collision between two regions where T > 0 that move independently and collide at a particular point. Such a collision generically occurs at a point where the curvature of both interfaces at the contact point is nonzero. We have analyzed the way in which both interfaces move after the contact. The two regions that were evolving independently become connected after the collision by means of a hole of size I a (t" + t) 2 , where a > 0 is a parameter that depends on the distribution of temperature as well as the curvature at the point of collision. The smoothing after nongeneric collisions has also been considered in (1]. Another singularity mechanism that can take place for the classical Stefan problem is the cusp formation . In [1] we have discussed in detail different mechanisms for such a phenomenon, both in two and three space dimensions. We should point out that the mechanisms of the cusp formation found in (1] for the classical Stefan problem are rather unstable, since they correspond to a transition between a non-graph-like interface yielding in its evolution one or several holes of ice and a smooth interface after a while, and a graph-like interface that, when evolving, remains smooth all the time. We have described in detail the manner of the singularity formation and the asymptotics of the interface and the temperature at different scales in (1]. We point out that the behaviour of these functions varies drastically at different length scales, although it can be described by a sequence of nested boundary layers. As t-+ W)-, the tip of the interface can be approximated by a Ivantsov's parabola, as it happened in the undercooled case. In (2] we provide a rigorous construction of one of the simplest cusp-forming solutions for (1)-(4) in the classical case. Another type of singular behaviour that can take place in the classical Stefan problem is the closing of a domain occupied by ice to a point. In [1] we have discussed in detail the existence of shrinking ellipses and ellipsoids (or circles and spheres in particular cases). In the two-dimensional case the characteristic length of the shrinking ellipses is (t*- t)l exp ( -Y!-Ilog (t•- t)ll) as t -+ (t•)-. In the three-dimensional case we find shrinking ellipsoids of size lt~(~·t!.~)l as t -+ (t*)-. Rigorous constructions of this type of solutions in the radially symmetric case can be found in (13]. There exist other singularity formation mechanisms that have been described in the same work, as for instance segment-like and disk-like domains shrinking to a point.

334

J.J.L. Vel&zquez

However, since those structures are rather unstable under small perturbations, we will

not pursue their discussion here.

4. STEFAN PROBLEM AND HOMEOIDS In this section we will describe some connections between the Stefan problem and the geometric figures known as homeoids. Let us consider a bounded domain D containing the origin. This last restriction is not completely needed, but we will assume it here for simplicity. We will say that D is a homeoid if the gravitational field generated by the density X.>.D - X.D in the component of RN \ (>.DAD) that contains the origin is identically zero. By x.c we mean the characteristic function of G and A stands for a symmetric difference of two sets. The first example of a homeoid is the ellipse (or ellipsoid in three dimensions) as was proved by Newton (cf. [5]). On the other hand, it was proved in [7) and [6) that the only examples of homeoids are just ellipses and ellipsoids. In the course of our analysis of shrinking ellipsoids (cf. [1]) we found a rather surprising rigidity result: Theorem 1. Assume that u

~

0 solves the problem:

Au= H (u) in JRN , N ~ 2.

(16)

Suppose that the set {u = 0} is bounded. Then the domain {u = 0} is an ellipsoid (an ellipse if N = 2).

In equation (16), H (u) is the Heaviside function (H (s) = 1 for s > 0, H (s) = 0 otherwise), and equation (16) has to be understood in the sense of variational inequalities. The fact that the only possible domains compatible with equation (16) are ellipsoids suggests some kind of relation with the homeoid problem. Actually, it turns out that equation (16) that arises in a natural way in the study of the Stefan problem is some kind of "integrated form" of the homeoid problem. Conversely, the homeoid problem is, in some sense, the first variation of (16). To make this statement more precise, notice that given any solution u (x) of (16), we can derive another solution of (16) by means of the rescaling

Let us define the function:

cp>. (x) = u>. (x) - u (x) This function solves the problem:

Acp>.. = X.>.D - X.D ' where D = { u = 0}. Notice also that cp>.. vanishes on the set >.DAD. A local analysis of the asymptotics of u (x) as lxl -+ oo shows that cp>.. is, up to an additive constant, the Newtonian potential associated to the density X.>.D - X.D· It then follows that D

Singularity Formation for the Stefan Problem

335

is a homeoid. Conversely, given any homeoid D, we can assume {after the addition of a constant) that the corresponding Newtonian potential vanishes at D. Rescaling the homeoid and adding the corresponding potentials, we can obtain a nonnegative solution of (16) that vanishes on the set D. Along the lines of this argument we have constructed a new proof of the "converse Newton's theorem" in (1) that states that all the homeoids are ellipsoids, taking as starting point the theorem stated above. REFERENCES [1] D. Andreucci, M.A. Herrero and J. J. L. Velazquez, The classical one-phase Stefan problem: a catalogue of behaviours. Preprint. [2] D. Andreucci, M. A. Herrero and J. J . L. Velazquez, Cusp formation for the classical one-phase two-dimensional Stefan problem, in preparation. [3] L. A. Caffarelli, The regularity of free boundaries in higher dimensions. Acta Mathematic& 139, 155-184 (1977). [4] L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5, 427-448 (1980) . [5] S. Chandrasekhar, Ellipsoidal figures of equilibrium. Dover Publishing (1969) . [6] E. DiBenedetto and A. Friedman, Bubble growth in porous media. Indiana Univ. Math. J. 35, 3, 573-606 (1986) . [7] P. Dive, Attraction des ellipsoides homogenes et reciproque d 'un tbeoreme de Newton, Bull. Soc. Mat. France 59, 128-140 (1931). [8] A. Fasano, M.Primicerio, S.H. Howison and J. R. Ockendon, On the singularities of onedimensional Stefan problem with supercooling. Mathematical models for Phase Change Problems (J. F. Rodrigues, ed.). International Series of Numerical Mathematics, 88, Birkhauser Verlag, 215-226 (1989) . [9] A. Friedman, Variational principles and free boundary problems. John Wiley&: Sons, New York (1982). [10] A. Friedman, Partial differential equations of parabolic type. Robert E. Krieger Pub. Co. (1983). [11] M. A. Herrero, E. Medina and J . J. L. Velazquez, The birth of a cusp in the two-dimensional, undercooled Stefan problem. Preprint. [12] M. A. Herrero and J . J. L. Velazquez, Singularity formation in the one-dimensional supercooled Stefan problem. European J . Appl. Math. 7, 119-150 (1996) . [13] M. A. Herrero and J. J. L. Velazquez, On the melting of ice balls. SIAM J. Math. Anal. 28, 1, 1-32 (1997) . [14] G. P. Ivaotsov, Temperature field around a spherical, cylindrical and acicular crystal growth in a supercooled melt. Doklady Akad. Nauk. USSR 58, 4, 567-569 (1947). [15] A. A. Lacey and J. R. Ockendon, lli-posed free boundary problems. Control and Cybernetics, 14, 275-296 (1985). [16] A. N. Meirmanov, The Stefan problem, De Gruyter Expositions in Mathematics, Walter de Gruyter (1992). [17] W. W. Mullins and R. F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Physics 35, 2, 444-451 (1964). [18] B. Shennan, A general one-phase Stefan problem, Quart. Appl. Matbs. 377-383 (1970). [19] J. J . L. Velazquez, Cusp formation for the undercooled Stefan problem in two and three dimensions. European J. Appl. Math. 8, 1-21 (1997). [20] J . J . L. Velazquez, Comer formation for the undercooled Stefan problem. Preprint.

J.J.L. Velazquez, Departamento Matematica Aplicada, Faculty of Mathematics,

Universidad Complutense de Madrid, Madrid 28040, Spain

On the Construction of

Interior Spike Layer Solutions to a Singularly Perturbed Semilinear Neumann Problem J.

WEI

Abstract. A singularly perturbed semilinear Neumann problem which models chemotaxis and pattern formation is considered. We establish the existence of interior spike layer solutions for any bounded domain and the location of the interior spike is given. Our construction uses the very properties of the distance function which arises in the estimate of the exponentially small terms. Keywords: Spike layer, Singular perturbation, Viscosity solution Classification: Primary 35B40, 35B45; Secondary 35J40

1.

INTRODUCTION

The aim of this paper is to construct a family of single-peaked interior spike layer solutions to the following singularly perturbed elliptic problem: e 2 ~u- u

{ u

+ u" =

0 inn,

> 0 in nand ~ = 0 on 00,

(1.1}

where~= L~t ~ is the Laplace operator, n is a bounded smooth domain in RN,

e > 0 is a constant,' the exponent p satisfies 1 < p < ~ for N ~ 3 and 1 < p < oo for N = 2 and v(x) denotes the normal derivative at x E 00. Equation (1.1} is known as the stationary equation of the Keller-Segal system in chemotaxis. It can also be seen as the limiting stationary equation of the so-called Gierer-Meinhardt system in biological pattern formation; see [11] for more details. In the pioneering papers of [7], [9) and [10), Lin, Ni and Takagi established the existence of least-energy solutions and showed that for e sufficiently small the leastenergy solution has only one local maximum point P. and P, E 00. Moreover, H(P.) -+ maxpeoo H(P) as e -+ 0, where H(P) is the mean curvature of Pat 00. In [11], Ni and Takagi constructed boundary spike solutions for axially symmetric domains. The author in [21) studied the general domain case and showed that for single boundary spike solutions, the boundary spike must approach a critical point of the mean curvature; on the other hand, for any nondegenerate critical point of H(P), one can construct boundary spike solutions with spike approaching that point. When p =~.similar results for the boundary spike layer solutions have been obtained by [1], [2], [3], [8), [18), [23), etc.

On the Construction of Interior Spike Layer Solutions to ...

337

In all the above papers, only boundary spike layer solutions are obtained and studied. It remains a question whether or not interior spike layer solutions exist for problem {1.1). It was proved in [22] that under some geometric conditions, one can construct single interior spike solutions. However the geometric conditions are very restricted. It is the purpose of this paper to construct solutions by using the very properties of the distance function. We note that the distance function has already appeared in the study of the corresponding Dirichlet problem in [12], [19]. In this paper, we shall construct a family of single-peaked interior spike solutions to problem (1.1) for f sufficiently small and any bounded smooth domain. We first state a more general theorem.

Theorem 1.1. Let A c fl be an open set such that maxd{P,OO)

PE8A

< maxd(P,OO) PEA

{1.2)

Then there exists Eo > 0 such that for f < fo, problem {1.1} has a solution u, with the property that u, has exactly one local maximum point P, in n, P, E A, u,(P,) -+ w(O) and u,( · + P.) -+ 0 in C1~(fi- P, \ {0}), where w is the unique solution of the following problem: {

Aw - w + wP = 0 in RN W > 0, w(Q) = m8.XsteRN w(y)

w(y) -+ 0 as

IYI -+ 0

(1.3)

Moreover, d(P., 00)-+ maxPeA d(P, 00) as f-+ 0. We have the following important corollary.

Corollary 1.2. Let n be a bounded smooth domain. Then for f sufficiently small, ( 1.1) has a single-peaked solution with interior peak near the most centered part of the domain {i.e., the points which achieve the maximum of the distance function). A particular example is a domain with k-handles (see Figure 1). In this case, Theorem 1.1 asserts that there are at least k solutions to problem (1.1) and each handle contributes a single-peaked interior spike solution. Note that in Theorem 1.1 we do allow some degeneracy of the distance function.

FIGURE 1. Domains With Handles

Remark: 1. By Theorem 1.1, if the function d(P,OO) has k strictly local maximum points, then for ~ sufficiently small, problem (1.1) has at least k solutions. An

J. Wei

338

interesting question is to construct multi-peaked solutions at multiple local maximum points of the distance function. Partial progress has been made in this direction. 2. By Corollary 1.2, interior single-peaked solutions always exist. We believe that this is the first result of this kind for problem {1.1). To introduce the main idea of the proof of Theorem 1.1, we first need to give some necessary notations and definitions. Let w be the unique solution of {1.2). It is known (see (5], (6]) that w is radially symmetric, decreasing and lim w(y)el•ljyl¥ =Co

lrl~oo

> 0.

For any smooth bounded domain U, we set Puw to be the unique solution of {

~u - u + wP

:: = 0 on U.

Let

{u, v}f =E-N(~ llull~ For P E 0, we set Of,P

= 0 in U,

{1.4)

In VuVv +In uv)

= {u, u}«.

= {y : EY + p

E,,p = { v E H 1 (0) : (v, Pn.,.,.w} f

E

0}

= {tJ, 8Pn. aA,.w} = 0,'. = 1, ... , N }. f

Our approach is based on establishing the existence of critical points of the functional

J.•(u ) -_

2 E

In 1Vul2 +In U2 UnuP+l);:h

,u E

Hl(n) u

{1.5)

via seeking critical points of

K,(P, v) = J«(Pn.,.,.w + v)

{1.6)

over

M = {(P,v) : P E A, v E E•.P• llvll«:::; 77}. To this end, we first solve v by minimizing K, for fixed P and obtain a solution v«,P which is C 1 in P . Then we maximize K, among PEA. This paper is organized as follows. In Section 2, we study the properties of Pn.,.,.w. We set up the technical framework in Section 3 and we solve tJ in Section 4. We study a maximization problem in Section 5. Finally we prove Theorem 1.1 in Section 6. The proofs of some technical lemmas are given in the Appendix. Throughout this paper, unless otherwise stated, the letter C will always denote various generic constants which are independent of E, for t: sufficiently small.

On the Construction of Interior Spike Layer Solutions to ...

339

2. PROJECTION OF w

In this section, we study properties of the function Pn,,Pas introduced in Section 1. Recall that Pn,,P w is the unique solution of

{ where Set

n,,p := {YIEY + P !p,,p(x)

~v- v + wP :: = 0 on

= 0 in n,,p,

(2.1)

an,,p,

E n}.

lx-PI = w(-f-)Pn,,Pw(y),

EY + P

= x.

Then !p,,p(x) satisfies

{

E 2 ~v-v=O

inn, on an.

= l!.w( !z-P!)

8v 811

an

It is immediately seen that on



811

(2.2)

!_w(lx- PI)= ~w'(lx- PI)< x- P,v > av f f f lx - PI

=

-~ f

(1x- PI-(N-l)/

2 • f+¥e-I•:P!

(1 + O(E))) < x- P, 11 >

p::.

~-~

= -E¥e-I•:PI (1

(2.3)

+ O(E)) < xlx-PI

To analyze Pn,,Pw, we introduce another linear problem. Let Pf?..Pw be the unique solution of f 2 ~v- v+w" = 0 inn, v =0 on an.

{

Set

.

Then t/Jfp satisfies

Note that for

f~V - 1Vvl 2 + 1 = 0 in n,

X

E

{v =

an

-flog(w(lz~PI))

on

an.

D lx-PI ¥ e- ~ 1/J.,p(x) =-flog ( (-f-)• (1 + O(E)) )

lx-PI lx- PI+ -N-1 -Elog(-f-) +0(£2 ) 2 lx-PI = lx- PI+ -N -1 -Elog(-f-) +0(£).

=

2

By the results of Section 4 of [12] and Section 3 in [20], we have

J. Wei

340

Lemma 2.1.

(1)

81/Jf,p = -( 1 O( )) < x- P, v > av + f lx- PI I

(2) 1/Jfp(x) ~ 1/Jo(x) '

= zEBfl inf (lz- xi+ lz- PI) as f--+ 0

uniformly in fi. In particular 1/Jo(P)

= 2d(P, an).

Let us now compare 0, fo > 0 such that for f < fo, we have < 'P 0 such that (x - P, Vz} 2: Co > 0 for all X E an, where Vz is the unit normal at X E an. Then on an we have

Co

o a'IJ.,P

av

._o

1 = e-~(--)VY' 0 such that Bd(&,8fi)(Z) ni is convex. Moreover

(3)

nan= Bd(&,8fi)(Z) n ani

< z- P, liz>~ 110 >

0

for lz- PI < ~. ll:z; being the outer normal of X at Then on ani nan = r it we have

011

an\r It we have

for

lz- PI < ~.

ani.

::; -(1 + O(t:)) o~.pf.p.

a~.p.,p On

341

011

Ia:~p I :5 ce-(1+2ao)Pd(P,8fl)

01./),,P

< Ce-(1+2ao)Pd(P,8fl) < Ce-(l+Q())/ld(P,8fi)8U,.

& -

-

&

By comparison principle, we get the inequality. Lemma 2.2 is thus proved. o By Lemma 2.2, we have that

,P,(P) := -t: log ( - 0 some suitable constant and 1

Er,P = { v E H (f2) l(v, Pn.,,.w)1

= (v, a~ Pn.,,.w) = 0, i = 1, ..., N}. 1

(3.5)

Let us now define the functional

K::

M, -+R m =(a, P, v)-+

and

K 1 (P, v) where

{ I

f-~ J1 (aPn.,,.w + v)

= K:(l, P, v)

(P,v) E {l x H 1 (f2), } (P, v) E M := (P, v) d(P, 00) > ao, v E Er.P• llvllr < 17 .

Since

that

K: is homogeneous of degree 0, it follows from Lemma 3.1 and Lemma 3.2

J. Wei

344

Proposition 3.3. (P, v) E M is a critical point of K. if and only if u = Pn.,,. w + v is a critical point of J., i.e. if and only if there exists (a, /3} E R x RN such that

(Ep )

8K. _ ~ (IPPn,,,.w 8P.· - ~ /3; 8P.·8P.· , v).

(Ev)

8K. ~ 8Pn_,,. w 8v = aPn_,,.w + ~ /3; BP.· .



J=l



(3.6)

J

J=l

(3.7)

J

Therefore to find a critical point of K., it is enough to find (P, v) satisfying (Ev) and (Ep). In Section 4, we first solve v by minimizing and then solve P by maximizing. 4. MINIMIZING v

In this section, we solve (E.,) for fixed P E 7i. and

f

sufficiently small.

Proposition 4.1. There exist 71o > 0, fo > 0 such that for£ ~ £o, 11 ~ f/0 there exists a smooth map which to any (f, P) such that (P, 0} E Mf/0 associates v.,P E E,,p, llv.,PII• < 11 such that (E.,) is satisfiedforsome (a,{J) E RxRN. Such av,,p is unique, minimizes K,(P,v) with respect to v in {v E E.,PIIIvll. < 17} and we have the estimate 8v,,p = 0 on 00, llv.,PII• 8v where u =min (p- 1, 1}.

~ O(lt,e.,p(P)Il¥),

(4.1)

Proof: We first expand K.(P,v) at (P,O}; we have K,(P, v}

where K e (P' 0} = f,,p(v)

0.,P

+(p +

(4.2)

f(p- 1)Nf(p+l) J. (Rn w} e c,P

= -2(10..P wPPn_,,.w)(10.,1' (Pn_,,.w}p+1}-$[10..P (Pw)Pv)

Q.,a,P(v}

x{1

= K,(P, 0} + f,.P(v) + Q,,p(v} + R.,p(v},

= 0 such that for small enough, and 17 small enough, we have

~:

Q.,p(v) ~ pllvll~, for all v E E.,p.

(4.8)

Therefore L.,P is a coercive operator whose modulus of coercivity is bounded from below independently on~:, P . The derivative of K. with respect to v on E.,P may be written F.,P

+ 2L.,pv + O(llvll~).

Using the implicit function theorem, we derive the existence of a 02-map T which to each (~:, P) associated v.,P E E.,P such that

a:v• (P, v)IE.,,. = 0 and (4.9) Moreover, since v.,P minimizes K. over E.,p, "';;/ = 0 on 00. We now claim that

lf.,p(v)l :5 O(lc,o.,p(P)Il¥)11vll.,

(4.10)

which by (4.6), (4.9) and (4.10) proves Proposition 4.1. By definition Since

i

(Pn.,,,.w)Pv =

i

n.,,,. n.,,. We now calculate, by Lemma 2.3,

((Pn.,,,.w)P- wl')v (since v E E.,p).

2

ln.,,.I(Pn.,,.w)P- wl'l :5 Thus (4.10) is established.

O(lc,o.,p(P)I 1 ~).

0

5. A

MAXIMIZING PROBLEM

In this section, we study a maximizing problem. We shall prove Proposition 5.1. Let v = v.,P be the solution given by Proposition small, the following maximizing problem max{K.(P,v.,p): PEA} has a solution P.

E

A.

-4.1. Then for

f

J. Wei

346

Proof: Since Je(Pn..rw + 11e,P) is continuous in P, the maximizing problem has a solution. Let le(Pn..r. w + Ve,P.) be the maximum. Then by Proposition 4.1, we have

Ke(Pe, Ve,P.}

= Ke(P., 0} + /e,P.(Ve,P.} + O(llve,P.II~}

= Ke(P., 0} + O(e-C1+ ~ max:reo d(x, an)}. 7. APPENDIX A. DECOMPOSITION LEMMA

In this appendix, we shall prove the decomposition lemma 3.1 in Section 3. We start with some lemmas.

l.imt-+oo tt =

Lemma 7.1. Let (tt) be a sequence with tt > 0, 11410 , Ot 1 ak E (~, 2) be such that

0. Let Pt, f>t E

a. wll •• = 0. ••·.-a

(7.1)

lim llakPn.,. w- citPo • •

k-+oo

Then we have lim lak- akl = 0,

(7.2)

A I = o.

(7.3)

k-+oo

lim IPk -

(k

k-+oo

Proof: We have

...

.

...,..

llakPn. ,. w - citPo

A

wll ••

pk- pk = llakPn.,. )II •• • • w- akPo• • •,.. w(·f A

= ll(ak- ak)Po ,. w + cit(Pn.,. w- Po ••· •

' •

A

,,,,.,

w(·- pk- pk))ll •• E

~ lakiiP•• ,P.wll ••.o -lphakiiP•• ,P.wll •• .nl· Since both Pk, pk E nao, IIP•• ,P.wll •• -+ llwiiH1(RN) and liP•• ,;;. wll •• .n -+ llwiiH1(RN)· Hence lak - akl = o(1). On the other hand, suppose (7.3) is not satisfied; we have that by passing to a subsequense, IP•;/•1 -+ a with 0 < a ~ co.

.

< a < co, then II Po.,,.. w- Po, •.,., w( · - ;;.~P· )ll •••o -+ llw- w( ·-a) IIHl(RN) =1- 0. If a= oo, then IIPn.,. .. w(·- i>.-P. )ll •• .n-+ 2llw11Hl(RN) =1- 0. In any case, ' w- Po ,.. If 0

...

we reach a contradiction with (7.1).

(

0

J . Wei

348

We now prove Lemma 3.1. We will follow closely to Appendix B of [20] . We argue by contradiction. Suppose there exists t~e --+ 0, Tile --+ 0 such that inf

ueA.,

llu -

and (a,., P,.) , (a,., P,.) E A,,, such that if v"'

vii •• < Tlko

... v" =

= u,.- a,.Pn.

(v", Pn.,.r, w)., = 0,

,. a

(v , aPi Pn.,.r, w) ••

p

= 0,

w,

...

u,.- a,.Pn ,... .. w ,

{7.4) (7.5)

{7.6) {7.7)

a,.

Let = p,;i\, 1-'le = a1e - ii~e . Then by Lemma 7.1, latl = o{l), 11-'~el = o(l). We denote ~ as various constants which do not depend on k. We first observe that

lw"(y)- w"(yBy the Maximum Principle

.•.

.

.

a~~:) I ~

IPn r w- Pn ........ wl

Clatlw"(y).

(7.8)

~ Cla~~:lw(y).

{7.9)

The rest of the proof is exactly the same as those of Appendix B in [20]; we omit the details. o Acknowledgment. This research is supported by an Earmarked Grant from RGC of Hong Kong. REFERENCES

(1] Adimurthi, G. Mancinni and S.L. Yadava, The role of mean curvature in a semilinear Neumann problem involving the critical Sobolev exponent, Comm. Partial Differential Equatioru 20 (1995), 591~31. (2] Adimurthi, F. Pacella and S.L. Yadava, Interaction between the geometry of the boundary and p011itive solutions of a semilinear Neumann problem with critical nonlinearity. J. JiUnct. Anal. 31 (1993) , 8-350. (3] Adimurthi, F. Pacella and S.L. Yadava, Characterization of concentration points and L 00 estimates for solutions involving the critical Sobolev exponent, Differential Integral Equatioru 8(1) (1995), 41~8. (4] W.H. Fleming and P.E. Souganidis, Asymptotic series and the method of vanishing viscosity, Indiana Univ. Math. J. 35 (2) (1986) , 425-447. (5] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of p011itive solutions of nonlinear elliptic equations in R" , Mathematical AnalJIIU and Applicatioru, Part A, Adv. Math. Suppl. Studies 1 A (1981) , Academic Press, New York, 369-402. (6] M. K. Kwong, Uniqueness of p011itive solutions of ~u- u + u" = 0 in R", Arch. Rational Mech. Anal. 105(1989}, 243-266. (7] C.-S.Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to chemotaxis systems, J. Differential Equatioru 72(1988), 1-27. (8] W.-M. Ni, X. Pan and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67(1992}, 1-20.

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J. Wei, Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong Email: weiCmath.cuhk.edu.hk