Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

Подождите немного. Документ загружается.

254

Chapter 15: Large-time Behavior of Semi-linear Diffusion Equations

More precisely, v(x, t) E C([0, oo); E), where E is a suitable space of functions

ofxEll.

The norm of v is

0 < t < oo}.

(15.43)

The main issue concerns the bilinear operators

BB,k,t(f, 9) = f R,RkS(t - s),9t(fg) ds,

(15.44)

where j, k, i belong to 11, 2, 3}, f = f (x, s), g = g(x, s).

If f and g belong to C([0, oo); E) then two difficulties might arise.

In many instances which will be treated, the pointwise product between

two functions f and g in E does not have any meaning (even in a weak sense).

This "divergence" needs to be fixed.

A second divergence is coming from the competition between the differ-

entiation ay, = A, which destroys smoothness, and S(t - s), which improves

smoothness. As s tends to t, the differentiation will win!

-a,oo

The Banach space which will be considered is our friend Bq .

With

this choice the bilinear operators (15.44) are not continuous. This difficulty

was already present in §15.5 and was solved by introducing the "artificial

norm"'1ujjT = IIuUUx + 1+uUU. in (15.17).

T. Kato applied the same program in the Navier-Stokes case and could

prove the existence of positive constants i3 and y such that, if div vo = 0 and

IIvo,13 < 0, there exists a solution v(x, t) to the Navier-Stokes equations with

the following properties:

V(-, t) E C([0, oo); L3(R3)), (15.45)

t114IIv(', t)116 < y.

(15.46)

Moreover, such a solution is unique.

We will improve on this result by relaxing the hypothesis 11veI13 <,6 into

a much weaker one. The following theorem was obtained in a collaboration

with M. Cannone when 3 < q < 6 and then improved by F. Planchon in his

Ph.D. dissertation, where he got rid of the limitation q < 6.

Theorem 15.9. There exists a continuous function q(q), defined on (3, oo),

with values in (0, oo) with the following property: if vo E L3(R), div vo = 0

and if, for some q E (3, oo) we have

lIvoIIB-a-3ia).oo

< rl(q),

(15.47)

Meyer

255

then there exists a unique solution v(x, t) to the Navier-Stokes equations such

that

E C([0, oo); L3(R3))

(15.48)

limn' lu(.,t)II.=0.

(15.49)

-(1-3/q),-

Since the Besov spaces Bq are increasing with q one might be

tempted to believe that the largest q yields the best result in (15.47). How-

ever, q (q) = 0 as q -4 +oo, and this limitation is due to the proof.

15.11 Self-Similar Solutions to Navier-Stokes

Equations

A self-similar solution is a solution v(x, t) such that v = va for 0 < A < oo.

This implies v(x, t) = V (7--t) as before. Now V is vector-valued.

It is well known that for any solution to (15.40) for which it makes sense

we have AlIvIlz < 0. It means that V(x) cannot belong to L2(R3).

We will construct self-similar solutions to Navier-Stokes equations by solv-

ing an initial value problem. Therefore we want uo(x) to be the weak limit

of I V(I) as ttends to0.

Then uo(x) will be homogeneous of degree -1, and the functional space

which will be used should contain such functions. Once more we are led

. -(1-3/q),-

to using our Besov spaces Bq = Bq

The first construction of self-

similar solutions to Navier-Stokes equations was obtained in a joint work

with M. Cannone and F. Planchon ([4], [5]).

Then F. Planchon could relax the condition imposed upon q and proved

the following theorem ([18]).

Theorem 15.10. There exist two continuous functions q(q) > 0 and /3(q) >

0, defined on (3, oc), with the following property: if uo(x) belongs to Bq for

some q. E (3, oo) and satisfies V uo = 0, uo(Ax) = A-luo(x), A > 0 and

IIuoUls9 < 77(q), then there exists a unique self-similar solution u(x,t) _

7 U

(I) to the Navier-Stokes equations such that

u(x,t) = S(t)[uo](x) +

W (3=),

(15.50)

IIWII3 < fi(q), (15.51)

. 1,2

and, more precisely, W E B3/2.

256

Chapter 15: Large-time Behavior of Semi-linear Diffusion Equations

15.12

Convergence to Self-Similar Solutions

This section is almost identical to what we obtained for the nonlinear heat

equation. As already stressed, these similarities come from the following

two points (a) the scaling invariance is the same in the two cases, (b) the

nonlinearity is treated by a perturbation argument without taking in account

the subtle cancellations, which played a crucial role in the Navier-Stokes

equations.

We want to know the large-scale behavior of the Navier-Stokes equations,

which is related to studying

Jim Av(.\x, alt) = Jim va (x, t). (15.52)

At+oo

More precisely, we want to find necessary and sufficient conditions for the

existence of v(x, t) such that

lim flva(., t) - v(', t)f lq = 0 (15.53)

At+oo

uniformly on t E [to, ti} for t1 > to > 0.

As we already observed, this limit field v(x, t) is self-similar: v(x, t) _

7 V

(7} if it exists.

Then our problem amounts to deciding whether

IIvcv(fx, t) -

0 (t -, +00).

(15.54)

If it is the case, we necessarily have

I(I/GV(VLx,t)jjq < C for t > T. (15.55)

Instead of (15.55) we will assume the stronger condition

fv(fx,

t)q < C for 0 < t < oo. (15.56)

We then have ([18]).

Lemma 15.11. If v(x, t) is a global solution to the Navier-Stokes equations,

if (15.56) holds for some q E (3, oo) and if limbo v(x, t) = vo(x) exists in the

-(1-3/q),oo

weak sense, then vg(x) E Bq

Our next step will be to decide, for a given initial value vo(x), whether the

corresponding solution to the Navier-Stokes equations is global and converges

to a self-similar solution. This issue is answered by the following theorem

(obtained in collaboration with F. Planchon).

Meyer

257

For simplifying the notation, we will denote by Dt : Lq(R3) -+ Lq(1R3) the

normalized dilation defined as Dt f (x) = V if (f x) and Dt will also act on

functions v(x, t) defined on JR3 x (0, oo) by [Dtv] (x, t) = f v(f x, t).

Our issue is to know whether

lim [Dtv](x,t) = V(x) in Lq(1R3).

(15.57)

tt+oo

Theorem 15.12. There exists a continuous positive function ij(q) defined

. -(1-3/q),oo

on (3, oo) with the following property: for every vo E Bq

such that

1kvoIIBq <,q(q) and V vo = 0, (15.58)

the following two properties are equivalent ones:

II [Dtv](x, t) - V(x)IIq - 0, as t -> +oo,

(15.59)

II [DtS(t)vo](x, t) - Vj(x)Ilq - 0, as t -* +oo.

(15.60)

Theorem 15.10 means that the asymptotic behavior of the full solution to

(15.12) is governed by the asymptotic behavior of the linear evolution. It is

easy to check that (15.60) implies limAt+oo )vo(.x) = Vo(x) in the weak-star

topology o,(Bq

-(1-3/q) ,oo, Bpl-3/q),1)'

where 1/p + 1/q = 1. However, as was

mentioned before, (15.60) is a stronger statement. The relationship between

V1 and Vo is given by V1 = S(1)Vo.

Moreover, Vo(x) is homogeneous of

degree -1, V Vo = 0, and the corresponding self-similar solution to (15.12)

is precisely

V (* ) .

A final observation is limat+,,. Av(Ax, alt) = 7 V

If vo(x) belongs to L3(1 ), then Vo = 0 and the theorem yields the well-

known decay estimates for solutions of the Navier-Stokes equations.

15.13 The Nonlinear Schrodinger Equation

We will consider the following evolution equation

i + Du = ryujuj2,

u(x, 0) = uo(x) (15.61)

where u = u(x, t) is a complex-valued function of x E R3 and t > 0. The

constant ry is real-valued.

T. Cazenave and F. Weissler [6] succeeded in constructing self-similar

solutions to (15.61) by applying the general organization which was used

258

Chapter 15: Large-time Behavior of Semi-linear Diffusion Equations

before. However, the functional spaces which will play a crucial role are no

longer Besov spaces.

Let us denote by S(t) = exp(itA) the linear Schrodinger group. These

operators are not smoothing but act unitarily on L2(1[3). Then (15.61) can

be rewritten

u(t) = S(t)uo - iy

S(t - s)uIul2(s) ds,

(15.62)

and we are looking for solutions u(t) to (15.61) which belong to C([0, oo); E)

where E is a suitable Banach space.

We then mimic our previous approach and define E by the following

condition

SUPt118IIS(t)[f]114

= IIf IIE < 00.

(15.63)

t>o

One easily checks that testing functions belong to E. An equivalent norm

on E is defined by the condition

I I.

(e'

' 2 f(x))114

CA5/6, (15.64)

where A E (0, oo) and

[Ff](x) =

f e-tx'Y f(y)

dy. (15.65)

It is easy to check that E does not contain any LP space for 1 < p < oo.

But E contains some nonsmooth functions since Cazenave and Weissler [5]

observed that IxI-I E E. What is most surprising is the following observation

by F. Oru [15]: f (x) E E does not imply 7(x) E E. Indeed, e'IxI2IxI-1 belongs

to E but e-ilkI2IxI-1 does not. This is related to the fact that (15.62) is valid

only for positive t. A final observation is the following: E is isometrically

invariant under translations and modulations (i.e., multiplication by e'-x).

This implies that quite irregular functions may belong to E.

Cazenave and Weissler proved the following theorem [6]:

Theorem 15.13. There exist two positive constants q and Q with the fol-

lowing properties: let uo(x) E S'(R3) satisfy the condition

sup t118IIS(t)uo114 < tl.

(15.66)

t>O

Then there exists a unique solution u E C([0, oo); E) to (15.62) such that

SUP 018114,0114 < Q. (15.67)

t>o

Meyer

259

We should warn the reader that E is not a separable Banach space, which

implies that u E C([0, oo); E) means that u(-, t) is a continuous function of t

when E is equipped with its weak-star topology (as it was the case for the

Besov spaces). However, the solution which is being constructed will also be

a continuous function of t E (0, oo) with values in L4 (where L4 is given its

strong topology defined by the L4 norm).

After proving Theorem 15.9, Cazanave and Weissler applied Theorem 15.9

to the construction of self-similar solutions to (15.61). The rescaling is once

more the same as in the nonlinear heat equation or as in the Navier-Stokes

equations.

To obtain self-similar solutions to (15.61), it suffices to start with uo(x)

with the following properties: uo(x) E CO°(1l3\{0}) and uo(Ax) =.-luo(x),

A > 0. Then it is easily checked that such a uo belongs to the Banach space

defined by (15.66).

If the norm of uo in this Banach space exceeds 77, it

suffices to multiply uo by a small constant. We end with a large family of

self-similar solutions to the nonlinear Schrodinger equation.

Addendum: With grief and sorrow we are returning to these notes which were

written three years ago when A. Calder6n was still among us. We would like

to mention two important theorems which have been obtained recently. The

first one is an improvement on Theorem 15.9. G. Furioli, P. G. Lemarie-

Rieusset, and E. Terraneo obtained uniqueness without the artificial condi-

tion (15.49). The best reference is [14]. In contrast, E. Terraneo built two dis-

tinct solutions u(x, t) and v(x, t) to (15.1) such that u(x, 0) = v(x, 0) = uo(x)

and u, v E C([0, oo), L3(R3)). In her counterexample, u is the solution given

by Theorem 15.3 while I Iv(-, t)116 = +oo fort > 0 [19].

References

[1] J. M. Ball, Remarks on blow-up and nonexistence theorems for non-linear

parabolic evolution equations, Quart J. Math. Oxford Ser.

28 (1977),

473-486.

[210. Barraza, Self-similar solutions in weak LP spaces of the Navier-Stokes

equations, Revista Matematica Iberoamericana, vol.

12, no.

2 (1996),

411-439.

[3] M. Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot Editeur,

20 rue N. D. de Nazareth, 75003 Paris (1995).

[4] M. Cannone, Y. Meyer, and F. Planchon, Solutions autosimilaires des

equations de Navier-Stokes, Seminaire Equations aux Derivees Partielles,

Centre de Mathematiques Ecole Polytechnique (1993-1994).

260

Chapter 15: Large-time Behavior of Semi-linear Diffusion Equations

[5] M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes

equations in R3, Comm. in P. D. E. 21 (1996), 179-193.

[6] T. Cazenave and F. Weissler, Asymptotically self-similar global solutions

of the non-linear Schrodinger and heat equations, Math. Z. 228 (1998),

no. 1, 83-120.

[7] M. Escobedo and 0. Kavian, Variational problems related to self-similar

solutions to the heat equation, Nonlinear analysis, theory, methods, and

applications, 11 (1987), 1103-1133.

[8] M. Escobedo, O. Kavian, and H. Matano, Large time behavior of solutions

of a dissipative semilinear heat equation, Comm. in P. D. E. 20 (1995),

1427- 1452.

[9] A. Haraux and F. Weissler, Non-uniqueness for a semilinear initial value

problem, Indian Univ. Math. J. 31 (1982), 166-189.

[10] T. Kato, Strong LP solutions of the Navier-Stokes equations in R1 with

application to weak solutions, Math. Zeit. 187 (1982), 167-189.

[11] 0. Kavian, Remarks on the large time behavior of a nonlinear diffusion

equation, Analyse non-lineaire, Annales IHP 4 (1987), 423-452.

[12] P. G. Lemarie and Y. Meyer, Ondelettes et bases hilbertiennes, Revista

Matematica Iberoamericana, vol. 2 (1986), 1

18.

[13] H. A. Levine, Some nonexistence and instability theorems for solutions

of formally parabolic equations of the form P(d/dt)u = -Au + F(u), Arch.

Rat. Mech. Anal. 51 (1973), 371-386.

[14] Y. Meyer, Wavelets, paraproducts, and Navier-Stokes equations, in Cur-

rent Developments in Mathematics, International Press, Cambridge, MA,

1996, 105- 212.

[15] F. Oru, Role des oscillations dans quelques problemes d'analyse non-

lineaire, Ph.D. (June 9, 1998), CMLA, ENS-CACHAN (France).

[16] L. Payne, personal communication to F. Weissler.

[17] F. Planchon, Solutions globales et comportement asymptotique pour les

equations de Navier-Stokes, Thesis, Ecole Polytechnique, Centre de Mathe-

matiques (May 28, 1996).

[18]

, Asymptotic behavior of global solutions to the Navier-Stokes

equations in 1R3, Revista Matematica Iberoamericana, vol.

14, no. 1

(1998), 71-93.

Meyer

261

[19] E. Terraneo, On the non-uniqueness of weak solutions of the nonlinear

heat equation with nonlinearity u3, Comptes Rendus de 1'Academie des

Sciences de Paris (1999), to appear.

Chapter 16

Estimates for Elliptic Equations

in Unbounded Domains and

Applications to Symmetry and

Monotonicity

Louis Nirenberg

The maximum principle for second-order elliptic equations plays a fundamen-

tal role in their study. In recent years it has been used to derive properties

of solutions of nonlinear equations, such as symmetry and monotonicity in

some direction. For simplicity I will discuss only problems of the form

u>0, Du+f(u) =0 inSl,

u = 0

on O!

(16.1)

Here f

is a domain (open connected set) in R", and f is a Lipschitz contin-

uous function. I begin with the simplest result, one in [14]:

Theorem 16.1. Suppose u is a solution of (16.1) for Q a ball: {jxj < R}.

Then u is radially symmetric and ur < 0 for 0 < r < R.

The proof uses the method of Moving Planes introduced by A. D. Alexan-

droff [2] and then used by J. Serrin (17].

It involves showing monotonicity and symmetry in some direction, say

the x1-direction. I describe it briefly: For -R < A < 0, let Ea = {x E SI

xl < A}; one shows that if x = (x1, x'), x' = (x2i ... ,

is in Ex then

Research partially supported by grant ARO-DAAL-03-92-6-0143 and by NSF grant DMS-

9400912.

263