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Jean Bourgain is Professor of Mathematics at the Institute for Advanced Study in Princeton. In 1994, he won the Fields Medal. He is the author of "Green's Function Estimates for Lattice Schrodinger Operators and Applications" (Princeton). Carlos E. Kenig is Professor of Mathematics at the University of Chicago. He is a fellow of the American Academy of Arts and Sciences and the author of "Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems". S. Klainerman is Professor of Mathematics at Princeton University. He is a MacArthur Fellow and Bocher Prize recipient. He is the coauthor of "The Global Nonlinear Stability of the Minkowski Space" (Princeton).

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Mathematical Aspects of Nonlinear Dispersive Equations

PRINCETON UNIVERSITY PRESS

Contents

Preface................................................................................................................................................................viiChapter 1. On Strichartz's Inequalities and the Nonlinear Schrdinger Equation on Irrational Tori J. Bourgain.........................................................1Chapter 2. Diffusion Bound for a Nonlinear Schrdinger Equation J. Bourgain and W.-M.Wang............................................................................21Chapter 3. Instability of Finite Difference Schemes for Hyperbolic Conservation Laws A. Bressan, P. Baiti, and H. K. Jenssen..........................................43Chapter 4. Nonlinear Elliptic Equations with Measures Revisited H. Brezis, M. Marcus, and A. C. Ponce.................................................................55Chapter 5. Global Solutions for the Nonlinear Schrdinger Equation on Three-Dimensional Compact Manifolds N. Burq, P. Grard, and N. Tzvetkov.........................111Chapter 6. Power Series Solution of a Nonlinear Schrdinger Equation M. Christ........................................................................................131Chapter 7. Eulerian-Lagrangian Formalism and Vortex Reconnection P. Constantin........................................................................................157Chapter 8. Long Time Existence for Small Data Semilinear Klein-Gordon Equations on Spheres J.-M. Delort and J. Szeftel................................................171Chapter 9. Local and Global Wellposedness of Periodic KP-I Equations A. D. Ionescu and C. E. Kenig....................................................................181Chapter 10. The Cauchy Problem for the Navier-Stokes Equations with Spatially Almost Periodic Initial Data Y. Giga, A. Mahalov, and B. Nicolaenko.....................213Chapter 11. Longtime Decay Estimates for the Schrdinger Equation on Manifolds I. Rodnianski and T. Tao...............................................................223Chapter 12. Dispersive Estimates for Schrdinger Operators: A Survey W. Schlag........................................................................................255Contributors...........................................................................................................................................................287Index..................................................................................................................................................................291

Chapter One

J. Bourgain

1.0 INTRODUCTION

Strichartz's inequalities and the Cauchy problem for the nonlinear Schrdinger equation are considerably less understood when the spatial domain is a compact manifold M, compared with the Euclidean situation M = [R.sup.d]. In the latter case, at least the theory of Strichartz inequalities (i.e., moment inequalities for the linear evolution, of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is basically completely understood and is closely related to the theory of oscillatory integral operators. Let M = [T.sup.d] be a flat torus. If M is the usual torus, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.1)

a partial Strichartz theorywas developed in [B1], leading to the almost exact counterparts of the Euclidean case for d = 1, 2 (the exact analogues of the p = 6 inequality for d = 1 and p = 4 inequality for d = 2 are false with periodic boundary conditions). Thus, assuming supp [??] [subset] B(0,N),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.3)

For d = 3, we have the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.4)

but the issue:

Problem. Does one have an inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [epsilon] > 0 and supp [??] [subset] B(0,N)?

is still unanswered.

There are two kinds of techniques involved in [B1]. The first kind are arithmetical, more specifically the bound

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.5)

which is a simple consequence of the divisor function bound in the ring of Gaussian integers. Inequalities (1.0.2), (1.0.3), (1.0.4) are derived from that type of result.

The second technique used in [B1] to prove Strichartz inequalities is a combination of the Hardy-Littlewood circle method together with the Fourier-analytical approach from the Euclidean case (a typical example is the proof of the Stein-Tomas [L.sup.2]-restriction theorem for the sphere). This approach performs better for larger dimension d although the known results at this point still leave a significant gap with the likely truth.

In any event, (1.0.2)-(1.0.4) permit us to recover most of the classical results for NLS

i[u.sub.t] + [DELTA]u - u[[absolute value of u].sup.p-2] = 0,

with u(0) [member of] [H.sup.1]([T.sup.d]), d [less than or equal to] 3 and assuming p < 6 (subcriticality) if d = 3.

Instead of considering the usual torus, we may define more generally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.6)

with Q(n) = [[theta].sub.1][n.sup.2.sub.1] + ... [[theta].sub.d][n.sup.2.sub.d] and, say, 1/ITLITL [less than or equal to] [[theta].sub.i] < C (1 [less than or equal to] 1 [less than or equal to] d) arbitrary (what we refer to as "(irrational torus)."

In general, we do not have an analogue of (1.0.5), replacing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is an interesting question what the optimal bounds are in N for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.8)

valid for all 1/2 < [[theta].sub.i] < 2 and A.

Nontrivial estimates may be derived from geometric methods such as Jarnick's bound (cf. [Ja], [B-P]) for the number of lattice points on a strictly convex curve. Likely stronger results are true, however, and almost certainly better results may be obtained in a certain averaged sense when A ranges in a set of values (which is the relevant situation in the Strichartz problem). Possibly the assumption of specific diophantine properties (or genericity) for the [[theta].sub.i] may be of relevance.

In this paper, we consider the case of space dimension d = 3 (the techniques used have a counterpart for d = 2 but are not explored here).

Taking 1/ITLITL <[[theta].sub.i] < C arbitrary and defining [DELTA] as in (1.0.6), we establish the following:

Proposition 1.1 Let supp [??] [subset] B(0, N). Then for p > 16/3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.9)

where [L.sup.p.sub.t] refers to [L.sup.p.sub.[0,1]](dt).

Proposition 1.3'. Let supp [??] [subset] B(0,N). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.10)

The analytical ingredient involved in the proof of (1.0.9) is the well-known inequality for the squares

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.11)

The proof of (1.0.10) is more involved and relies on a geometrical approach to the lattice point counting problems, in the spirit of Jarnick's estimate mentioned earlier. Some of our analysis may be of independent interest. Let us point out that both (1.0.9), (1.0.10) are weaker than (1.0.4). Thus,

Problem. Does (1.0.4) hold in the context of (1.0.6)?

Using similar methods as in [B1, 2] (in particular [X.sub.s,b]-spaces), the following statements for the Cauchy problem for NLS on a 3D irrational torus may be derived.

Proposition 1.2 Let [DELTA] be as in (1.0.6). Then the 3D defocusing NLS

[iu.sub.t] + [DELTA]u - u[absolute value of u].sup.p-2] = 0

is globally wellposed for 4 [less than or equal to] p < 6 and [H.sup.1]-data.

Proposition 1.4'. Let [DELTA] be as in (1.0.6). Then the 3D defocusing cubic NLS [iu.sub.t] + [DELTA]u u[absolute value of u].sup.2] = 0

is locally wellposed for data u(0) [member of] [H.sup.s](T3), s > 2/3.

This work originates from discussion with P. Gerard (March, 04) and some problems left open in his joint paper [B-G-T] about NLS on general compact manifolds. The issues in the particular case of irrational tori, explored here for the first time, we believe, unquestionably deserve to be studied more. Undoubtedly, further progress can be made on the underlying number theoretic problems.

1.1 AN INEQUALITY IN 3D

Q(n) = [[theta].sub.1][n.sup.2.sub.1] + [[theta].sub.2][n.sup.2.sub.2] + [[theta].sub.3][n.sup.2.sub.3], (1.1.1)

where the [[theta].sub.i] are arbitrary, [[theta].sub.i] and [[theta].sup.-1.sub.i] assumed bounded. Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.2)

Proposition 1.1 For p > 16/3, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.3)

assuming supp [??] [subset] B(0, N). Here [L.sup.p.sub.t] denotes [L.sup.p.sub.t] (loc).

Remark. Taking [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we see that (1.1.3) is optimal.

Proof of Proposition 1.1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.4)

since p [greater than or equal to] 4.

Denote [c.sub.n] = [absolute value of [??](n)]. Applying Hausdorff-Young,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.5)

Rewrite [absolute value of Q(n) + Q(a - n) - k] [less than or equal to] 1/2 as [absolute value of Q(2n - a) + Q(a) - 2k] [less than or equal to] 1 and hence

2n [member of] a + [[??].sub.l],

where

l = 2k - Q(a) and [[??].sub.l] = {m [member of] [Z.sup.3]| [absolute value of Q(m) - l] [less than or equal to] 1}. (1.1.6)

Clearly (1.1.5) may be replaced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.6')

and an application of Hlder's inequality yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.7)

(since the [[??].sub.l] are essentially disjoint).

Substitution of (1.1.7) in (1.1.4) gives the bound

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.8)

Next, write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.8')

where [psi] is compactly supported and [??] [greater than or equal to] 0, [psi] [greater than or equal to] 1 on [-1, 1].

Assume p [less than or equal to] 8, so that p/p-4 [greater than or equal to] 2 and from the Hausdorff-Young inequality again

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.9)

Since p > 16/3, q = 3p/4 > 4 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.10)

(immediate from Hardy-Littlewood).

Therefore,

(1.1.9) [??] [N.sup.3- 8/p],

and substituting in (1.1.8), we obtain (1.1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for p [less than or equal to] 8. For p > 8, the result simply follows from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.11)

This proves Proposition 1.

Remarks.

1. For p = 16/3, we have the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.12)

assuming supp [??] [subset] B(0, N).

2. Inequalities (1.1.3) and (1.1.12) remain valid if supp [??] [subset] B(a,N) with a [member of] [Z.sup.3] arbitrary.

Indeed,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.2 APPLICATION TO THE 3D NLS

Consider the defocusing 3D NLS

[iu.sub.t] + [DELTA]u - u[absolute value of u].sup.p-2] = 0 (1.2.1)

on [T.sup.3] and with [DELTA] as in (1.1.2).

Assume 4 [less than or equal to] p < 6.

Proposition 1.2 (1.2.1) is locally and globally wellposed in [H.sup.1] for p < 6.

Sketch of Proof. Using [X.sub.s,b]-spaces (see [B1]), the issue of bounding the nonlinearity reduces to an estimate on an expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[parallel] [[phi].sub.1] [parallel].sub.2], [[parallel] [[phi].sub.2] [parallel].sub.2] [less than or equal to] 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we need to estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.2)

By dyadic restriction of the Fourier transform, we assume further

supp [[??].sub.1] [subset] B(0, 2M)\B(0,M) (1.2.3)

supp [??] [subset] B(0, 2N)\B(0,N) (1.2.4)

for some dyadic M,N > 1.

Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.5)

where [(1 + [[absolute value of z].sup.2]).sup.p/4-1] is a smooth function of z.

If in (1.2.3), (1.2.4), M > N, partition [Z.sup.3] in boxes I of size N and write

[[phi].sub.1] = [summation over (I)] [P.sub.I] [[phi].sub.1],

and by almost orthogonality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.6)

For fixed I, estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.7)

and in view of (1.1.12) and Remarks (1), (2) above and (1.2.4),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.9)

To bound the last factor in (1.2.7), distinguish the cases

(A) 4 [less than or equal to] p [less than or equal to] 16/3

Then 8(p/2 - 2) [less than or equal to] 16/3 and by (1.2.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.10)

Substitution of (1.2.8)-(1.2.10) in (1.2.7) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.11)

hence

(1.2.6) < [N.sup.-1/6+].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.12)

and

(1.2.7) [less than or equal to] [N.sup.p/4 - 3/2] + [[parallel] [P.sub.I]][phi].sub.1] [parallel].sub.2] (1.2.13)

(1.2.6) < [N.sup.p/4 - 3/2+].

This proves Proposition 1.2.

1.3 IMPROVED [L.sup.4]-BOUND

It follows from (1.1.12) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.1)

In this section, we will obtain the following first improvement:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.2)

Restrict [??] to a one level set, thus

[??] = [??][chi][[omega].sub.[mu]] (1.3.3)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.4)

In what follows, we assume f of the form (1.3.3).

Lemma 1.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.5)

Proof. From estimates (1.1.4) and (1.1.5') with p = 4 and letting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we get the following bound on [[parallel] [e.sup.it[DELTA]f [parallel].sup.2.sub.4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.6)

Recall also estimate (1.1.9) for p = 16/3,

[([summation] [[absolute value of [[??].sub.l]].sup.4).sup.1/4] < [N.sup.3/2+]. (1.3.7)

Hence, if we denote for L [greater than or equal to] 1 (a dyadic integer)

[L.sub.L] = {l [member of] Z | [absolute value of [[??].sub.l] ~ [N.sup.3/2] + [L.sup.-1/4]}, (1.3.8)

it follows that

[absolute value of [absolute value of [L.sub.L]] < L. (1.3.9)

Estimate (1.3.6) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.10)

and restrict in (1.3.10) the l-summation to [L.sub.L].

There are the following two bounds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.11)

and also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.12)

Taking the minimum of (1.3.11), (1.3.12), we obtain [.sup.1/3][N.sup.1+]. Summing over dyadic values of L [??] [N.sup.2], the estimate follows.

Next, we need a discrete maximal inequality of independent interest.

Lemma 1.2 Consider the following maximal function on [Z.sup.3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.14)

For

[lambda] > [N.sup.1/2] [[parallel] F [parallel].sub.2] (1.3.14)

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.15)

([[parallel] F [parallel].sub.2] denotes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let A = [[F.sup.*] > [lambda]] [subset] [Z.sup.3]. Thus for x [member of] A, there is [l.sub.x] s.t.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Estimate as usual

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.16)

Use the crude bound [absolute value of [[??].sub.l] < [N.sup.3/2+] from (1.3.7) and denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.17)

From (1.3.16), we conclude that

[absolute value of A] < [N.sup.3/2+] [[parallel] F [parallel].sup.2.sub.2] [[lambda].sup.-2] (1.3.18)

if

[lambda] > [[parallel] F [parallel].sub.2] [K.sup.1/2]. (1.3.19)

It remains to evaluate K.

(Continues...)

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