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Edited by Jean Bourgain, Carlos E. Kenig & S. Klainerman

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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...