**Memoirs of the American Mathematical Society**

2018;
108 pp;
Softcover

MSC: Primary 35; 76;

**Print ISBN: 978-1-4704-3203-4
Product Code: MEMO/256/1229**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4921-6
Product Code: MEMO/256/1229.E**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

# Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations

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*T. Alazard; N. Burq; C. Zuily*

This memoir is devoted to the proof of a well-posedness result for
the gravity water waves equations, in arbitrary dimension and in fluid
domains with general bottoms, when the initial velocity field is not
necessarily Lipschitz. Moreover, for two-dimensional waves, the authors
consider solutions such that the curvature of the initial free surface
does not belong to \(L^2\).

The proof is entirely based on
the Eulerian formulation of the water waves equations, using microlocal
analysis to obtain sharp Sobolev and Hölder estimates. The authors first prove
tame estimates in Sobolev spaces depending linearly on Hölder norms
and then use the dispersive properties of the water-waves system,
namely Strichartz estimates, to control these Hölder norms.

#### Table of Contents

# Table of Contents

## Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Strichartz estimates 1320
- 2.1. Symmetrization of the equations 1320
- 2.2. Smoothing the paradifferential symbol 1623
- 2.3. The pseudo-differential symbol 2027
- 2.4. Several reductions 2027
- 2.5. Straightening the vector field 2229
- 2.6. Reduction to a semi-classical form 2431
- 2.7. The parametrix 3037
- 2.8. The dispersion estimate 4350
- 2.9. The Strichartz estimates 4855

- Chapter 3. Cauchy problem 5360
- Appendix A. Paradifferential calculus 7380
- Appendix B. Tame estimates for the Dirichlet-Neumann operator 7986
- Appendix C. Estimates for the Taylor coefficient 93100
- Appendix D. Sobolev estimates 97104
- Appendix E. Proof of a technical result 103110
- Bibliography 105112
- Back Cover Back Cover1120