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

Author: T Alazard; N Burq; C Zuily
Publisher: Providence : American Mathematical Society, 2019.
Series: Memoirs of the American Mathematical Society Ser.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Alazard, T.
Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations.
Providence : American Mathematical Society, ©2019
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: T Alazard; N Burq; C Zuily
ISBN: 9781470449216 1470449218
OCLC Number: 1083457987
Description: 1 online resource (120 pages)
Contents: Cover; Title page; Chapter 1. Introduction; 1.1. Equations and assumptions on the fluid domain; 1.2. Regularity thresholds for the water waves; 1.3. Reformulation of the equations; 1.4. Main result; 1.5. Paradifferential reduction; 1.6. Strichartz estimates; Chapter 2. Strichartz estimates; 2.1. Symmetrization of the equations; 2.2. Smoothing the paradifferential symbol; 2.3. The pseudo-differential symbol; 2.4. Several reductions; 2.5. Straightening the vector field; 2.6. Reduction to a semi-classical form; 2.7. The parametrix; 2.8. The dispersion estimate; 2.9. The Strichartz estimates Chapter 3. Cauchy problem3.1. A priori estimates; 3.2. Contraction estimates; 3.3. Passing to the limit in the equations; 3.4. Existence and uniqueness; Appendix A. Paradifferential calculus; A.1. Notations and classical results; A.2. Symbolic calculus; A.3. Paraproducts and product rules; Appendix B. Tame estimates for the Dirichlet-Neumann operator; B.1. Scheme of the analysis; B.2. Parabolic evolution equation; B.3. Paralinearization; Appendix C. Estimates for the Taylor coefficient; Appendix D. Sobolev estimates; D.1. Introduction; D.2. Symmetrization of the equations D.3. Sobolev estimatesAppendix E. Proof of a technical result; Bibliography; Back Cover
Series Title: Memoirs of the American Mathematical Society Ser.

Abstract:

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 l.

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