Strichartz estimates and the Cauchy problem for the gravity water waves equations

Cover......Page 1
Title page......Page 2
Chapter 1. Introduction......Page 8
 1.1. Equations and assumptions on the fluid domain......Page 10
 1.2. Regularity thresholds for the water waves......Page 12
 1.4. Main result......Page 13
 1.5. Paradifferential reduction......Page 14
 1.6. Strichartz estimates......Page 16
 2.1. Symmetrization of the equations......Page 20
 2.2. Smoothing the paradifferential symbol......Page 23
 2.4. Several reductions......Page 27
 2.5. Straightening the vector field......Page 29
 2.6. Reduction to a semi-classical form......Page 31
 2.7. The parametrix......Page 37
 2.8. The dispersion estimate......Page 50
 2.9. The Strichartz estimates......Page 55
 3.1. A priori estimates......Page 60
 3.2. Contraction estimates......Page 66
 3.3. Passing to the limit in the equations......Page 75
 3.4. Existence and uniqueness......Page 78
 A.1. Notations and classical results......Page 80
 A.2. Symbolic calculus......Page 81
 A.3. Paraproducts and product rules......Page 82
 B.1. Scheme of the analysis......Page 86
 B.2. Parabolic evolution equation......Page 89
 B.3. Paralinearization......Page 92
Appendix C. Estimates for the Taylor coefficient......Page 100
 D.1. Introduction......Page 104
 D.3. Sobolev estimates......Page 105
Appendix E. Proof of a technical result......Page 110
Bibliography......Page 112
Back Cover......Page 120