Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations.
- 1st ed.
- 1 online resource (120 pages)
- Memoirs of the American Mathematical Society Series ; v.256 .
- Memoirs of the American Mathematical Society Series .
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 problem -- 3.1. 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 estimates -- Appendix E. Proof of a technical result -- Bibliography -- Back Cover.
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.
9781470449216
Water waves-Mathematical models. Waves-Mathematical models. Streamflow velocity-Mathematical models. Inequalities (Mathematics).