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Strichartz estimates for geophysical fluid equations using Fourier restriction theory

Corentin Gentil, Côme Tabary

Abstract

We prove Strichartz estimates for the semigroups associated to stratified and/or rotating inviscid geophysical fluids using Fourier restriction theory. We prove new results for rotating stratified fluids, and recover results from [16] for rotation only, and from [17] for stratification only. Our restriction estimates are obtained by the slicing method [20], which relies on the well-known Tomas-Stein theorem for 2-dimensional spheres. To our knowledge, such a method has never been used in this setting. Moreover, when the fluid is stratified, our approach yields sharp estimates, showing that the slicing method captures all the available curvature of the surfaces of interest.

Strichartz estimates for geophysical fluid equations using Fourier restriction theory

Abstract

We prove Strichartz estimates for the semigroups associated to stratified and/or rotating inviscid geophysical fluids using Fourier restriction theory. We prove new results for rotating stratified fluids, and recover results from [16] for rotation only, and from [17] for stratification only. Our restriction estimates are obtained by the slicing method [20], which relies on the well-known Tomas-Stein theorem for 2-dimensional spheres. To our knowledge, such a method has never been used in this setting. Moreover, when the fluid is stratified, our approach yields sharp estimates, showing that the slicing method captures all the available curvature of the surfaces of interest.
Paper Structure (20 sections, 11 theorems, 142 equations)

This paper contains 20 sections, 11 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strichartz estimates
  4. Restriction theory
  5. Geophysical fluids
  6. Standard method for geophysical fluid equations
  7. Further results
  8. Layout and Notations
  9. On the slicing method
  10. The Primitive System
  11. Linear analysis
  12. Proof of Theorem \ref{['th:pri']}
  13. Sharpness: proof of Proposition 1
  14. Stably stratified Boussinesq system
  15. Derivation of the system and linear analysis
  16. ...and 5 more sections

Key Result

theorem 1

There exists a universal constant $C>0$ such that the following homogeneous Strichartz estimate holds for all $g\in \dot{H}^1 (\mathbb{R}^3)$: Moreover, for all $f\in L^1_t\dot H^1_x$, the following inhomogeneous estimate holds:

Theorems & Definitions (17)

  • theorem 1
  • proposition 1
  • theorem 2: tsb, ts
  • theorem 3
  • theorem 4
  • theorem 5: stri
  • lemma 1
  • proof : Theorem 5
  • lemma 2
  • proof : Lemma 2
  • ...and 7 more