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Increased lifespan for 3D compressible Euler flows with rotation

Haram Ko, Benoit Pausader, Ryo Takada, Klaus Widmayer

Abstract

We consider the compressible Euler equation with a Coriolis term and prove a lower bound on the time of existence of solutions in terms of the speed of rotation, sound speed and size of the initial data. Along the way, we obtain precise dispersive decay estimates for the linearized equation. In the incompressible limit, this improves current bounds for the incompressible Euler-Coriolis system as well.

Increased lifespan for 3D compressible Euler flows with rotation

Abstract

We consider the compressible Euler equation with a Coriolis term and prove a lower bound on the time of existence of solutions in terms of the speed of rotation, sound speed and size of the initial data. Along the way, we obtain precise dispersive decay estimates for the linearized equation. In the incompressible limit, this improves current bounds for the incompressible Euler-Coriolis system as well.

Paper Structure

This paper contains 23 sections, 15 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Long time existence for the compressible Euler system with rotation
  3. Main strategy
  4. Comparison to previous works
  5. Structure of the paper
  6. Notation
  7. Energy estimates
  8. Linear analysis
  9. Dispersive unknowns and their boundedness
  10. Decay and Strichartz estimates
  11. Quantified stationary phase theorem
  12. Dispersive decay estimate
  13. Decay of $\Sigma$
  14. Low frequency, $\kappa 2^k \ll 1$.
  15. High frequency, $\kappa 2^k \gg 1$.
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $q > 2$. For $m > \frac{7}{2}$ there exists $M=M(m,q)$ such that the solutions to the Cauchy problem to eq:CER2 with initial data $(\rho_0, u_0) \in H^m$ exist at least up to time In particular, When $m \in (\frac{5}{2}, \frac{7}{2})$, the solutions exist at least up to time In particular,

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 12 more