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Dispersion of compressible rotating Euler equations with low Mach and Rossby numbers

Pengcheng Mu

Abstract

In this paper, we consider the low Mach and Rossby number singular limits and longtime existence of strong solution to the initial value problem of 3D compressible rotating Euler equations with ill-prepared initial data. We establish the Strichartz decay estimates that are uniform to the Mach number, the Rossby number, and the ratio of these two parameters for the associated linear propagator without any restrictions on the frequency. In particular, difficulties arisen from the degeneracy of the phase function and the vanishing of the ratio of the two parameters are addressed by elaborately designed splitting techniques and discussions for each frequencies. Using the decay estimates, we prove the longtime existence and obtain a rate of convergence to zero of strong solution to the compressible rotating Euler equations with initial data of finite energy in $\mathbb{R}^3$.

Dispersion of compressible rotating Euler equations with low Mach and Rossby numbers

Abstract

In this paper, we consider the low Mach and Rossby number singular limits and longtime existence of strong solution to the initial value problem of 3D compressible rotating Euler equations with ill-prepared initial data. We establish the Strichartz decay estimates that are uniform to the Mach number, the Rossby number, and the ratio of these two parameters for the associated linear propagator without any restrictions on the frequency. In particular, difficulties arisen from the degeneracy of the phase function and the vanishing of the ratio of the two parameters are addressed by elaborately designed splitting techniques and discussions for each frequencies. Using the decay estimates, we prove the longtime existence and obtain a rate of convergence to zero of strong solution to the compressible rotating Euler equations with initial data of finite energy in .

Paper Structure

This paper contains 15 sections, 5 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th-main']}
  3. Littlewood-Paley decomposition
  4. Study of the propagator
  5. Dispersion and longtime existence of the solution
  6. Proof of Proposition \ref{['pro1']}
  7. Estimates for $I_1$
  8. Estimates for high frequencies
  9. Estimates for middle frequencies
  10. Estimates for low frequencies
  11. Estimates for $I_2$
  12. Estimates for high frequencies
  13. Estimates for middle frequencies
  14. Estimates for low frequencies
  15. A result of TT* argument

Key Result

Proposition 1.1

Assume that $U_0\in H^{m}(\mathbb{R}^3)$ with $m\geq3$. Then for any $\varepsilon>0$ and $\delta>0$, the initial value problem CEL-in admits a unique strong solution $U=(b,u)^{\top}\in C([0,T^*];H^{m}(\mathbb{R}^{3}))$ for some $T^*>0$, and the following estimate holds: where the constant $M_1$ and the time $T^*$ are independent of $\varepsilon$ and $\delta$.

Theorems & Definitions (7)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 4.2
  • proof