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Low regularity solutions of two-dimensional compressible Euler equations with dynamic vorticity

Huali Zhang

Abstract

By establishing a sharp Strichartz estimate for the velocity and density, we prove the local well-posedness of solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial velocity, density, and specific vorticity $(\bv_0, ρ_0, \varpi_0) \in H^{s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2) \times H^2(\mathbb{R}^2), s>\frac{7}{4}$. Our strategy relies on Smith-Tataru's work \cite{ST} for quasi-linear wave equations.

Low regularity solutions of two-dimensional compressible Euler equations with dynamic vorticity

Abstract

By establishing a sharp Strichartz estimate for the velocity and density, we prove the local well-posedness of solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial velocity, density, and specific vorticity . Our strategy relies on Smith-Tataru's work \cite{ST} for quasi-linear wave equations.

Paper Structure

This paper contains 26 sections, 36 theorems, 440 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Background
  4. Motivation
  5. Statement of the result
  6. Some definitions
  7. Statement of Results.
  8. A sketch of the proof
  9. Notations
  10. Outline of the paper
  11. Basic energy estimates and stability theorem
  12. The reduction to a hyperbolic system and a wave-transport system
  13. Commutator estimates
  14. New transport equations
  15. Energy estimates
  16. ...and 11 more sections

Key Result

Lemma 1.1

LS1 For 2D compressible Euler equations CEE0, it can be reduced to the following equation: where $\mathbf{T}=\partial_t + \mathbf{v} \cdot \nabla$.

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • ...and 62 more