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

TL;DR

This paper proves local well-posedness of 2D compressible Euler equations with low regularity initial data by establishing sharp Strichartz estimates, extending understanding of solutions with minimal smoothness.

Contribution

It introduces a novel approach using sharp Strichartz estimates and Smith-Tataru's framework to handle low regularity initial conditions in compressible Euler equations.

Findings

Established local well-posedness for initial data in low regularity spaces.

Extended the applicability of Strichartz estimates to compressible Euler equations.

Demonstrated the effectiveness of Smith-Tataru's method for quasi-linear wave equations in fluid dynamics.

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 $(\bv_0, \rho_0, \varpi_0) \in H^{s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2) \times H^2(\mathbb{R}^2), s>\frac{7}{4}$. Our strategy relies on Smith-Tataru's work \cite{ST} for quasi-linear wave equations.