Improved Local Well-Posedness in Sobolev Spaces for Two-Dimensional Compressible Euler Equations

TL;DR

This paper proves local well-posedness for 2D compressible Euler equations in Sobolev spaces with lower regularity requirements, using wave-transport analysis and Strichartz estimates, improving previous regularity thresholds.

Contribution

It introduces a novel approach applying Smith-Tataru method and wave-transport structure to lower the regularity needed for well-posedness in 2D Euler equations.

Findings

Established local existence and uniqueness in lower Sobolev spaces.

Applied Smith-Tataru method to compressible Euler equations.

Achieved a 0.25-order improvement in regularity requirements.

Abstract

We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $\rho_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, \rho_0, w_0, \nabla w_0)\in H^{\frac74+}(\mathbb{R}^2)\times H^{\frac74+}(\mathbb{R}^2) \times H^{\frac32}(\mathbb{R}^2) \times L^{8}(\mathbb{R}^2)$. The proof applies Smith-Tataru method \cite{ST} and the inherent wave-transport structure of the two-dimensional compressible Euler equations. The key observation is that Strichartz estimates hold when the regularity requirement for vorticity is lower than that for velocity and density, even though the gradient of vorticity appears as a source term in the velocity wave equation. Furthermore, our result presents an improvement of $\frac{1}{4}$-order regularity compared to previous results \cite{Z1} and \cite{Z2}.