Well-posedness for rough solutions of the 3D compressible Euler equations

TL;DR

This paper establishes local well-posedness for the 3D compressible Euler equations with rough initial data, including velocity, density, and vorticity, and introduces new results on continuous dependence and entropy.

Contribution

It improves regularity conditions for well-posedness and proves continuous dependence for rough solutions, extending previous results to include entropy considerations.

Findings

Proves local existence, uniqueness, and continuous dependence for rough initial data.

Extends well-posedness results to include solutions with entropy.

Demonstrates continuous dependence on initial data under lower regularity conditions.

Abstract

In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity $(\mathbf{v}_0, \rho_0, \mathbf{w}_0) \in H^{2+} \times H^{2+} \times H^{2}$, improving on the regularity conditions of \cite{WQEuler}. The continuous dependence on initial data for rough solutions of the compressible Euler system is new, even with the same regularity conditions as in \cite{WQEuler}. In addition, we prove new local well-posedness results for the 3D compressible Euler system with entropy.