Well-posedness for rough solutions of the 3D compressible Euler equations
Lars Andersson, Huali Zhang
TL;DR
The paper establishes local well-posedness for the 3D compressible Euler equations with rough initial data, achieving existence, uniqueness, and continuous dependence in Sobolev spaces where $( extbf{v}_0, ho_0)$ lie in $H^{2+}$ and the vorticity $ extbf{w}_0$ lies in $H^2$. Central to the approach is a reduction to a hyperbolic-wave-transport system using an acoustic metric $g$, a velocity decomposition $ extbf{v}= extbf{v}_++ extbf{v}_-$, and Strichartz-type estimates for the linear wave equation with respect to $g$, which enable control of dispersive effects despite rough vorticity. The authors develop energy estimates that couple hyperbolic energy with transport/vorticity dynamics, prove small-data existence via smooth-approximation arguments, and then extend to large data by a continuity/compactness framework, including continuous dependence on initial data for rough solutions. Additional results cover local well-posedness with entropy and refined isentropic cases, demonstrating that the acoustic-geometry-based Strichartz framework can handle low-regularity vorticity and entropy transport. The work advances the edge of low-regularity well-posedness for compressible flows and provides tools potentially applicable to broader hyperbolic-transport systems with dispersive features.
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 $(\mathbf{v}_0, ρ_0, \mathbf{w}_0) \in H^{2+} \times H^{2+} \times H^{2}$, 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.