Global well-posedness and stability of three-dimensional isothermal Euler equations with damping

TL;DR

This paper proves the global existence, stability, and decay rates of solutions to three-dimensional isothermal Euler equations with damping, even for some large initial data, by exploiting the equations' structural properties.

Contribution

It establishes the global well-posedness and stability for large initial data in certain norms for the 3D isothermal Euler equations with damping, extending previous results beyond small data regimes.

Findings

Global well-posedness and stability for large initial data

Optimal algebraic decay rates obtained

Reduction to a symmetrically hyperbolic system with partial damping

Abstract

The global well-posedness and stability of solutions to the three-dimensional compressible Euler equations with damping is a longstanding open problem. This problem was addressed in \cite{WY, STW} in the isentropic regime (i.e. $\gamma>1$) for small smooth solutions. In this paper, we prove the global well-posedness and stability of smooth solutions to the three-dimensional isothermal Euler equations ($\gamma=1$) with damping for some partially large initial values, i.e., $\|(\rho_0-\rho_*,u_0)\|_{L^2}$ could be large, but $\|D^3(\rho_0-\rho_*,u_0)\|_{L^2}$ is necessarily small. Moreover, the optimal algebraic decay rate is also obtained. The proof is based on the observation that the isothermal Euler equations with damping possess a good structure so that the equations can be reduced into a symmetrically hyperbolic system with partial damping, i.e., \eqref{au}. In the new system, all desired a priori estimates can be obtained under the assumption that $\int_0^T(\|\nabla \mathrm{ln}\rho\|_{L^{\infty}}+\|\nabla u\|_{L^{\infty}}) \mathrm{d}t $ is small. The assumption can be verified through the low-high frequency analysis via Fourier transformation under the condition that $\|D^3(\rho_0-\rho_*,u_0)\|_{L^2}$ is small, but $\|(\rho_0-\rho_*,u_0)\|_{L^2}$ could be large.