Well-Posedness and Asymptotic Decay of Solutions to the Three-Dimensional Euler Equations with Damping

Abstract

The global well-posedness of the multi-dimensional compressible Euler equations with damping remains a longstanding open problem. This problem has been partially resolved in the isentropic regime ({\it i.e.}, the adiabatic exponent \(γ>1\)) for small smooth initial data (see \cite{WY, STW}). In this paper, we establish the global well-posedness and asymptotic decay of smooth solutions of the Cauchy problem of the three-dimensional compressible Euler equations with damping for the isentropic regime \(γ>1\) and the isothermal regime \(γ=1\), allowing for partially large initial data. More precisely, the \(L^2\)-norm of the initial data is allowed to be large, while the third-order Sobolev norm of the initial data is assumed to be small. For the isentropic case, we develop a new analytical framework in which all required {\it a priori} estimates of solution $(\rho ,u)$ can be derived under the condition that ${\int }_0^T(\Vert \nabla \rho {\Vert }_{L^{\infty }}+\Vert \nabla u{\Vert }_{L^{\infty }})\mathrm{d}t$ remains sufficiently small. Moreover, we obtain the optimal algebraic decay rates of global solutions. Furthermore, we study the isothermal limit of solutions of the isentropic regime as $\gamma \to 1$, and establish the global well-posedness and asymptotic decay of solutions to the isothermal Euler equations with damping.