Well-posedness and blow-up criterion for strong solutions of the compressible Navier-Stokes/Allen-Cahn system with vacuum

Abstract

This paper is devoted to the study of strong solutions for the compressible Navier-Stokes/Allen-Cahn system in bounded domain $\Omega \subset {\mathbb{R}}^3$, allowing for the presence of initial vacuum. A characteristic of this system is the strong coupling between density and the Allen-Cahn equation, which leads to strong degeneracy in vacuum regions. Under a compatibility condition on the initial phase-field variable, we establish the local existence and uniqueness of strong solutions for $0\le {\rho }_0\in W^{1,q}$ with $q\in (3,6)$, $u_0\in H_0^1$ and ${\chi }_0\in H^2$. Owing to time-weighted estimates, no compatibility condition is required for the velocity, but these estimates introduce a singularity in proving uniqueness. We then establish a criterion for the possible breakdown of such a local strong solution at finite time in terms of blow-up of the quantities $\Vert \nabla u{\Vert }_{L_t^1{L}_x^{\infty }}$, $\Vert u{\Vert }_{L_t^2{L}_x^{\infty }}$ and $\Vert \nabla \chi {\Vert }_{L_t^2{L}_x^{\infty }}$.