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 $Ω\subset\mathbb R^3$, 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 $0\leρ_0\in W^{1,q}$ with $q\in(3,6)$, $u_0\in H_0^1$ and $χ_0\in H^2$. 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 $\|\nabla u\|_{L_t^{1} L_x^{\infty}}$, $\|u\|_{L_t^{2} L_x^{\infty}}$ and $\|\nabla χ\|_{L_t^{2} L_x^{\infty}}$.