A well-posedness result for the compressible two-fluid model with density-dependent viscosity

Abstract

In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space $\mathbb R^d\;(d\in \{2,3\}$). The two phases of the mixture are separated by a $\mathscr{C}^{1+\alpha}$-regular sharp interface $\mathcal{C}$ across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is $\alpha$-H\"older continuous on both sides of $\mathcal{C}$. The initial velocity belongs to the Sobolev space $H^1(\mathbb R^d)$, and the divergence of the initial stress tensor belongs to $L^2(\mathbb R^d)$. The later assumption expresses somehow the continuity of the stress tensor. This result is more general than the one by Tani [32], as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.