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 \(\alpha \)-Hölder 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 normal component of the stress tensor. This result is more general than the one by Tani [Two-phase free boundary problem for compressible viscous fluid motion. Journal of Mathematics of Kyoto University 24(2): 243–267, 1984] 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.