Global-in-time Discontinuous Solutions for the Two-Phase Model of Compressible Fluids with Density-Dependent Viscosity
We are concerned with a model describing the motion of two compressible, immiscible fluids with density-dependent viscosity in the whole $\mathbb R^3$. The phases of the flow may have different pressure and viscosity laws and are separated by a sharp interface, across which the (total) density is discontinuous. Our goal is to study the persistence of the regularity of this sharp interface over time. More precisely, the dynamics of the flow are governed by three coupled equations: two hyperbolic equations (for the volume fraction of one phase and for the density) and a parabolic equation for the velocity field. We assume that, at the initial time, the density is $\alpha$-H\"older continuous on both sides of a $\mathscr C^{1+\alpha}$-regular surface across which it may be discontinuous. We prove the existence and uniqueness of a global-in-time weak solution in an intermediate regularity class that ensures the persistence of the piecewise H\"older regularity of the density and the $\mathscr C^{1+\alpha}$ regularity of the sharp interface.