Global-in-time Discontinuous Solutions for the Two-Phase Model of Compressible Fluids with Density-Dependent Viscosity

TL;DR

The paper proves global-in-time existence and uniqueness of weak solutions for a 3D two-phase compressible flow with density-dependent viscosity, where immiscible phases are separated by a sharp interface and the total density is piecewise Hölder with a $\mathscr{C}^{1+\alpha}$ interface. An intermediate regularity framework is developed so that a flow map for the velocity is well defined, and the interface regularity persists over time. Central to the argument are piecewise Hölder estimates for the density and velocity, and time-weighted energy bounds (via $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{A}_3$) that control dissipation and enable a bootstrap that yields global weak solutions. Under structural conditions that enforce exponential decay of pressure and velocity-jump across the interface, the jumps vanish exponentially, ensuring long-time stability of the sharp interface. This advances the mathematical theory of density patches with density-dependent viscosity in a fully 3D compressible setting, providing a robust global existence/uniqueness result and interface persistence.

Abstract

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 $α$-Hölder continuous on both sides of a $\mathscr C^{1+α}$-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 Hölder regularity of the density and the $\mathscr C^{1+α}$ regularity of the sharp interface.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof : Proof of \ref{['sec2:pwholder']}