Maximal Regularity for Compressible Two-Fluid System

TL;DR

This paper establishes local and global existence of regular solutions for a compressible two-fluid Navier-Stokes system using maximal regularity techniques, with implications for understanding complex fluid interactions.

Contribution

It introduces a novel approach combining transformations and maximal regularity estimates to analyze a two-fluid system with algebraic pressure closure.

Findings

Existence of local-in-time regular solutions.

Global solutions under small initial data.

Application of Lagrangian coordinates for analysis.

Abstract

We investigate a compressible two-fluid Navier-Stokes type system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. We are interested in regular solutions in a $L_p-L_q$ maximal regularity setting. We show that such solutions exists locally in time and, under additional smallness assumptions on the initial data, also globally. Our proof rely on appropriate transformation of the original problem, application of Lagrangian coordinates and maximal regularity estimates for associated linear problem.