Maximal Regularity for Compressible Two-Fluid System

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 Lp-Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p-L_q$$\end{document} 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.