On strong dynamics of compressible two-component mixture flow

TL;DR

This paper establishes local and global existence of strong solutions for a compressible two-component mixture flow system, using entropic variables and maximal regularity estimates, under small initial data assumptions.

Contribution

It introduces a symmetric reformulation of the system via entropic variables and proves existence results in an $L_p-L_q$ framework for the first time.

Findings

Proved local existence of solutions using Lagrangian coordinates.

Established global existence under small initial data conditions.

Derived exponential decay estimates for solutions.

Abstract

We investigate a system describing the flow of a compressible two-component mixture. The system is composed of the compressible Navier-Stokes equations coupled with non-symmetric reaction-diffusion equations describing the evolution of fractional masses. We show the local existence and, under certain smallness assumptions, also the global existence of unique strong solutions in $L_p-L_q$ framework. Our approach is based on so called entropic variables which enable to rewrite the system in a symmetric form. Then, applying Lagrangian coordinates, we show the local existence of solutions applying the $L_p$-$L_q$ maximal regularity estimate. Next, applying exponential decay estimate we show that the solution exists globally in time provided the initial data is sufficiently close to some constants. The nonlinear estimates impose restrictions $2<p<\infty, \ 3<q<\infty$. However, for the purpose of generality we show the linear estimates for wider range of $p$ and $q$.