On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

TL;DR

This paper proves the global existence, uniqueness, and decay rates of strong solutions for a multi-dimensional non-conservative viscous compressible two-fluid system, using Fourier analysis and energy methods.

Contribution

It establishes well-posedness and decay rates in critical spaces for a complex two-fluid model, extending previous results to a non-conservative, multi-dimensional setting.

Findings

Global unique strong solutions near equilibrium exist.

Solutions decay over time under low-frequency data assumptions.

Decay rates are explicitly characterized using Fourier analysis.

Abstract

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.