The maximal regularity and its application to a multi-dimensional non-conservative viscous compressible two-fluid model with capillarity effects in $L^{ p}$-type framework

TL;DR

This paper develops maximal regularity estimates for a multi-dimensional non-conservative viscous compressible two-fluid model with capillarity effects, enabling global well-posedness and decay rates in a critical $L^p$-type Besov space framework.

Contribution

It extends previous work to a critical functional setting using maximal regularity, allowing analysis with large, oscillatory initial data in Besov spaces with negative regularity.

Findings

Established maximal regularity estimates for the linearized system.

Proved global well-posedness for initial data near equilibrium in $L^p$-Besov spaces.

Derived optimal decay rates for solutions under low-frequency decay assumptions.

Abstract

The present paper is the continuation of work \cite{XC}, devoted to extending it to a critical functional framework which is not related to the energy space. Employing the special dissipative structure of the non-conservative viscous compressible two-fluid model with capillarity effects, we first exploit the maximal regularity estimates for the corresponding linearized system in all frequencies which behaves like the heat equation. Then we construct the global well-posedness for the multi-dimensional model when the initial data are close to a stable equilibrium state in the sense of suitable $L^{ p}$-type Besov norms. As a consequence, this allows us to work in the framework of Besov space with negative regularity indices and this fact is particularly important when the initial data are large highly oscillating in physical dimensions $N= 2, 3$. Furthermore, based on a refined time weighted inequalities in the Fourier spaces, we also establish optimal time decay rates for the constructed global solutions under a mild additional decay assumption involving only the low frequencies of the initial data.