The unique global solvability and optimal time decay rates for a multi-dimensional compressible generic two-fluid model with capillarity effects

TL;DR

This paper proves the unique global existence and optimal decay rates of solutions for a multi-dimensional compressible two-fluid model with capillarity effects, using advanced analytical techniques in critical spaces.

Contribution

It establishes the first global solvability results in critical spaces for this model with capillarity, and derives optimal decay rates under mild conditions.

Findings

Global unique solvability in critical spaces

Optimal time decay rates for solutions

Applicability to multi-dimensional models with capillarity

Abstract

The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension $N\geq2$. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show 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 optimal time decay rates for the constructed global solutions.