Global well--posedness and large time behavior of classical solutions to a generic compressible two-fluid model

TL;DR

This paper proves the global existence and analyzes the large-time behavior of classical solutions to a 3D compressible two-fluid model with common pressure, overcoming previous challenges due to the system's partial dissipation and complex nonlinear structure.

Contribution

It establishes the global well-posedness and large-time decay of classical solutions for the compressible two-fluid model with common pressure, using novel energy estimates and velocity combinations.

Findings

Proved global existence of classical solutions.

Analyzed large-time decay rates.

Introduced a new velocity combination for better estimates.

Abstract

In this paper, we investigate a generic compressible two-fluid model with common pressure ($P^+=P^-$) in $\mathbb{R}^3$. Under some smallness assumptions, Evje-Wang-Wen [Arch Rational Mech Anal 221:1285--1316, 2016] obtained the global solution and its optimal decay rate for the 3D compressible two-fluid model with unequal pressures $P^+\neq P^-$. More precisely, the capillary pressure $f(\alpha^-\rho^-)=P^+-P^-\neq 0$ is taken into account, and is assumed to be a strictly decreasing function near the equilibrium. As indicated by Evje-Wang-Wen, this assumption played an key role in their analysis and appeared to have an essential stabilization effect on the model. However, global well-posedness of the 3D compressible two-fluid model with common pressure has been a challenging open problem due to the fact that the system is partially dissipative and its nonlinear structure is very terrible. In the present work, by exploiting the dissipation structure of the model and making full use of several key observations, we establish global existence and large time behavior of classical solutions to the 3D compressible two-fluid model with common pressure. One of key observations here is that to closure the higher-order energy estimates of non-dissipative variables (i.e, fraction densities $\alpha_{\pm}\rho_\pm$), we will introduce the linear combination of two velocities ($u^\pm$): $v=(2\mu^++\lambda^+)u^+-(2\mu^-+\lambda^-)u^-$ and explore its good regularity, which is particularly better than ones of two velocities themselves.