The non-conservative compressible two-fluid system with common pressure: Global existence and sharp time asymptotics

TL;DR

This paper establishes the global existence and sharp asymptotic behavior of solutions for a non-conservative compressible two-fluid system with common pressure, using a novel energy method independent of spectral analysis.

Contribution

It introduces a new energy method in Besov spaces to prove global solutions without spectral analysis or $L^1$ smallness assumptions, and characterizes the asymptotic convergence of non-dissipative components.

Findings

Global classical solutions exist for initial data close to equilibrium.

Optimal time decay rates are achieved for lower regularity initial data.

First characterization of asymptotic convergence of non-dissipative components.

Abstract

This paper concerns the global-in-time evolution of a generic compressible two-fluid model in $\mathbb{R}^d$ ($d\geq3$) with the common pressure law. Due to the non-dissipative properties for densities and two different particle paths caused by velocities, the system lacks the usual symmetry structure and is partially dissipative in the sense that the Shizuta-Kawashima condition is violated, which makes it challenging to study its large-time stability. By developing a pure energy method in the framework of Besov spaces, we succeed in constructing a unique global classical solution to the Cauchy problem when the initial data are close to their constant equilibria. Compared to the previous related works, the main novelty lies in that our method is independent of the spectral analysis and does not rely on the $L^1$ smallness of the initial data. Furthermore, if additionally the initial perturbation is bounded in $\dot{B}^{\sigma_0}_{2,\infty}$ type spaces with lower regularity, the optimal time convergence rates are also obtained. In particular, the asymptotic convergence of the non-dissipative components toward their equilibrium states is first characterized.