Global existence and decay rates for a generic compressible two-fluid model

TL;DR

This paper establishes the global existence and optimal decay rates of solutions for a generic compressible two-fluid model, revealing detailed convergence behaviors of densities, velocities, and fractional densities over time.

Contribution

It provides the first comprehensive analysis of decay rates for a general compressible two-fluid model, including convergence of densities, velocities, and fractional densities with optimal rates.

Findings

Densities and velocities converge to equilibrium at rate (1+t)^(-3/4)

Fractional densities converge at a slower rate (1+t)^(-1/4)

Decay rates are proven to be optimal with matching lower bounds

Abstract

We investigate global existence and optimal decay rates of a generic non-conservative compressible two--fluid model with general constant viscosities and capillary coefficients.The main novelty of this work is three--fold: First, for any integer $\ell\geq3$, we show that the densities and velocities converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and the $k$($\in [1, \ell]$)--order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, which are the same as ones of the compressible Navier--Stokes system, Navier--Stokes--Korteweg system and heat equation. Second, the linear combination of the fraction densities ($\beta^+\alpha^+\rho^++\beta^-\alpha^-\rho^-$) converges to its corresponding equilibrium state at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and its $k$($\in [1, \ell]$)--order spatial derivative converges to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, but the fraction densities ($\alpha^\pm\rho^\pm$) themselves converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{1}{4}}$, and the $k$($\in [1, \ell]$)--order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{1}{4}-\frac{k}{2}}$, which are slower than ones of their linear combination ($\beta^+\alpha^+\rho^++\beta^-\alpha^-\rho^-$) and the densities. We think that this phenomenon should owe to the special structure of the system. Finally, for well--chosen initial data, we also prove the lower bounds on the decay rates, which are the same as those of the upper decay rates. Therefore, these decay rates are optimal for the compressible two--fluid model.