Decay Rates for a Nonconservative Compressible Generic Two-Fluid Model

Abstract

In this paper, we are concerned with a nonconservative viscous compressible generic two-fluid model in $\mathbb{R}^3$, which is commonly used in industrial applications. The decay rates of classical solutions are established. Precisely, for any integer $s\geq 3$, we show that the velocities converge to the equilibrium states at the $L^2$-rate $(1+t)^{-\frac{3}{4}}$, and the $k(\in [1, s-2])$ order spatial derivatives of velocities converge to zero at the $L^2$-rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$ as the compressible Navier--Stokes system, Navier--Stokes--Korteweg system, etc., but the fraction densities converge to the equilibrium states at the $L^2$-rate $(1+t)^{-\frac{1}{4}}$, and the $k(\in [1, s-1])$ order spatial derivatives of the fraction densities converge to zero at the $L^2$-rate $(1+t)^{-\frac{1}{4}-\frac{k}{2}}$, which are slower than the $L^2$-rate $(1+t)^{-\frac{3}{4}}$ and $L^2$-rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$ for the compressible Navier--Stokes system, Navier--Stokes--Korteweg system, etc. See [R. J. Duan et al., Math. Models Methods Appl. Sci., 17 (2007), pp. 737--758], [T. P. Liu and W. K. Wang, Comm. Math. Phys., 196 (1998), pp. 145--173], and [Y. Wang and Z. Tan, J. Math. Anal. Appl., 379 (2011), pp. 256--271]. This is caused by the structure of the system itself, and we can prove that the convergence rates above are the same as its linearized system. The proof is based on detailed analysis of the Green's function to the linearized system and on elaborate energy estimates to the nonlinear system.