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

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, and our main purpose is three–fold: First, for any integer \(\ell \ge 3\), 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–Korteweg system. This can be regarded as non-straightforward generalization from the compressible Navier–Stokes–Korteweg system to the two–fluid model. Compared to the compressible Navier–Stokes–Korteweg system, many new mathematical challenges occur since the corresponding model is non-conservative, and its nonlinear structure is very terrible, and the corresponding linear system cannot be diagonalizable. One of key observations here is that to tackle with the strongly coupling terms, we will introduce the linear combination of the fraction densities (\(\beta ^+\alpha ^+\rho ^++\beta ^-\alpha ^-\rho ^-\)), and explore its good regularity, which is particularly better than ones of two fraction densities (\(\alpha ^\pm \rho ^\pm \)) themselves. 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.