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

We investigate global existence and optimal decay rates of a generic non-conservative compressible two–fluid model with general constant viscosities and capillary coefficients. Bresch, et al. in the seminal work (Arch Rational Mech Anal 196:599–629, 2010) considered the compressible two–fluid model with a special type of density–dependent viscosities (μ±(ρ±) = μ±ρ±, λ± = 0). However, as indicated by themselves, their methods cannot deal with the case of constant viscosity coefficients. Besides, Cui, et al. (SIAM J Math Anal 48:470–512, 2016) studied the same model with a more special type of density-dependent viscosity (μ±(ρ±) = νρ±, λ± = 0) and equal capillary coefficients (σ+ = σ− = σ ). Since their analysis relies heavily this special choice for viscosities and capillary coefficients, the case of general constant viscosities and capillary coefficients cannot be handled in their settings. The main novelty of this work is three–fold: First, for any integer l≥ 3, we show that the densities and velocities converge to their corresponding equilibrium states at the L rate (1+ t)− 3 4 , and the k(∈ [1,l])–order spatial derivatives of them converge to zero at the L rate (1+t)− 3 4 − 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 (β+α+ρ+ + β−α−ρ−) converges to its corresponding equilibrium state at the L rate (1+ t)− 3 4 , and its k(∈ [1,l])–order spatial derivative converges to zero at the L rate (1+ t)− 3 4 − k 2 , but the fraction densities (α±ρ±) themselves converge to their corresponding equilibrium states at the L rate (1+ t)− 1 4 , and the k(∈ [1,l])–order spatial derivatives of them converge to zero at the L rate (1+ t)− 1 4 − k 2 , which are slower than ones of their linear combination (β+α+ρ++β−α−ρ−) 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.