The non-conservative compressible two-fluid system with common pressure: Global existence and sharp time asymptotics
Ling-Yun Shou, Jiayan Wu, Lei Yao, Yinghui Zhang
TL;DR
This work analyzes global-in-time stability for a compressible two-fluid system with a common pressure in $\mathbb{R}^d$ ($d\ge3$), where the dynamics are partially dissipative and the Shizuta–Kawashima condition fails. A pure energy method in Besov spaces is developed to obtain global existence and uniqueness for small perturbations around a constant equilibrium, without relying on $L^1$ data or spectral analysis. Under additional low-regularity Besov assumptions on the low-frequency part of the initial data, the authors establish optimal time-decay rates and characterize the asymptotic behavior of the non-dissipative components $\alpha^{\pm}\rho^{\pm}$, showing convergence to a profile $R_{\infty}^{\pm}$. A key innovation is the introduction of the two-phase effective velocity $v$ and a hierarchical set of Lyapunov functionals, enabling control of cross-derivative terms and maximal regularity estimates for the Lamé-type equations, which together circumvent the lack of SK symmetry and $L^1$-smallness requirements.
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 $\dot{B}^{σ_0}_{2,\infty}$ 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.