Optimal Large-Time Behavior of the Two-Phase Fluid Model in the Whole Space

Abstract

We investigate the large-time behavior of strong solutions to a two-phase fluid model in the whole space $\mathbb R^3$. This model was first derived by Choi [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122] by taking the hydrodynamic limit from the Vlasov--Fokker--Planck/isentropic Navier--Stokes equations with strong local alignment forces. Under the assumption that the initial perturbation around an equilibrium state is sufficiently small, the global well-posedness issue has been established in [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122]. However, as indicated by Choi, the large-time behavior of these solutions has remained an open problem. In this article, we resolve this problem by proving convergence to its associated equilibrium with the optimal rate which is the same as that of the heat equation. Particularly, the optimal convergence rates of the higher-order spatial derivatives of the solutions are also obtained. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates. Our method is based on Hodge decomposition, low-frequency and high-frequency decomposition, delicate spectral analysis, and energy methods.