SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5748-5774, January 2020.
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.

中文翻译：

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两相流模型在整个空间中的最优长时间行为

SIAM数学分析期刊，第52卷，第6期，第5748-5774页，2020年1月。
我们研究了在整个空间$ \ mathbb R ^ 3 $中两相流体模型的强解的长时间行为。该模型首先由Choi [SIAM J. Math。Anal。，48（2016），pp。3090--3122]，采用具有强大局部对准力的Vlasov-Fokker-Planck / isentropic Navier-Stokes方程的流体动力极限。假设围绕平衡态的初始扰动足够小，则在[SIAM J. Math。Anal。，48（2016），pp。3090--3122]。但是，正如Choi所指出的，这些解决方案的长时间行为仍然是一个未解决的问题。在本文中，我们通过证明以最佳速率收敛到其关联的平衡来解决该问题，该最优速率与热方程相同。尤其，还获得了解的高阶空间导数的最优收敛速度。此外，对于精心挑选的初始数据，我们还显示了收敛速度的下界。我们的方法基于Hodge分解，低频和高频分解，精细的频谱分析和能量方法。