论文标题
全球适应性和经典解决方案对通用可压缩两流体模型的大量时间行为
Global well--posedness and large time behavior of classical solutions to a generic compressible two-fluid model
论文作者
论文摘要
在本文中，我们在$ \ mathbb {r}^3 $中调查了具有常见压力（$ p^+= p^ - $）的通用压缩两流体模型。在某些较小的假设下，Evje-Wang-wen [Arch Ronication Mech Anal 221：1285--1316，2016]获得了全局溶液及其对3D可压缩两流体模型的最佳衰减速率，该模型$ P^+\ \ \ \ \ \ \ neq P^ - $。更准确地说，毛细管压力$ f（α^-ρ^ - ）= p^+ - p^ - \ neq 0 $被考虑在内，被认为是在平衡附近的严格降低函数。如Evje-Wang-Wen所示，该假设在他们的分析中起着关键作用，并且似乎对模型具有必不可少的稳定作用。然而，由于系统部分耗散，其非线性结构非常可怕，因此3D可压缩的两流体模型的全球适应性是一个挑战性的开放问题。在当前的工作中，通过利用模型的耗散结构并充分利用几种关键观察，我们建立了经典解决方案的全球存在和庞大的时间行为，并以常见压力为3D可压缩的两流体模型。这里的主要观察结果之一是，要关闭非隔离变量的高阶能量估计（即，分数密度$α_ {\ pm}ρ_\ pm $），我们将介绍两个速度的线性组合（$ u^^\ pm $）：探索其良好的规律性，这比两个速度本身的尤其好。
In this paper, we investigate a generic compressible two-fluid model with common pressure ($P^+=P^-$) in $\mathbb{R}^3$. Under some smallness assumptions, Evje-Wang-Wen [Arch Rational Mech Anal 221:1285--1316, 2016] obtained the global solution and its optimal decay rate for the 3D compressible two-fluid model with unequal pressures $P^+\neq P^-$. More precisely, the capillary pressure $f(α^-ρ^-)=P^+-P^-\neq 0$ is taken into account, and is assumed to be a strictly decreasing function near the equilibrium. As indicated by Evje-Wang-Wen, this assumption played an key role in their analysis and appeared to have an essential stabilization effect on the model. However, global well-posedness of the 3D compressible two-fluid model with common pressure has been a challenging open problem due to the fact that the system is partially dissipative and its nonlinear structure is very terrible. In the present work, by exploiting the dissipation structure of the model and making full use of several key observations, we establish global existence and large time behavior of classical solutions to the 3D compressible two-fluid model with common pressure. One of key observations here is that to closure the higher-order energy estimates of non-dissipative variables (i.e, fraction densities $α_{\pm}ρ_\pm$), we will introduce the linear combination of two velocities ($u^\pm$): $v=(2μ^++λ^+)u^+-(2μ^-+λ^-)u^-$ and explore its good regularity, which is particularly better than ones of two velocities themselves.
