发表机构
Università degli Studi di Parma（帕尔马大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Navier-Stokes/平均曲率流系统，提出基于De Giorgi能量耗散原理的varifold弱解概念，首次证明二维和三维中无条件全局时间存在性，并利用相对熵方法证明经典解在该弱解类中的唯一性。
AI 中文摘要
Navier-Stokes/平均曲率流系统描述了由尖锐界面分隔的不相溶、粘性、不可压缩两相流，其界面的演化由对流平均曲率流方程控制，并与考虑表面张力的两相Navier-Stokes方程耦合。该模型除其他应用外，还描述了干泡沫的动力学，并且作为具有非消失迁移率的Navier-Stokes/Allen-Cahn系统的尖锐界面极限出现。尽管应用广泛，但任何类型的弱解的全局时间存在性仍然是一个具有挑战性的开放问题。在这项工作中，我们基于De Giorgi式的尖锐能量耗散原理，在二维和三维环境维度中引入了一种新颖的varifold弱解概念，并首次证明了此类弱解的无条件全局时间存在性。我们还通过相对熵方法证明了，Navier-Stokes/平均曲率流系统的任何经典解在我们的新弱varifold解类中是唯一的。
英文摘要
The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.
Comments50 pages, comments are welcome
