Navier-Stokes/Allen-Cahn方程弱解的尖锐界面极限问题  

On a sharp interface limit for weak solutions of the Navier-Stokes/Allen-Cahn equations

作　　者：梁之磊 王德华 Zhilei Liang;Dehua Wang

机构地区：[1]西南财经大学数学学院,成都611130 [2]Department of Mathematics,University of Pittsburgh,Pittsburgh,PA 15260,USA

出　　处：《中国科学:数学》2025年第3期685-702,共18页Scientia Sinica:Mathematica

基　　金：美国国家自然科学基金(批准号:DMS-2219384);四川省科技计划(批准号:2023ZYD0003);四川省社会科学哲学基金(批准号:SCJJ23ND196);中央高校基本科研业务费(批准号:JBK2304082)资助项目。

摘　　要：宏观上不混溶的复合黏性流体的运动规律主要由尖锐界面模型和扩散界面模型来描述.NavierStokes/Allen-Cahn(NS/AC)耦合方程是描述扩散交界过程最主要的微观连续介质模型之一,本文研究NS/AC方程弱解的尖锐界面极限问题.具体地,考虑3维有界光滑域中NS/AC方程弱解,假设连续介质模型的扩散层厚度ϵ(>0)趋向于0,本文证明弱解序列(u^(ϵ),c^(ϵ))在弱拓扑意义下收敛到极限函数(u^(0),c^(0)),并讨论极限函数所满足的方程与平均曲率流方程之间的联系.证明主要基于NS/AC方程弱解关于ϵ的一致能量估计和测度意义下的紧性性质.The diffusion interface model describes the motion of two macroscopically immiscible,viscousfluids.This article concerns the sharp interface limit for the diffusion interface model.We focus on the NavierStokes/Allen-Cahn(NS/AC)equations,one of the most important diffusion interface models.In particular,we study the zeroϵ-limit for the weak solutions(u^(ϵ),c^(ϵ))to NS/AC equations in a bounded domain in dimension three,whereϵ(>0)is the width of the diffuse interface.We prove that the solution sequence(u^(ϵ),c^(ϵ))converges to a limit(u^(0),c^(0))in the weak sense;in addition,we discuss the relationship between(u^(0),c^(0))and the solutions to the sharp interface model.Our approach is based on the uniform inϵestimates and the compactness results in Radon measures.

关 键 词：复合流体 弱解 Navier-Stokes/Allen-Cahn方程 尖锐界面极限 

分 类 号：O175[理学—数学]