论文标题
Mullins-Sekerka流动的弱解决方案作为希尔伯特空间梯度流量
Weak solutions of Mullins-Sekerka flow as a Hilbert space gradient flow
论文作者
论文摘要
我们为Mullins-Sekerka方程提出了一种新颖的弱解理论，主要是从梯度流动的角度进行的。先前的存在导致由于勒克豪斯和Sturzenhecker引起的弱解决方案（Calc。Var。Pde3，1995）或Röger（Siam J. Math。Anal。37，2005），使既有尖锐的能量耗散原理，又包含了界面和结构域交汇处的接触角度较弱的表述。为了结合这些内容，我们引入了一个功能框架，编码弱的解决方案概念，用于Mullins-Sekerka流量基本上仅依靠（i）（i）（i）de〜giorgi精神的单个尖锐的能量耗散不平等，以及（ii）弱配方，用于通过下属能量的首次分布分布的任意固定接触角来进行任意固定接触角。两种成分都是不断发展的相位指示剂界面的固有的，并且可以从其电势得出带有电势的显式分布PDE公式。弱解的存在是通过自然相关的最小化运动方案的随后极限点建立的。平滑的解决方案与经典的mullins-sekerka流相一致，更进一步的是，我们希望我们的解决方案概念至少在原则上是可正常的，与最近开发的相对熵方法用于曲率驱动的界面演化。
We propose a novel weak solution theory for the Mullins-Sekerka equation primarily motivated from a gradient flow perspective. Previous existence results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3, 1995) or Röger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both a sharp energy dissipation principle and a weak formulation of the contact angle at the intersection of the interface and the domain boundary. To incorporate these, we introduce a functional framework encoding a weak solution concept for Mullins-Sekerka flow essentially relying only on (i) a single sharp energy dissipation inequality in the spirit of De~Giorgi, and (ii) a weak formulation for an arbitrary fixed contact angle through a distributional representation of the first variation of the underlying capillary energy. Both ingredients are intrinsic to the interface of the evolving phase indicator and an explicit distributional PDE formulation with potentials can be derived from them. Existence of weak solutions is established via subsequential limit points of the naturally associated minimizing movements scheme. Smooth solutions are consistent with the classical Mullins-Sekerka flow, and even further, we expect our solution concept to be amenable, at least in principle, to the recently developed relative entropy approach for curvature driven interface evolution.
