{"title":"作为希尔伯特空间梯度流的穆林斯-塞克尔卡流的弱解","authors":"","doi":"10.1007/s00205-023-01950-0","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions <span> <span>\\(d=2\\)</span> </span> and 3, 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 <em>(i)</em> a single sharp energy dissipation inequality in the spirit of De Giorgi, and <em>(ii)</em> 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. The 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.</p>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weak Solutions of Mullins–Sekerka Flow as a Hilbert Space Gradient Flow\",\"authors\":\"\",\"doi\":\"10.1007/s00205-023-01950-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3>Abstract</h3> <p>We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions <span> <span>\\\\(d=2\\\\)</span> </span> and 3, 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 <em>(i)</em> a single sharp energy dissipation inequality in the spirit of De Giorgi, and <em>(ii)</em> 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. The 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.</p>\",\"PeriodicalId\":55484,\"journal\":{\"name\":\"Archive for Rational Mechanics and Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-01-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archive for Rational Mechanics and Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00205-023-01950-0\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00205-023-01950-0","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
摘要 我们主要从梯度流的角度出发,提出了Mullins-Sekerka方程在维数\(d=2\)和3的新的弱解理论。之前由 Luckhaus 和 Sturzenhecker (Calc. Var. PDE 3, 1995) 或 Röger (SIAM J. Math. Anal. 37, 2005) 提出的关于弱解的存在性结果并没有包含尖锐的能量耗散原理和界面与域边界交点处接触角的弱表述。为了纳入这些内容,我们引入了一个函数框架,该框架编码了 Mullins-Sekerka 流动的弱解概念,基本上只依赖于:(i) 一个符合 De Giorgi 精神的单一尖锐能量耗散不等式;(ii) 通过基础毛细管能第一次变化的分布表示,对任意固定接触角进行弱表述。这两个要素都是演化相指示器界面的固有要素,可以从它们推导出带有势的显式分布 PDE 公式。弱解的存在是通过自然相关的最小化运动方案的后续极限点确定的。平滑解与经典的 Mullins-Sekerka 流一致,甚至更进一步,我们希望我们的解概念至少在原则上适合于最近开发的曲率驱动界面演化的相对熵方法。
Weak Solutions of Mullins–Sekerka Flow as a Hilbert Space Gradient Flow
Abstract
We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions \(d=2\) and 3, 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. The 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.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.