Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

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• 作者：Daniel Loibl ; Daniel Matthes ; Jonathan Zinsl
• 关键词：Fourth ; order equations ; Gradient flow ; Wasserstein distance ; Weak solution ; Minimizing movement scheme
• 刊名：Potential Analysis
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：45
• 期：4
• 页码：755-776
• 全文大小：438 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Potential Theory
  Probability Theory and Stochastic Processes
  Geometry
  Functional Analysis
  
• 出版者：Springer Netherlands
• ISSN：1572-929X
• 卷排序：45

文摘

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L2-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.