偏微分方程式セミナー: The fourth-order total variation flow in $\mathbf{R}^n$, Michał Łasica ・Time periodic profile on a diffusion equation model to describe a bubble cluster with rupture, 上田 祐暉

Event Date: May 31, 2024


Place:理学部4号館4-501室 (hybrid)

Organizer:黒田 紘敏、浜向 直

Speaker:Michał Łasica 氏 (ポーランド科学アカデミー), 16:30-17:10

Title:The fourth-order total variation flow in $\mathbf{R}^n$

Abstract:Gradient flows of the total variation (TV) functional are of interest due to their applications in image processing and connections to models of crystal growth. The relatively well-known $L^2$-gradient flow, corresponding to a second-order parabolic PDE, has a notable property of preserving the form of characteristic functions of the class of so-called calibrable sets. In this talk we consider the gradient flow of TV with respect to the $H^{-1}$ metric. Motivated by providing a natural setting to investigate the evolution of characteristic functions and calibrability of sets, we choose $\mathbf{R}^n$ as the spatial domain. We give a rigorous formulation of the flow, which is a bit tricky in low dimensions. We characterize flow paths in terms of a Cahn-Hoffman vector field. We define a notion of calibrability in our setting and investigate calibrability of balls and annuli. We obtain explicit description of flow paths emanating from piecewise constant, radially symmetric data in terms of a system of ODEs. In particular, we compute flow paths emanating from characteristic functions of balls, whose qualitative behavior turns out to vary considerably by dimension. This is joint work with Y. Giga and H. Kuroda.

Speaker:上田 祐暉 氏 (北海道大学), 17:10-17:50

Title:Time periodic profile on a diffusion equation model to describe a bubble cluster with rupture

Abstract:In this talk, we propose a simple model to describe a bubble cluster with rupture. The shape of bubbles and the liquid layer are described by diffusion equations. Furthermore, under several assumptions, our model can be simplified into a single parabolic equation, which represents the evolution of thickness of the liquid layer. This enables us to analyze the model rigorously. We can estimate the rupture time, and prove the time periodic profile of the solution. Numerical examples demonstrate the time periodic behavior approximately.