On total variation flow type equations

Abstract

The classical total variation flow is the $L^2$ gradient flow of the total variation. The total variation of a function u is one-Dirichlet energy, i.e.,$ \int |Du| dx$. Different from the Dirichlet energy $\int |Du|^2 dx/2$, the energy density is singular at the place where the slope of the function u equals zero. Because of this structure, its gradient flow is actually non-local in the sense that the speed of slope zero part (called a facet) is not determined by infinitesimal quantity. Thus, the definition of a solution itself is a nontrivial issue even for the classical total variation flow. This becomes more serious if there is non-uniform driving force term. Recently, there need to study various types of such equations.A list of examples includes the total variation map flow as well as the classical total variation flow and its fourth order version in image de - noising, crystalline mean curvature flow or fourth order total variation flow in crystal growth problems which are important models in materials science below roughening temperature. In this talk, we survey recent progress on these equations with special emphasis on a crystalline mean curvature flow whose solvability was left open more than ten years.We shall give a global - in -time unique solvability in the level - set sense.It includes a recent extension when there is spatially non - uniform driving force term which is going to be published in the journal SN Partial Differential Equations.These last well - posedness results are based on my joint work with N.Požár(Kanazawa University) whose basic idea depends on my earlier joint work with M. - H.Giga(The University of Tokyo) and N.Požár.