Abstract:
The classical total variation flow is the L2 gradient flow of the total variation. The variation of a function is a singular energy at the place where the slope of the function equals zero. Because of this structure, its gradient flow is actually nonlocal 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.
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 variation in image denoising, crystalline mean curvature flow or fourth order total variation flow of exponential type in crystal growth problems which are special important problems in materials science.
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. This last well-posedness result is based on my joint work with N. Požár (Kanazawa University) whose basic idea depends on my early joint work with M.-H. Giga (The University of Tokyo) and N. Požár.
Biography:
Yoshikazu Giga is a Professor of Mathematics at the University of Tokyo. He has contributed to many areas such as the Navier-Stokes equations, blow-up phenomena for semilinear heat equations, defect energy surface evolution equations and singular diffusion equations based on the theory of viscosity solutions. Professor Giga received many prizes and honors. For example, he was awarded the Medal of Honour with Purple Ribbon in 2010 by the Government of Japan for his contribution to mathematics. He was selected as a fellow of American Mathematical Society in 2012. He was invited to be a plenary lecturer at the International Congress on Industrial and Applied Mathematics (ICIAM) in 2003. He was a section lecturer at the International Congress of Mathematicians (ICM) in 2018.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai