BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Yoshikazu Giga (University of Tokyo) DTSTART:20201019T130000Z DTEND:20201019T140000Z DTSTAMP:20260423T052620Z UID:SNPDEA/9 DESCRIPTION:Title: O n total variation flow type equations\nby Yoshikazu Giga (University o f Tokyo) as part of "Partial Differential Equations and Applications" Webi nar\n\n\nAbstract\nThe classical total variation flow is the $L^2$ gradien t 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 en ergy $\\int |Du|^2 dx/2$\, the energy density is singular at the place whe re the slope of the function u equals zero. Because of this structure\, it s 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. T hus\, the definition of a solution itself is a nontrivial issue even for t he classical total variation flow. This becomes more serious if there is n on-uniform driving force term.\n\nRecently\, there need to study various t ypes of such equations. A list of examples includes the total variation ma p flow as well as the classical total variation flow and its fourth order version in image de-noising\, crystalline mean curvature flow or fourth or der total variation flow in crystal growth problems which are important mo dels in materials science below roughening temperature.\n\nIn this talk\, we survey recent progress on these equations with special emphasis on a cr ystalline mean curvature flow whose solvability was left open more than te n years. We shall give a global-in-time unique solvability in the level-se t sense. It includes a recent extension when there is spatially non-unifor m driving force term which is going to be published in the journal SN Part ial Differential Equations. These last well-posedness results are based o n my joint work with N. Požár (Kanazawa University) whose basic idea dep ends on my earlier joint work with M.-H. Giga (The University of Tokyo) an d N. Požár.\n LOCATION:https://researchseminars.org/talk/SNPDEA/9/ END:VEVENT END:VCALENDAR