17 apr 2026 -- 11:00
BANG (Bridging ANalysis & Geometry) online seminar
organized by Esther Cabezas-Rivas & Salvador Moll (Universitat de València)
Abstract.
Consider the following initial value problem: suppose that we are given a hypersurface and a time-dependent vector field (such as a flow field). Let the hypersurface evolve by the motion law that the velocity of the hypersurface is equal to the mean curvature plus the given vector field. Obviously, if all of the given quantities are smooth enough, the hypersurface should evolve nicely according to the rule for a while until some singularities occur. For the existence of such evolution, on the other hand, what is the minimal regularity assumption that one should impose on the vector field? Do we have a weak notion of evolution? Do we have some nice regularity theory for such evolving hypersurface?
The problem came from some two-phase flow problem involving the Allen-Cahn equation but it is interesting independent of the origin. In the talk I would like to describe what we can and cannot say currently. The work is joint effort with many people that I mention along the way.