Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion

Abstract

The existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory, a prominent example being the fundamental solutions of the linear theory. In nonlinear equations the existence of such solutions may find special obstacles, that can be either essential, or otherwise they may lead to more general concepts of solution. We give a particular review of results in the field of nonlinear diffusion. As a new contribution, we study in detail the case of logarithmic diffusion, associated with Ricci flow in the plane, where we can prove existence of measure-valued solutions. The surprising thing is that these solutions become classical after a finite time. In that general setting, the standard concept of weak solution is not adequate, but we can solve the initial-value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions. We prove that the problem is well-posed. The phenomenon of blow-down in finite time is precisely described: initial point masses diffuse into the medium and eventually disappear after a finite time T i = M i / 4 π .