Very singular diffusion equations: second and fourth order problems

Abstract

This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H −1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove by giving an explicit example that the solution may instantaneously develop a jump discontinuity for the fourth order total variation flow.