Energy solutions to one-dimensional singular parabolic problems with BVBV data are viscosity solutions

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in $BV$, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.Comment: 15 page

A caricature of a singular curvature flow in the plane

We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak solutions to the semi-discretization of the problem as well as convergence of the approximating sequences. The semi-discretization shows that facets must form. For a class of initial data we are able to study in details the facet formation and interactions and their asymptotic behavior. We notice that our qualitative results may be interpreted with the help of a special composition of multivalued operators

Consistency in evolutions by crystalline curvature

Motion of curves by crystalline energy is often considered for "admissible" piecewise linear curves. This is because the evolution of such curves can be described by a simple system of ordinary differential equations. Recently, a generalized notion of solutions based on comparison principle is introduced by the authors. In this note we show that a classical admissible solution is always a generalized solution in our sense

Stability for evolving graphs by nonlocal weighted curvature

A general stability and convergence theorem is established for generalized solutions of a family of nonlinear evolution equations with non­local diffusion in one space dimension. As the first application motion by nonlocal weighted curvature is approximated by solutions of regular problem, when initial curve is given as the graph of a continuous periodic function. This justifies the motion by crystalline energy as a limit of regularized prob­lems. As the second application the motion by crystalline energy is shown to approximate the motion by regular interfacial energy if the crystalline energy approximates the regular energy. This gives the convergence of crys­talline algorithm for general curvature flow equations. Our general results are also important to explain that geometric evolution of crystals depends continuously on temperature even if facets appear

Crystalline and level set flow - Convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane

Recently, a level set formulation is extended by the authors to handle evolution of curves driven by singular interfacial energy including crystalline energy. In this paper as an application of this theory a general convergence result is established for a crystalline algorithm for a general anisotropic curvature flow

Very Singular Diffusion Equations-Second and Fourth Order Problems

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 that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example

Convergence of the Allen-Cahn equation with Neumann boundary conditions

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.Comment: 26 pages, 1 figur