33 research outputs found
Renormalization study of two-dimensional convergent solutions of the porous medium equation
In the focusing problem we study a solution of the porous medium equation
whose initial distribution is positive in the exterior of a
closed non-circular two dimensional region, and zero inside. We implement a
numerical scheme that renormalizes the solution each time that the average size
of the empty region reduces by a half. The initial condition is a function with
circular level sets distorted with a small sinusoidal perturbation of wave
number . We find that for nonlinearity exponents m smaller than a
critical value which depends on k, the solution tends to a self-similar regime,
characterized by rounded polygonal interfaces and similarity exponents that
depend on m and on the discrete rotational symmetry number k. For m greater
than the critical value, the final form of the interface is circular.Comment: 26 pages, Latex, 13 ps figure
Complete Embedded Self-Translating Surfaces under Mean Curvature Flow
We describe a construction of complete embedded self-translating surfaces
under mean curvature flow by desingularizing the intersection of a finite
family of grim reapers in general position.Comment: 42 pages, 8 figures. v2: typos correcte
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 Hâ1 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
A Comparison Principle for Singular Diffusion Equations with Spatially Inhomogeneous Driving Force for Graphs
On the motion by singular interfacial energy
Anisotropic curvature flow equations with singular interfacial energy are important for good unders_tanding of motion of phase-boundaries. If the energy and the interfacial surface were smooth, then the speed of the interface would be equal to the gradient of the energy. However, this is not so simple in the case of non-smooth crystalline energy. But it's well-known that a unique gradient characÂterization of the velocity is possible if the interface is a curve in the two-dimensional space. In this paper we propose a notion of solution in the three-dimensional space by introducing geometric subdifferentials and characterizing the speed. We also give a counterexample to a problem concerning the Cahn-Hoffman vector field on a facet, a flat portion of the interface