We present a numerical scheme for radially symmetric solutions to curvature driven moving boundary problems governed by a local law of motion, e.g. the mean curvature flow, the surface diffusion flow, and the Willmore flow. We then present several numerical experiments for the Willmore flow. In particular, we provide numerical evidence that the Willmore flow can develop singularities in finite time.
This modified (two-sided) Mullins-Sekerka model is a nonlocal evolution model for closed hypersurfaces, which appears as a singular limit of a modified Cahn-Hilliard equation describing microphase separation of diblock copolymer.
Under this evolution the propagating interfaces maintain the enclosed volumes of the two phases. We will show by means of an example that this model does not preserve convexity in two space dimensions.
The surface diffusion flow is a moving boundary problem that has a gradient flow structure. This gradient flow structure suggests an implicit finite differences approach to compute numerical solutions.
The resulting numerical scheme will allow to compute the flow for any smooth orientable immersed initial surface. Observations include the loss of embeddedness for some initially embedded surface, the creation of singularities, and the long term behavior of solutions.
An example of an embedded curve is presented which under numerical simulation of the averaged mean curvature flow develops first a loss of embeddedness, and then a singularity where the curvature becomes infinite, all in finite time.
This leads to the conjecture that not all smooth embedded curves persist for all times under the averaged mean curvature flow.
We prove existence and uniqueness of classical solutions for the motion of hypersurfaces driven by mean curvature and diffusion of a solute along the surface. This free boundary problem involves solving a coupled system of fully nonlinear partial differential equations.
We consider a sharp interface model which describes diffusion-induced grain-boundary motion in a poly-crystalline material. This model leads to a fully nonlinear coupled system of partial differential equations. We show existence and uniqueness of smooth solutions.
The notion of gradient flows is generalized to a metric space setting without any linear structure. The metric spaces considered are a generalization of Hilbert spaces. The properties of such metric spaces are used to set up a finite-difference scheme of variational form.
The proof of the Crandall-Liggett generation theorem is adapted to show convergence. The resulting flow generates a strongly continuous semigroup of Lipschitz-continuous mappings, is Lipschitz continuous in time for positive time, and decreases the energy functional along a path of steepest descent.
In case the underlying metric space is a Hilbert space, the solutions resulting from this new theory coincide with those obtained by classical methods. As an application, the harmonic map flow problem for maps from a manifold into a nonpositively curved metric space is considered, and the existence of a solution to the initial boundary value problem is established.