Curvature and the evolution of fronts

The evolution of a front propagating along its normal vector field with speedF dependent on curvatureK is considered. The change in total variation of the propagating front is shown to depend only ondF/dK only whereK changes sign. Analysis of the caseF(K)=1−εK, where ε is a constant, shows that curvature plays a role similar to that of viscosity in Burgers equation. For ε=0 and non-convex initial data, the curvature blows up, corners develop, and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity. We then numerically show that the solution as ε goes to zero converges to the constructed weak solution. Numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds. As a remedy, we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution.