Vanishing Curvature Viscosity for Front Propagation

In this paper we study the front propagation with constant speed and small curvature viscosity. We first investigate two related problems of conservation laws, one of which is on the nonlinear viscosity methods for the conservation laws, and the other one is on the structure of solutions to conservation laws with L1 initial data. We show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first-order rate of L1-convergence. The solutions of conservation laws with L1 initial data are shown to be bounded after t>0 if all singular points of initial data are from shocks. These results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed, as the small parameter goes to zero. After the front breaks down, the cusps will disappear promptly and corners will be formed.