The $epsiv;-Regularized Mean Curvature Flow in One Dimension Space

The evolutionary motion of surfaces or curves by their meancurvature has found much interest during the last years. The problem withmean curvature flow is that singularities can appear during the evolutioneven if the initial surface is convex. To prove the existence of a viscositysolution u of the mean curvature flow, Evans and Spruck 4] builtthe -regularized mean curvature flow. For practicalpurposes, i.e., numerical computations, it would be interesting to knowhow fast the solution of the regularized problem converges to the viscocitysolution of the original problem. The goal in this paper is to presentsome results concerning the -regularized mean curvatureflow in the one-dimension space. It is proved that there exists anasymptotic expansion of the solution of the regularized problem, in powers of theparameter , such that the first term of the asymptoticexpansion is the viscosity solution of the mean curvature flow problem.Moreover, that this asymptotic expansion is true in appropriate topologies,in particular in weighted Sobolev spaces is proved. Finally, an estimateof the rate of convergence in these topologies is given.