Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy

Abstract

In this paper we study evolution of plane curves satisfying a geometric equation $v= \beta(k, \nu)$, where v is the normal velocity and k and $\nu$ are the curvature and tangential angle of a plane curve $\Gamma$. We follow the direct approach and we analyze the so-called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation. The intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional $\alpha$. We show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities. From an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when $\beta(k,\nu)=\gamma(\nu) k^m$, $0 < m\le 2$, and the governing system of equations includes a nontrivial tangential velocity functional.