Evolution of plane curves driven by a nonlinear function of curvature and anisotropy

In this paper we study evolution of plane curves satisfying a geometric equ ation v = beta (k,v), where v is the normal velocity and k and are the curv ature and tangential angle of a plane curve. We follow the direct approach and we analyze the so-called intrinsic heat equation governing the motion o f plane curves obeying such a geometric equation. The intrinsic heat equati on is modi ed to include an appropriate nontrivial tangential velocity func tional. We show how the presence of a nontrivial tangential velocity can pr event numerical solutions from forming various instabilities. From an analy tical 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,v) = gamma (v)k(m), 0 < m <less than or equal to> 2, and the governing system of equations includes a nontrivial tangential velocity functional.