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 equation v= β(k, ν), where v is the normal velocity and k and ν are the curvature and tangential angle of a plane curve Γ. 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 α. 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 β(k,ν)=γ(ν) kᵐ, 0 < m≤ 2, and the governing system of equations includes a nontrivial tangential velocity functional.