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

• Published in: SIAM journal on applied mathematics Vol. 61; no. 5; pp. 1473 - 1501
• Main Authors: Mikula, Karol, Ševčovič, Daniel
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 2001
• Subjects: Anisotropy; Applied mathematics; Curvature; Curves; Heat equation; Interfaces; Kinetics; Mathematical functions; Parameterization; Plane curves; Velocity
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/s0036139999359288

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 v 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.