Solution of nonlinearly curvature driven evolution of plane curves. (English) Zbl 0938.65145

Mikula, Karol; Ševčovič, Daniel

Appl. Numer. Math. 31, No. 2, 191-207 (1999).

Authors’ abstract: The evolution of plane curves obeying the equation \(v=\beta(k)\), where \(v\) is normal velocity and \(k\) curvature of the curve is studied. Morphological image and shape multiscale analysis of L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel [Axioms and fundamental equations of image processing, Arch. Ration. Mech. Anal. 123, No. 3, 199-257 (1993; Zbl 0788.68153)] and affine invariant scale space of curves introduced by G. Sapiro and A. Tannenbaum [J. Funct. Anal. 119, No. 1, 79-120 (1994; Zbl 0801.53008)] as well as isotropic motions of plane phase interfaces studied by S. B. Angenent and M. E. Gurtin [Multiphase thermomechanics with an interfacial structure. II: Evolution of an isothermal interface, Arch. Rat. Mech. Anal. 108, 323-391 (1989); J. Reine Angew. Math. 446, 1-47 (1994; Zbl 0784.35124)] are included in the model. We introduce and analyze a numerical scheme for solving the governing equation and present numerical experiments.

Reviewer: R.Gorenflo (Berlin)

MSC:

65P10	Numerical methods for Hamiltonian systems including symplectic integrators
37C10	Dynamics induced by flows and semiflows
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
35K05	Heat equation

Keywords:

curve evolution; image and shape multiscale analysis; phase interface; nonlinear degenerate parabolic equations; numerical experiments