×

zbMATH — the first resource for mathematics

On the evolution of curves via a function of curvature. I: The classical case. (English) Zbl 0771.53003
Summary: The problem of curve evolution as a function of its local geometry arises naturally in many physical applications. A special case of this problem is the curve shortening problem which has been extensively studied. Here, we consider the general problem and prove an existence theorem for the classical solution. The main theorem rests on lemmas that bound the evolution of length, curvature, and how far the curve can travel.

MSC:
53A04 Curves in Euclidean and related spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Angenent, S; Angenent, S, Parabolic equations for curves and surfaces, II, University of wisconsin-Madison technical summary reports, nos. 88-19, University of wisconsin-Madison technical summary reports, nos. 89-24, (1989)
[2] Ben-Jacobi, E; Goldenfield, N; Langer, J; Schon, G, Dynamics of interfacial pattern formation, Phys. rev. lett., 51, No. 21, 1930-1932, (1983)
[3] Brower, R.C; Kessler, D.A; Koplik, J; Levine, H, Geometrical models of interface evolution, Phys. rev. A, 29, 1335-1342, (1984)
[4] Gage, M; Hamilton, R.S, The heat equation shrinking plane curves, J. differential geom., 23, 69-96, (1986) · Zbl 0621.53001
[5] Gage, M, On an area-preserving evolution equation for plane curves, Contemp. math., 51, 51-62, (1986)
[6] Grayson, M.A, The heat equation shrinks embedded plane curves to round points, J. differential geom., 26, 285-314, (1987) · Zbl 0667.53001
[7] Kimia, B.B; Tannenbaum, A; Zucker, S.W, Toward a computational theory of shape: an overview, (), 402-407
[8] Kimia, B.B, Conservations laws and a theory of shape, ()
[9] Langer, J.S, Instabilities and pattern formation in crystal growth, Rev. modern phys., 52, No. 1, 1-28, (1980)
[10] Lax, P.D, Shock waves and entropy, (), 603-634
[11] Lax, P.D, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, () · Zbl 0108.28203
[12] Osher, S; Sethian, J, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[13] Sethian, J.A, Curvature and the evolution of fronts, Comm. math. phys., 101, 487-499, (1985) · Zbl 0619.76087
[14] Sivashinsky, G.I, Nonlinear analysis of hydrodynamic instability in laminar flames. I. derivation of basic equations, Acta astronautica, 4, 1177-1206, (1977) · Zbl 0427.76047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.