Acta mathematica scientia, Series B >
HYPERBOLIC MEAN CURVATURE FLOW: EVOLUTION OF PLANE CURVES
Received date: 2008-07-01
Online published: 2009-05-20
Supported by
The work of Kong and Wang was supported in part by the NSF of China (10671124) and the NCET of China (NCET-05-0390)and the work of Liu was supported in part by the NSF of China.
In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves F: S1× [0,T) → R2 which satisfies the following evolution equation
∂2 F/ ∂ t 2 (u, t) = k(u, t) N (u, t) - ∧ρ (u, t), ∨(u, t) ∈S1 × [0, T)
with the initial data
F(u,0)=F0(u) and ∂ F/ ∂ t (u, 0)=f(u) N0,
where k is the mean curvature and N is the unit inner normal vector of the plane curve F(u, t), f(u) and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ∨ρ is given by
∨ρ ＝(∂2F/ ∂s ∂t, ∂F/ ∂t ) T,
in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampère equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax ) and when t goes to Tmax, either the solution converges to a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R1,1.
KONG De-Xin , LIU Ke-Feng , WANG Ceng-Gui . HYPERBOLIC MEAN CURVATURE FLOW: EVOLUTION OF PLANE CURVES[J]. Acta mathematica scientia, Series B, 2009 , 29(3) : 493 -514 . DOI: 10.1016/S0252-9602(09)60049-7
[1] Albrecht A, Tuok, N. Evolution of cosmic string. Physics Review Letters, 1985, 54: 1868--1871
[2] Alvarez L, Guichard F, Lions P L, Morel J M. Axioms and fundamental equations of image processing. Arch Rational Mech Anal, 1993, 123: 199--257
[3] Angenent S, Gurtin M E. Multiplhase thermomechanics with an interfacial structure 2. evolution of an isothermal interface. Arch Rational Mech Anal, 1989, 108: 323--391
[4] Cao F. Geometric Curve Evolution and Image Processing. Lecture Notes in Mathematics 1805. Berlin: Springer, 2003
[5] Christodoulou D. Global solution of nonlinear hyperbolic equations for small initial data. Comm Pure Appl Math, 1986, 39: 367--282
[6] DeTurck D. Some regularity theorems in Riemannian geometry. Ann Scient Ecole Norm Sup Paris, 1981, 14: 249--260
[7] Gage M, Hamilton R. The heat equation shrinking convex plane curves. J Diff Geom, 1986, 23: 417--491
[8] Grayson M. Shortening embedded curves. Ann Math, 1989, 101: 71--111
[9] Gurtin M E, Podio-Guidugli P. A hyperbolic theory for the evolution of plane curves. SIAM J Math Anal, 1991, 22: 575--586
[10] Hadamard J. Le problème de Cauchy et lesèquations aux dèrivèes partielles linèaries hyperboliques. Paris: Hermann, 1932
[11] He C -L, Kong D -X, Liu K -F. Hyperbolic mean curvature flow. J Differ Equs, 2009, 246: 473--390
[12] Hòmander L. Lectures on Nonlinear Hyperbolic Differential Equations. Math\'{e}matiques And Applications 26. Berlin: Springer-Verlag, 1997
[13] Huisken G. Asymptotic behavior for singularities of the mean curvature flow. J Diff Geom, 1990, 31: 285--299
[14] Huisken G, Ilmanen T. The inverse mean curvature flow and the Riemannian Penrose inequality.
J Diff Geom, 2001, 59: 353--437
[15] Klainerman S. Global existence for nonlinear wave equations. Comm Pure Appl Math, 1980, 33: 43--101
[16] Kong D -X. Maximum principle in nonlinear hyperbolic systems and its applications. Nonlinear Analysis, Theory, Method \& Applications, 1998, 32: 871--880
[17] Kong D -X, Hu H -R. Geometric approach for finding exact solutions to nonlinear partial differential equations. Physics Letters A, 1998, 246: 105--112
[18] Kong D -X, Zhang Q, Zhou Q. The dynamics of relativistic string moving in the Minkowski space R1+n. Commun Math Phys, 2007, 269: 153--174
[19] LeFloch P G, Smoczyk K. The hyperbolic mean curvature flow. http://arxiv.org/arXiv:0712.0091v1, 2007
[20] Lewy H. Ueber das Anfangswertproblem einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhangigen Veranderlichen. Math Annal, 1928, 98: 179--191
[21] Liu K. Hyperbolic geometric flow. Lecture at International Conference of Elliptic and Parabolic Differential Equations, Hangzhou, August 20, 2007. Available at preprint webpage of Center of Mathematical Science, Zhejiang University
[22] Nishida T. Nonlinear hyperbolic equations and related topics in fluid dynamics. Publications Mathèmatiqèes Dòsay 78--02, Paris-Sud, 1978
[23] Protter M H, Weinberger H F. Maximum Principles in Differential Equations. New York: Springer-Verlag, 1984
[24] Rotstein H G, Brandon S, Novick-Cohen A. Hyperbolic flow by mean curvature. Journal of Crystal Growth, 1999, 198/199: 1256--1261
[25] Schneider R. Convex Bodies: The Brum-Minkowski Theory. Cambridge: Cambridge University Press, 1993
[26] Tso K. Deforming a hypersurface by its Gauss-Kronecker curvature. Commu Pure Appl Math, 1985, 38: 867--882
[27] Tsuji M. Formation of singularties for Monge-Ampère equations. Bull Sci Math, 1995, 119: 433--457
[28] Turok N, Bhattacharjee P. Stretching cosmic strings. Physical Review D, 1983, 29: 1557--1562
[29] Tynitskii D V. The Cauchy problem for a hyperbolic Monge-Ampère equation. Mathmatical Notes, 1992, 51: 582--589
[30] Yau S -T. Review of geometry and analysis. Asian J Math, 2000, 4: 235--278
[31] Zhu X -P. Lectures on Mean Curvature Flows. Studies in Advanced Mathematics 32. AMS/IP, 2002
/
| 〈 |
|
〉 |