In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the followin...In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the following evolution equation δ^2F /δt^2 （u, t） = k（u, t）N（u, t）-▽ρ（u, t）, ∨（u, t） ∈ S^1 × [0, T ） with the initial data F （u, 0） = F0（u） and δF/δt （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 Δ=（δ^2F /δsδt ,δF/δt） 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-Ampere 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 convergesto 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 R^1,1.展开更多
基金Kong and Wang was supported in part by the NSF of China (10671124)the NCET of China (NCET-05-0390)the work of Liu was supported in part by the NSF of China
文摘In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the following evolution equation δ^2F /δt^2 （u, t） = k（u, t）N（u, t）-▽ρ（u, t）, ∨（u, t） ∈ S^1 × [0, T ） with the initial data F （u, 0） = F0（u） and δF/δt （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 Δ=（δ^2F /δsδt ,δF/δt） 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-Ampere 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 convergesto 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 R^1,1.
