ncku-talk2015-06-04

NCTS /NCKU Math Colloquium

DATE 2015-06-04　15:10-16:00

PLACE 數學系3174

SPEAKER 林育竹助理教授（成大數學系）

TITLE Contracting Convex Immersed Closed Curves with Speed of Powers of Curvature

ABSTRACT Abstract: In this talk we are concerned with the contraction of a convex immersed (i.e., the curve may have self-intersections) plane curve along its inward normal vector direction with speed function (1/α)k^{α}, where α∈(0,1] is a constant and k is the curvature. The behavior of the contracting flow with fast speed, i.e., α∈(1,∞), has been studied by Poon-Tsai; they showed that the blow-up rate of the curvature is always of type one (i.e., the blow-up rate is given by the corresponding ODE of the PDE) and the rescaled solution will converge to a limit that may or may not be degenerate (a limit is degenerate has zero curvature somewhere). For the case α∈(0,1], we may have either type one blow-up or type two blow-up of the curvature. The generic behavior is type-two blow-up of the curvature. In type one blow-up, we shall show that the curve will converge to a positive homothetic self-similar solution. For type two blow-up, we consider a special symmetric curve and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α=1), this translational self-similar solution is the famous "grim reaper solution".

NCTS /NCKU Math Colloquium

DATE	2015-06-04　15:10-16:00

PLACE	數學系3174

SPEAKER	林育竹助理教授（成大數學系）

TITLE	Contracting Convex Immersed Closed Curves with Speed of Powers of Curvature

ABSTRACT	Abstract: In this talk we are concerned with the contraction of a convex immersed (i.e., the curve may have self-intersections) plane curve along its inward normal vector direction with speed function (1/α)k^{α}, where α∈(0,1] is a constant and k is the curvature. The behavior of the contracting flow with fast speed, i.e., α∈(1,∞), has been studied by Poon-Tsai; they showed that the blow-up rate of the curvature is always of type one (i.e., the blow-up rate is given by the corresponding ODE of the PDE) and the rescaled solution will converge to a limit that may or may not be degenerate (a limit is degenerate has zero curvature somewhere). For the case α∈(0,1], we may have either type one blow-up or type two blow-up of the curvature. The generic behavior is type-two blow-up of the curvature. In type one blow-up, we shall show that the curve will converge to a positive homothetic self-similar solution. For type two blow-up, we consider a special symmetric curve and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α=1), this translational self-similar solution is the famous "grim reaper solution".