NCTS /NCKU Math Colloquium

DATE2015-06-04 15:10-16:00


SPEAKER林育竹 助理教授(成大數學系

TITLEContracting 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".