The evolution of immersed locally convex plane curves driven by anisotropic curvature flow

In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:mfrac> <m:mi>ψ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>κ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> V=1/α ψ (x){κ }α for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>0</m:mn> </m:math> α < 0 or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> α > 1 , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mi>m</m:mi> <m:mi>π</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> x∈ [0,2mπ ] is the tangential angle at the point on evolving curves. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>0</m:mn> </m:math> -1≤ α < 0 , we show the flow exists globally and the rescaled flow has a full-time convergence. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:math> α < -1 or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> α > 1 , we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ψ</m:mi> </m:math> ψ and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.