Contracting convex immersed closed plane curves with slow speed of curvature

Abstract

We study the contraction of a convex immersed plane curve with speed 1 α k α , where α ∈ (0, 1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up 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 familiar "Grim Reaper" (a terminology due to M. Grayson [GR]). * AMS Subject Classifications: 35K15, 35K55. We thank Ben Andrews for giving us this summary.rescaled about the final point (the isoperimetric ratio approaches infinity); and the exceptional ones where the isoperimetric ratio remains bounded converge to homothetic solutions, which have been classified. For α > 1/3, the rescaled solutions converge to circles; and for α = 1/3, they converge to ellipses.Remark 2 As a consequence of Theorem 1, we have the following interesting elliptic result. For 0 < λ < 3 (here λ = 1/α), the only positive 2π-periodic solution to the equation

Evolution of highly symmetric curves under the shrinking curvature flow

“…If α ≠ 1, the classifications of self‐shrinking curves are characterized by Andrews and Urbas . The general results on the asymptotic behavior of locally convex curves are obtained by Tsai et al in . Recently, Cortissoz–Murcia investigate the asymptotic stability of multi‐circles in the flow with .…”

“…A classical model studies the evolution of locally convex, immersed curves X. , t/ with inner normal speed V D k˛, (1.1) If˛¤ 1, the classifications of self-shrinking curves are characterized by Andrews [6] and Urbas [17]. The general results on the asymptotic behavior of locally convex curves are obtained by Tsai et al in [18][19][20]. Recently, Cortissoz-Murcia [21] investigate the asymptotic stability of multi-circles in the flow (1.1) with˛2 Z C .…”

On length-preserving and area-preserving nonlocal flow of convex closed plane curves

“…When f = 1 α k α (α = 0 is a constant) and X 0 is a convex 1 simple closed curve, the flow (1) has been investigated in [And1,And2]. Also in [LPT1,LPT2,PT,U1,U2], they studied the case when X 0 is a locally convex non-simple closed curve. For more about (1) and its counterpart in high-dimensional space (more precisely, the mean curvature flow ) one can find literature in the book [Z].…”

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

“…Next, the mathematical problem in (1.1) is explained. There have been many studies on blow-up solutions in (1.1) for the case δ = 0 (see [2,3,4,5,15,19,21]). According to these references, the solution has the blow-up of Type I when 0 < p < 2.…”