Convergence for global curve diffusion flows

Grim Raindrop：曲线扩散流的平移解

Grim Raindrop: A Translating Solution to Curve Diffusion Flow

本文证明了平面中曲线扩散流存在一个正则浸入的平移解，并指出该流是曲线缩短流的高阶版本。

On the isoperimetric inequality and surface diffusion flow for multiply winding curves

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $H^2$-close to a multiply covered circle and sufficiently rotationally symmetric.

A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle $α\in (0, π)$ with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.

Short Time Existence for the Curve Diffusion Flow with a Contact Angle

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $α\in (0, π)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^γ$ with $γ\in (\tfrac{3}{2}, 2]$. For the proof the evolving curve is represented by a height function over a reference curve: The local well-posedness of the resulting quasilinear, parabolic, fourth-order PDE for the height function is proven with the help of contraction mapping principle. Difficulties arise due to …

On the curve diffusion flow of closed plane curves

In this paper we consider the steepest descent $H^{-1}$-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves which develop at least one singularity in finite time and initially embedded curves which self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised $L^2$ oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficien…

固定长度夹持曲线的保长弹性流的存在性与收敛性

Existence and convergence of the length-preserving elastic flow of clamped curves

本文研究固定长度夹持曲线在弹性能量负$L^2$-梯度流下的演化，证明了解的存在性、抛物光滑化及长时间收敛到临界点。