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An analytic study of bi-harmonic flow with a forcing term

Mohammad Javad Habibi Vosta Kolaei

TL;DR

The paper analyzes the evolution of smooth, closed planar curves under a biharmonic flow with forcing, establishing global existence under suitable forcing constraints. By reformulating the flow with the support function, it derives a Monge–Ampère–type scalar equation for $S(\theta,t)$ and identifies conditions under which this formulation exhibits hyperbolic behavior, while the underlying flow remains parabolic. The authors prove convexity preservation and long-time convergence to steady states satisfying $S_{\theta\theta\theta\theta} + 2S_{\theta\theta} + S = F(S)$, with the convergence and final shape strongly influenced by the forcing term. Through analytic energy methods and several forcing scenarios, they show that forcing can drive the curve toward a circle, yield anisotropic equilibria, or even cause collapse to a point, highlighting the stabilizing or destabilizing effects of forcing on high-order curvature flows.

Abstract

In this paper, we study the evolution of smooth, closed planar curves under a fourth order biharmonic flow with an external forcing term. Such flows arise naturally in the theory of biharmonic maps and geometric variational problems involving bending energy. We first establish the global existence of smooth solutions to the associated initial value problem, assuming appropriate conditions on the forcing term. The analysis is performed through a reformulation of the geometric flow using the support function, enabling a scalar PDE characterization of the evolution. Under specific geometric constraints, we demonstrate that the governing equation admits a Monge Ampére type structure that can exhibit hyperbolic behavior. Furthermore, we prove that convexity is preserved during the evolution and derive sufficient conditions ensuring long time convergence to steady-state solutions. Our results extend recent developments in geometric analysis by clarifying the role of forcing terms in stabilizing high order curvature flows and enhancing their qualitative behavior.

An analytic study of bi-harmonic flow with a forcing term

TL;DR

The paper analyzes the evolution of smooth, closed planar curves under a biharmonic flow with forcing, establishing global existence under suitable forcing constraints. By reformulating the flow with the support function, it derives a Monge–Ampère–type scalar equation for and identifies conditions under which this formulation exhibits hyperbolic behavior, while the underlying flow remains parabolic. The authors prove convexity preservation and long-time convergence to steady states satisfying , with the convergence and final shape strongly influenced by the forcing term. Through analytic energy methods and several forcing scenarios, they show that forcing can drive the curve toward a circle, yield anisotropic equilibria, or even cause collapse to a point, highlighting the stabilizing or destabilizing effects of forcing on high-order curvature flows.

Abstract

In this paper, we study the evolution of smooth, closed planar curves under a fourth order biharmonic flow with an external forcing term. Such flows arise naturally in the theory of biharmonic maps and geometric variational problems involving bending energy. We first establish the global existence of smooth solutions to the associated initial value problem, assuming appropriate conditions on the forcing term. The analysis is performed through a reformulation of the geometric flow using the support function, enabling a scalar PDE characterization of the evolution. Under specific geometric constraints, we demonstrate that the governing equation admits a Monge Ampére type structure that can exhibit hyperbolic behavior. Furthermore, we prove that convexity is preserved during the evolution and derive sufficient conditions ensuring long time convergence to steady-state solutions. Our results extend recent developments in geometric analysis by clarifying the role of forcing terms in stabilizing high order curvature flows and enhancing their qualitative behavior.

Paper Structure

This paper contains 6 sections, 4 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A brief background of evolving curves under geometric flows
  3. Bi-harmonic flow and main aim
  4. Existence result and Monge-Ampère equation
  5. On the hyperbolic structure in the support function evolution
  6. Analytic behavior of the forcing term

Key Result

Lemma 2.1

If the evolving curve $X$ is closed, then there is a parameter change $\varphi$ for $X$ such that $X\circ \varphi$ is a normally evolving curve.

Figures (2)

  • Figure 1: Evolution of the curve under the proportional forcing term. The initial shape (left) evolves into a circular steady-state shape (right)
  • Figure 2: Evolution of the curve under the anisotropic forcing term. The initial shape (left) evolves into a non-circular steady-state shape (right)

Theorems & Definitions (12)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4
  • Theorem 3.5
  • proof
  • ...and 2 more