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

TL;DR

This paper analyzes the evolution of planar curves under a biharmonic flow with external forcing, establishing global existence, convexity preservation, and conditions for convergence to steady states, advancing understanding of high-order geometric flows.

Contribution

It introduces a scalar PDE reformulation using the support function and explores the impact of forcing terms on stability and long-term behavior of biharmonic flows.

Findings

Global existence of smooth solutions under certain forcing conditions

Convexity preservation during flow evolution

Conditions for convergence to steady-state solutions

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 Amp\'ere 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.