2025/11/21 by Mohammad Javad Habibi Vosta Kolaei, Kolaei, Mohammad Javad Habibi Vosta
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2511.16915
openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28
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.