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Rods in flows: the PDE theory of immersed elastic filaments

Dallas Albritton, Laurel Ohm

TL;DR

This work develops a rigorous PDE framework for immersed elastic filaments, modeling Kirchhoff rods in viscous, low-Reynolds-number flows as gradient flows of a rod energy under an anisotropic resistive-force metric. The authors derive coupled curve–frame evolution equations, establish energy-dissipation structures, and prove global and conditional well-posedness across planar and 3D settings, with dynamic convergence to elastica equilibria. The analysis combines linear parabolic theory, fixed-point arguments, and Lojasiewicz inequalities to handle long-time behavior, including convergence in free-end and periodic geometries, and large-time periodic forcing in 2D. In 3D, they show conditional global well-posedness and convergence under curvature bounds, and they develop a robust framework for energy propagation and stability that remains valid under vanishing rotational friction. Overall, the paper provides a comprehensive PDE-theoretic treatment of immersed rod dynamics with bend–twist coupling, establishing foundational results for the mathematical theory of filament flows and their equilibria.

Abstract

We investigate a family of curve evolution equations approximating the motion of a Kirchhoff rod immersed in a low Reynolds number fluid. The rod is modeled as a framed curve whose energy consists of the bending energy of the curve and the twisting energy of the frame. The equations we consider may be realized as gradient flows of the rod energy under a certain anisotropic metric coming from resistive force theory. Ultimately, our goal is to provide a comprehensive treatment of the PDE theory of immersed rod dynamics. We begin by analyzing the problem without the frame, in which case the evolution is globally well-posed and solutions asymptotically converge to Euler elasticae. Next, in the planar setting, we demonstrate the existence of large time-periodic solutions forced by intrinsic curvature relevant to undulatory swimming. Finally, the majority of the paper is devoted to the evolution of an immersed Kirchhoff rod in 3D, which involves a strong coupling between the curve and frame. Under a physically reasonable assumption on the curvature, solutions exist globally-in-time and asymptotically converge to rod equilibria.

Rods in flows: the PDE theory of immersed elastic filaments

TL;DR

This work develops a rigorous PDE framework for immersed elastic filaments, modeling Kirchhoff rods in viscous, low-Reynolds-number flows as gradient flows of a rod energy under an anisotropic resistive-force metric. The authors derive coupled curve–frame evolution equations, establish energy-dissipation structures, and prove global and conditional well-posedness across planar and 3D settings, with dynamic convergence to elastica equilibria. The analysis combines linear parabolic theory, fixed-point arguments, and Lojasiewicz inequalities to handle long-time behavior, including convergence in free-end and periodic geometries, and large-time periodic forcing in 2D. In 3D, they show conditional global well-posedness and convergence under curvature bounds, and they develop a robust framework for energy propagation and stability that remains valid under vanishing rotational friction. Overall, the paper provides a comprehensive PDE-theoretic treatment of immersed rod dynamics with bend–twist coupling, establishing foundational results for the mathematical theory of filament flows and their equilibria.

Abstract

We investigate a family of curve evolution equations approximating the motion of a Kirchhoff rod immersed in a low Reynolds number fluid. The rod is modeled as a framed curve whose energy consists of the bending energy of the curve and the twisting energy of the frame. The equations we consider may be realized as gradient flows of the rod energy under a certain anisotropic metric coming from resistive force theory. Ultimately, our goal is to provide a comprehensive treatment of the PDE theory of immersed rod dynamics. We begin by analyzing the problem without the frame, in which case the evolution is globally well-posed and solutions asymptotically converge to Euler elasticae. Next, in the planar setting, we demonstrate the existence of large time-periodic solutions forced by intrinsic curvature relevant to undulatory swimming. Finally, the majority of the paper is devoted to the evolution of an immersed Kirchhoff rod in 3D, which involves a strong coupling between the curve and frame. Under a physically reasonable assumption on the curvature, solutions exist globally-in-time and asymptotically converge to rod equilibria.

Paper Structure

This paper contains 34 sections, 25 theorems, 447 equations.

Table of Contents

  1. Introduction
  2. Additional literature
  3. Derivation of the models
  4. Resistive force theory
  5. Derivation of the coupled rod--frame evolution
  6. Energy balance
  7. Immersed filament evolution without forcing
  8. Linear theory
  9. Parabolic estimates
  10. Tension estimates
  11. Local well-posedness
  12. Energetics and global well-posedness
  13. Propagation estimates and decay to equilibrium
  14. Poincaré inequality for free ends
  15. Propagation estimates
  16. ...and 19 more sections

Key Result

Theorem 1.1

The classical elastohydrodynamics model eq:classical on $I = \mathbb{T}$ and $[0,1]$ is globally well-posed in $H^2_s$, and its solutions converge strongly to Euler elasticae as $t \to +\infty$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Remark 3.2: Solvability at $\bm{X}_s$ level
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 36 more