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A dynamic approach to heterogeneous elastic wires

Anna Dall'Acqua, Leonie Langer, Fabian Rupp

Abstract

We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.

A dynamic approach to heterogeneous elastic wires

Abstract

We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated -gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.
Paper Structure (24 sections, 27 theorems, 179 equations)

This paper contains 24 sections, 27 theorems, 179 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Local well-posedness and regularity
  4. Transformation to a periodic setting
  5. Short-time existence and continuous dependence
  6. Smoothness and uniqueness of solutions
  7. Proof of Theorem \ref{['thm:Main STE']}
  8. Global existence
  9. Controlling the matrix Pi-1
  10. Energy estimates
  11. Proof of Theorem \ref{['longtimeex']}
  12. Convergence result
  13. Analyticity
  14. The constrained Lojasiewicz-Simon gradient inequality
  15. Convergence
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $(\theta_0, \rho_0)\in h^{1+\alpha}([0,L])$ for some $\alpha\in (0,1)$ and assume Hypothesis hyp:intro. Then there exists $T_0>0$ and a unique solution $(\theta, \rho)\in C^\infty((0,T_0)\times[0,L])$ of eq:flow equation on $(0,T_0)\times [0,L]$ which satisfies Moreover, the solution depends continuously on the initial datum.

Theorems & Definitions (62)

  • Theorem 1.2: Local well-posedness
  • Theorem 1.3: Global existence
  • Theorem 1.4: Long-time behavior
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3: Existence and continuous dependence
  • ...and 52 more