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Session 13. Global existence versus blowup in nonlinear parabolic systems
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On a Willmore-Helfrich \(L^2\)-flow of open curves in \(\mathbb{R}^n\)
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Anna Dall'Acqua, Ulm University, Germany
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The talk is based on the joint work with Paola Pozzi (University of Duisburg-Essen)
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In the Bernoulli model of an elastic rod described by a curve \(f: I
\rightarrow \mathbb{R}^n\), \(n \geq 2\), the elastic energy is given by
\[ \mathcal{E}(f)= \int_{I} |\vec{\kappa}|^2 \, ds \, ,\]
where \(\vec{\kappa}\) is the curvature and \(s\) is the arc-length parameter.
A natural approach to get to minimisers is to study the associated
steepest descent flow. Due to the behavior of the energy under scaling,
one soon notices that the growth of length of the curve should be
penalized (alternatively one could impose that the length does not
change).
Motivated by the model of an elastic rod we study the evolution of open
curves with fixed end-points, subject to natural boundary conditions and
moving in time so that the energy decreases.
We show that, if the initial datum is smooth enough, the solution exists
globally in time and subconverges to a critical point. Our results apply
also to the more general situation where the energy functional contains a
so-called spontaneous curvature.
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References- A. Dall'Acqua, P. Pozzi,
A Willmore-Helfrich \(L^2\)-flow of curves with natural boundary conditions ,
Accepted for publication in Communications in Analysis and Geometry.
- G. Dziuk, E. Kuwert and R. Schätzle, Reiner,
Evolution of elastic curves in \(\Bbb{R^n}\) : existence and computation, SIAM J. Math. Anal. 33, 2002, 1228-1245.
- C.-C. Lin, \(L^2\)-flow of elatic curves with clamped boundary conditions, J. Differ.
Equ. 252, 2012, 6414-6428.
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