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On motion of an elastic wire in a Riemannian manifold and singular perturbation. (English) Zbl 1317.35251
Summary: R. E. Caflisch and J. H. Maddocks [Proc. R. Soc. Edinb., Sect. A, Math. 99, 1–23 (1984; Zbl 0589.73057)] analyzed the dynamics of a planar slender elastic rod. We consider a thin elastic rod \(\gamma\) in an \(N\)-dimensional Riemannian manifold. The former model represents an elastic rod with positive thickness, and the equation becomes a semilinear wave equation. Our model represents an infinitely thin elastic rod, and the equation becomes a 1-dimensional semilinear plate equation. We prove the short time existence of solutions. We also discuss the behaviour of the solution when the resistance goes to infinity, and find that the solution converges to a solution of a gradient flow equation.
MSC:
35Q74 PDEs in connection with mechanics of deformable solids
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A04 Curves in Euclidean and related spaces
35B25 Singular perturbations in context of PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B20 Perturbations in context of PDEs
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Full Text: Euclid
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