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Elastic flow of networks: short-time existence result. (English) Zbl 1470.35374

Summary: In this paper we study the \(L^2\)-gradient flow of the penalized elastic energy on networks of \(q\)-curves in \(\mathbb{R}^n\) for \(q\geq 3\). Each curve is fixed at one end-point and at the other is joint to the other curves at a movable \(q\)-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov’s theory on linear parabolic systems and Banach fixed point theorem in proper Hölder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

MSC:

35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35K51 Initial-boundary value problems for second-order parabolic systems
53E10 Flows related to mean curvature
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35Q74 PDEs in connection with mechanics of deformable solids
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