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Elastic flow of networks : Short-time existence result
- Publication Year :
- 2021
-
Abstract
- 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 $$q \ge 3$$ . 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 Holder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.
- Subjects :
- Banach fixed-point theorem
010102 general mathematics
Mathematical analysis
Elastic energy
01 natural sciences
Short interval
010101 applied mathematics
Parabolic system
Mathematics - Analysis of PDEs
Mathematics (miscellaneous)
Flow (mathematics)
Mathematik
FOS: Mathematics
primary 35K52, secondary 53C44, 35K61, 35K41
Boundary value problem
0101 mathematics
Balanced flow
Analysis of PDEs (math.AP)
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....6d00d558cc5ef10112b9585daf781710