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Flow of elastic networks

Subject Area Mathematics
Term from 2018 to 2023
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 404870139
 
Final Report Year 2024

Final Report Abstract

Networks and flow of networks arise naturally in the study of multiphase systems and of the dynamics of their interfaces. Here, the word network refers to a set of curves “glued” together at their end-points. Each junction point is free to move in a more favorable energy configuration with the condition that the topology of the system has to be conserved. One of the main contributions of this project is the understanding of the evolution of networks that are driven by energy steepest descent and that describe elastic objects. The energy leading the evolution is the elastic energy which is quadratic in the curvature of the curves. As a consequence, the resulting system of evolution equations is quasi-linear and of higher order. The novelty of the project lies in the combination of the study of networks with fourth order geometric partial differential equations. The main challenges are the higher order nature of the equations, the modelling and control of the junction points as well as the geometric nature of the problem. Thanks to a good understanding of the geometry involved and capturing many information from this, we establish a well-posedness result for the problem and discover sufficient conditions for the existence of a global solution for a large class of initial data. Our further investigations are motivated either by the study of networks, or by the study of variational problems (such as obstacle problems, problems in an anisotropic ambient space, etc.) and their associated evolution problems involving the elastic energy or some generalization thereof. Besides some numerical features, in the project we consider mainly analytical aspects with two main objectives. The first is the study of the asymptotic behavior of solutions and in establishing new techniques to reach this goal (like the Løjasiewicz-Simon inequality for constrained problem or a detailed study of the possible critical points). The second is the study of qualitative properties of solutions like regularity, preservation of convexity or embeddedness. Due to the higher order nature of the problems, such questions lead to interesting new theories and results.

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