On gradient flows of singular interaction energies on curves

Matt, Hannes; Westdickenberg, Michael (Thesis advisor); von der Mosel, Heiko (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

In this thesis we prove various results connected to integral energies with singular interaction densities on curves and rectifiable sets. Firstly, we consider a flow in the Wasserstein space of a singular energy whose gradient is, at least formally, the Riesz transform. We show that gradient flows exist for suitably regularised versions of the energy, and that these converge to a limit flow. We further prove the following conditional result. If the gradient flows of the regularised energies and the limit flow consist of uniformly rectifiable measures, then the limit flow satisfies a continuity equation driven by the Riesz transform. Using a smooth and symmetrical function, we cut off the singularity of the energy density. Then, we show that the Wasserstein-gradient flow of the integral energy exists by solving a corresponding continuity equation. Decreasing the amount that is cut off step by step, we obtain a family of gradient flows and their associated energy dissipation inequalities. The initial energy gives a uniform bound for the dissipation and thus implies the equi-continuity of the gradient flows. The theorem of Arzelà-Ascoli then yields a limit flow. The above-mentioned condition allows us to pass to the limit in the dissipation inequalities; this yields the result. Secondly, we consider a flow of the integral Menger curvature in a Banach space of knotted curves. Geometric curvature energies have been popular for the past three decades. In knot theory, they appear as so-called knot energies and are used as a classification tool. In the last decade, their dynamical behaviour has been investigated and gradient flows of various knot energies in Hilbert spaces have been shown to exist. In this thesis, we give a reinterpretation of one such a gradient flow as a flow of measures. Moreover, we leave the Hilbert space setting and prove the long-time existence of a gradient flow of the integral Menger curvature with an additional penalty term in a Banach space. Our approach is based on the theory of minimising movements in metric spaces. The additional penalty term is based on the logarithmic strain of a curve and thus uniform bounds on the energy give uniform control on the parametrisation of the curve. When combined with uniform bounds on the integral Menger curvature of the curve, we obtain uniform control over the bi-Lipschitz constants of the curves and thus preserve their injectivity. That way, the total energy is weakly lower semicontinuous on the set of injective and regular Sobolev-Slobodeckij curves and a minimising movement exists. By choosing a Banach space that is slightly smaller than the energy space of the integral Menger curvature, we get access to compact Sobolev-embeddings. As a consequence, the weak lower semicontinuity of the local slope is established and we conclude that the minimising movement is in fact a gradient flow. Lastly, we compute the first variation of the total Menger curvature on Lipschitz graphs and give a reinterpretation of a flow of curves as a flow of measures.