On optimal convergence rates of nonconvex gradient flows

• Über optimale Konvergenzraten nicht-konvexer Gradientenflüsse

Biesenbach, Sarah; Westdickenberg, Maria Gabrielle (Thesis advisor); von der Mosel, Heiko (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

This thesis examines (optimal) convergence rates for two nonconvex gradient flows. The central part of the present work is concerned with the one-dimensional Cahn-Hilliard equation as the $\dot{H}^{-1}$-gradient flow of the scalar Ginzburg-Landau energy. This gradient flow will be considered on both a one-dimensional torus of large system-size and the real line. The initial data is assumed to have a finite (not necessarily small) $L^1$-distance to an appropriately defined configuration with two transition layers (a "bump"-like state) and we will establish optimal algebraic-in-time relaxation rates. Our result extends the relaxation method developed in [F. Otto, S. Scholtes, Maria G. Westdickenberg, Optimal $L^1$-type relaxation rates for the Cahn-Hilliard equation on the line, SIAM J. Math. Anal. 51 (2019), no. 6, 4645-4682] for a single transition layer (a "kink"-like state) to the case of two transition layers. We use an energy-method that relies on the gradient flow structure and relationships between the energy and the energy-dissipation. Among other tools, Nash-type inequalities, duality arguments, and Schauder estimates play an important role. One challenge dealt with is that due to translation invariance of the energy gap to minimizers, energetic decay alone cannot identify to which member of the family of minimizers the solution converges. While for the kink, the conservation law for the Cahn-Hilliard equation determines the longtime limit, a different argument is required for the case of the two-layer configuration. On the torus, an exponential convergence rate for the long-time limit behavior is established. On the line, the bump-like states are only metastable and we describe the initial relaxation behavior at the optimal rate as mentioned above. In the long-time limit, the solution convergesto $-1$. In the second part of this thesis, we investigate the gradient flow of curves with respect to the $L^2$-metric of the Möbius energy. A Łojasiewicz-Simon gradient inequality and the gradient flow structure of the Möbius flow are used to establish a bound on the convergence rate, relying on results in [S. Blatt, The gradient flow of the M¨obius energy near local minimizers, Calc. Var. Partial Differential Equations, 43 (2012), no. 3-4, 403-439]. We will also explain how one might proceed in order to determine an optimal convergence rate. This includes an examination of the Hessian of the energy and its kernel around a critical point, in addition to an explicit representation of the former from [A. Ishizeki, T. Nagasawa, A decomposition theorem of the Möbius energy II: variational formulae and estimates, Math. Ann., 363 (2015), no. 1-2, 617-635], as well as a Fourier coefficient approach. A subset of the kernel is classified for the case of the critical point being a circle.