Vol. 3, No. 4, 2021

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Gradient flow structure of a multidimensional nonlinear sixth-order quantum-diffusion equation

Daniel Matthes and Eva-Maria Rott

Vol. 3 (2021), No. 4, 727–764
Abstract

A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics and also as the gradient flow of a second-order information functional with respect to the L2-Wasserstein metric. First, we prove global existence of weak solutions for initial conditions of finite entropy by means of the time-discrete minimizing movement scheme. Second, we calculate the linearization of the dynamics around the unique stationary solution, for which we can explicitly compute the entire spectrum. A key element in our approach is a particular relation between the entropy, the Fisher information and the second-order functional that generates the gradient flow under consideration.

Keywords
higher-order diffusion equations, quantum-diffusion model, Wasserstein gradient flow, flow interchange estimate, long-time behavior, linearization
Mathematical Subject Classification
Primary: 35K30
Secondary: 35B40, 35B45
Milestones
Received: 27 January 2021
Revised: 20 May 2021
Accepted: 21 August 2021
Published: 12 February 2022
Authors
Daniel Matthes
Zentrum für Mathematik
Technische Universität München
Garching
Germany
Eva-Maria Rott
Zentrum für Mathematik
Technische Universität München
Garching
Germany