Gradient Flow Structure of a Multidimensional Nonlinear Sixth Order Quantum-Diffusion Equation

TL;DR

This paper studies a sixth-order nonlinear parabolic equation from quantum mechanics, establishing global solutions and spectral properties by leveraging its gradient flow structure in the Wasserstein metric.

Contribution

It introduces a novel analysis of a sixth-order quantum-diffusion equation as a Wasserstein gradient flow, proving existence and spectral characteristics.

Findings

Global existence of weak solutions for finite entropy initial data

Explicit spectral computation around the stationary solution

Identification of a key relation between entropy, Fisher information, and the second order functional

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 $L^2$-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.