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Abstract
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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
-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.
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Keywords
higher-order diffusion equations, quantum-diffusion model,
Wasserstein gradient flow, flow interchange estimate,
long-time behavior, linearization
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Mathematical Subject Classification
Primary: 35K30
Secondary: 35B40, 35B45
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Milestones
Received: 27 January 2021
Revised: 20 May 2021
Accepted: 21 August 2021
Published: 12 February 2022
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