A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

TL;DR

This paper investigates the global existence and long-term behavior of solutions to a family of nonlinear fourth order gradient flow equations on Euclidean space, covering models from quantum drift to thin film equations.

Contribution

It introduces a unified analysis for a family of nonlinear fourth order equations as gradient flows of perturbed information functionals in Wasserstein space.

Findings

Proves global existence of solutions for the family of equations.

Analyzes the long-time asymptotic behavior of solutions.

Connects models from quantum drift diffusion to thin film equations.

Abstract

Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on $R^d$ are studied. These equations constitute gradient flows for the perturbed information functionals $F[u] = 1/(2\alpha) \int | D (u^\alpha) |^2 dx + \lambda/2 \int |x|^2 u dx$ with respect to the $L^2$-Wasserstein metric. The value of $\alpha$ ranges from $\alpha=1/2$, corresponding to a simplified quantum drift diffusion model, to $\alpha=1$, corresponding to a thin film type equation.