A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Matthes, Daniel; McCann, Robert J.; Savar'e, Giuseppe

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Authors

Matthes, Daniel

McCann, Robert J.

Savar'e, Giuseppe

Description

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 $F[u] = 1/(2α) \int | D (u^α) |^2 dx + λ/2 \int |x|^2 u dx$ with respect to the $L^2$-Wasserstein metric. The value of $α$ ranges from $α=1/2$, corresponding to a simplified quantum drift diffusion model, to $α=1$, corresponding to a thin film type equation.
33 pages, no figures