A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

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 $F[u]=1/(2\alpha )\int |D(u^{\alpha }){|}^{2}dx+\lambda /2\int |x{|}^{2}udx$ 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.

33 pages, no figures