A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Details A Family of Nonlinear Fourth Order Equations of Gradient Flow Type A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

2009-01-05

arxiv.org/abs/0901.0540v1

Mathematics

Analysis of PDEs

33 pages, no figures

Scientific paper

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.

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