Calculus of Variations and Geometric Measure Theory

D. Matthes - R. J. McCann - G. Savaré

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

created by savare on 12 Jan 2009


Submitted Paper

Inserted: 12 jan 2009

Year: 2009


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 $$ \frac1{2\alpha} \int{Rd} \big
D u\alpha\big
2 \,dx + \frac\lambda2 \int{Rd}
2 u\,dx $$ with respect to the $L^2$-Wasserstein metric. The value of $\alpha$ ranges from $1/2$, corresponding to a simplified quantum drift diffusion model, to $1$, corresponding to a thin film type equation.

Keywords: Fisher information, Wasserstein distance, entropy, Gradient flows, minimizing movements, Thin film equation, Asymptotic decay, Second order logarithmic-Sobolev inequalities