## D. Matthes - R. J. McCann - G. SavarÃ©

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

created by savare on 12 Jan 2009

[

BibTeX]

*Submitted Paper*

**Inserted:** 12 jan 2009

**Year:** 2009

**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
$$
\frac1{2\alpha}
\int_{{R}^{d}} \big

D u^{\alpha\big}^{2} \,dx + \frac\lambda2 \int_{{R}^{d}x}^{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

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