A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

By Daniel Matthes, Robert J. McCann, Giuseppe Savar'e. 2009

205 citations recorded by the source corpus.

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 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eff289bb5f3b49078fdac3627d6bd0f176ce5066

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Nonlinear mobility continuity equations and generalized displacement convexity (2009, methodology): “Convexity and dissipation inequalities for second order diffusion equations are also a crucial tool (see the “metric” techniques developed by [25,32] in the linear mobility case m(ρ) = ρ) for the study of nonnegative solutions to a certain class of fourth-order nonlinear diffusion equations ∂tρ + ∇ ”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, methodology): “Another possibility is the use of estimates obtained via a technique called flow interchange (where the optimality of a density in the JKO scheme is tested against its evolution via the gradient flow of another functional, see [66]), which allows to modify the exponent of in (4.”
SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence (2020, background): “This observation is made precise and more general in [MMS09] where it is noted that the gradient ﬂow of a functional U dissipates a diﬀerent functional V at the same rate that the gradient ﬂow of V dissipates the functional U .”
On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion (2013, unknown): “This class was ﬁrst identiﬁed in the seminal work by Jordan, Kinderlehrer, and Otto [JKO98, Ott01] and later shown to encompass a wide range of known and unknown partial diﬀerential equations [AGS08, MMS09, PP10, Mie11b, LM12].”