Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Lisini, Stefano

doi:10.1051/cocv:2008044

Cited by 38 publications

(73 citation statements)

References 23 publications

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“…Proof. We only need to show that, for T > 0, the functions f τ and g τ defined as 13) where C > 0 is a constant independent of τ . By Lemma 3.8, the summation in the right hand side of (5.13) is bounded, and therefore the total variation of f τ in [0, T ] is uniformly bounded.…”

Section: Considering the Function ϕ(S) = E(t(s)û(s)) And Using (54)mentioning

confidence: 99%

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Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Ferreira

Valencia-Guevara

2017

Monatsh Math

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We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the Wasserstein space P 2 (R d ) and apply the results for a large class of PDEs with time-dependent coefficients like confinement and interaction potentials and diffusion. Our results can be seen as an extension of those in Ambrosio- Gigli-Savaré (2005)[2] to the case of timedependent functionals. For that matter, we need to consider some residual terms, timeversions of concepts like λ-convexity, time-differentiability of minimizers for MoreauYosida approximations, and a priori estimates with explicit time-dependence for De Giorgi interpolation. Here, functionals can be unbounded from below and satisfy a type of λ-convexity that changes as the time evolves. AMS MSC2010: 35R20, 34Gxx, 58Exx, 49Q20, 49J40, 35Qxx, 35K15, 60J60, 28A33.

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Section: Considering the Function ϕ(S) = E(t(s)û(s)) And Using (54)mentioning

confidence: 99%

“…We also quote the paper [4] where a 1D non-local fluid mechanics model with velocity coupled via Hilbert transform was analyzed by using gradient flow theory in P 2 . In [13], the authors dealt with nonlinear diffusion equations in the form…”

Section: Introductionmentioning

confidence: 99%

Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Ferreira

Valencia-Guevara

2017

Monatsh Math

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“…For uniformly convex potentials V , or more generally under (24), the WJ inequality for (ν, A) is obtained without using the non-negative contribution φ(x) + φ * (∇φ(x)) − 2n in J , which stems from the diffusion term. Proposition 2.5 gave a first way of taking advantage of the diffusion term to consider nonuniformly convex cases and even non-convex cases.…”

Section: Lemma 33 If ν Is In P 2c (R N ) and A Is Such Thatmentioning

confidence: 99%

Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations

Bolley

Gentil

Guillin

2012

Journal of Functional Analysis

124

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We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.

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“…The first one is to explore a larger class of evolution equations that are Wasserstein (generalized/modified) gradient flows. Many equations are now proven to belong to this class [2,[11][12][13][14]20,21,23,24]. Recently attempts have been made to extend this theory to discrete settings [22,25] and to systems that also contain conservative behavior [10,15,16].…”

Section: The Porous Medium Equationmentioning

confidence: 99%

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Duong

2015

Asymptotic Analysis

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In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional K h , which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, K h is asymptotically equivalent, in the sense of Γ -convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.1 E 1 is the Boltzmann-Shannon entropy. 0921-7134/15/$35.00 © 2015 -IOS Press and the authors. All rights reserved 86 M.H. Duong / Asymptotic equivalence of the discrete variational functional with respect to the Wasserstein distance W 2 on the space of probability measures with finite second moment P 2 (R d ),where Γ (μ, ν) is the set of all probability measures on R d × R d with marginals μ and ν. This statement can be understood in a variety of different ways. In [27], Otto shows this from a differential geometry point of view by considering (P 2 (R d ), W 2 ) as an infinite dimensional Riemannian manifold. However, for the purpose of this paper, the most useful way is that the solution t → ρ(t, x) can be approximated by the JKO-discretization scheme as follows [17,31].

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Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Cited by 38 publications

References 23 publications

Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

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