Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient ﬂows

Portegies, Jacobus W.; Peletier, MA Mark

doi:10.4171/ifb/229

Cited by 6 publications

(4 citation statements)

References 23 publications

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“…A special subclass is formed by gradient flows with respect to a Wasserstein-type metric. 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]. For a class of evolution equations defined on discrete spaces such as graphs, various generalizations exist that unite both continuous and discrete spaces into the same structure [Maa11,Mie13b,CHLZ12].…”

Section: Introductionmentioning

confidence: 99%

On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

2014

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Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions L that induce a flow, given by L (ρ t ,ρ t ) = 0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when L is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropyWasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

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Section: Introductionmentioning

confidence: 99%

On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

2014

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“…Since the introduction of the Wasserstein gradient flows in 1997-8 [26,27,40,42] it has become clear that a very large number of well-known parabolic partial differential equations and other evolutionary systems can be written as gradient flows. Examples of these are nonlinear drift-diffusion equations [2], diffusion-drift equations with non-local interactions [8], higher-order parabolic equations [41,22,25,32,23], moving-boundary problems [41,44], and chemical reactions [35]. The parallel development of rate-independent systems introduced similar variational structures for friction [18], delamination [29], plasticity [33], phase transformations [38], hysteresis [37], and various other phenomena.…”

Section: On the Origin Of Wasserstein Gradient Flowsmentioning

confidence: 99%

“…(ii.) For the second convergence in (44), we can assume that the approximation of the end condition satisfies:…”

Section: From Large Deviations To Wasserstein Gradient Flowmentioning

confidence: 99%

Variational Formulation of the Fokker–planck Equation With Decay: A Particle Approach

Peletier

Renger

Veneroni

2013

Commun. Contemp. Math.

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We introduce a stochastic particle system that corresponds to the Fokker-Planck equation with decay in the many-particles limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Mosco-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker-Planck equation with decay.1 As suggested by one of the referees, this particular form (8)+(9) arises as the quenched largedeviation rate of a system of independent exponentially distributed decay processes, with a = 1 n #{non-decayed X i (h)} andā = 1 n #{non-decayed X i (0)}. P S 2 (R d ) := ρ ∈ P(R d ) : |x| 2 dρ < ∞, S(ρ) < ∞ .

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“…Strategy of the proof: Kantorovich duality and a variable-doubling technique. In order to prove Theorem 1.1 we develop a new strategy, generalizing [18]. It relies on the wellknown dual Kantorovich formulation [21] of the transportation cost (13):…”

Section: Corollary 13 (Strongly Monotone Operators and Invariant Meamentioning

confidence: 99%

Contraction of general transportation costs along solutions to Fokker–Planck equations with monotone drifts

Natile

Peletier

Savaré

2011

Journal de Mathématiques Pures et Appliquées

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We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in R d , when the drift is a monotone (or λ-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solutions without requiring stronger regularity and it extends the Wasserstein theory of Fokker-Planck equations with gradient drift terms started by Jordan-Kinderlehrer-Otto [14] to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.

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Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient ﬂows

Cited by 6 publications

References 23 publications

On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

Variational Formulation of the Fokker–planck Equation With Decay: A Particle Approach

Contraction of general transportation costs along solutions to Fokker–Planck equations with monotone drifts

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