Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

Arnrich, Steffen; Mielke, Alexander; Peletier, Mark A.; Savaré, Giuseppe; Veneroni, Marco

doi:10.1007/s00526-011-0440-9

Cited by 46 publications

(42 citation statements)

References 38 publications

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“…The similarities between these two approaches and ours is that all the methods hinge on duality structure of the relevant functionals, allow one to obtain both compactness and limiting results, and can work with approximate solutions, see e.g. [6] and the papers above for details. In addition, all methods assume some sort of well-prepared initial data, such as bounded initial free energy and boundedness of the functionals.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…We now consider the many-particle limit n → ∞ in (6). It is a well-known fact that the empirical measure…”

Section: Origin Of the Functional I ε : Large Deviations Of A Stochasmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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“…It has become clear, see e.g., Refs. [39][40][41] and a recent paper [42], that the formulation of dissipative equations via the entropy functional and the dissipation potential is compatible well with asymptotic limits. This is because many techniques in calculus of variations, such as Gamma convergence, can be exploited.…”

Section: Discussionmentioning

confidence: 70%

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

Duong

2015

Physica A: Statistical Mechanics and its Applications

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h i g h l i g h t s • GENERIC formulation of the relativistic heat equation. • GENERIC formulation of the relativistic kinetic Fokker-Planck equations. • Alternative computations of the stationary solution of the relativistic kinetic FP equations. a b s t r a c tIn this paper, we formulate the relativistic heat equation and the relativistic kinetic FokkerPlanck equations into the GENERIC (General Equation for Non-Equilibrium ReversibleIrreversible Coupling) framework. We also show that the relativistic Maxwellian distribution is the stationary solution of the latter. The GENERIC formulation provides an alternative justification that the two equations are meaningful relativistic generalizations of their non-relativistic counterparts.

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“…This question was answered affirmatively in two different ways [14,3], and we refer the reader to [3] for further discussion of these issues.…”

Section: Discussionmentioning

confidence: 94%

Chemical Reactions as $\Gamma$-Limit of Diffusion

Peletier¹,

Savaré²,

Veneroni³

2012

SIAM Rev.

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Abstract. We study the limit of high activation energy of a special Fokker-Planck equation known as the Kramers-Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential H/ε. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to the KS converges, in the limit of high activation energy (ε → 0), to the solution of a simpler system modeling the spatial diffusion of A and B combined with the reaction A B. With this result we give a rigorous proof of Kramers's formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, Γ-convergence. In this paper we prove that in the limit ε → 0 solutions of the parabolic partial differential equation in spatial variables x and ξ,with no-flux boundary conditions, converge to solutions of the system of reaction-diffusion equations in spatial variable x,, for x ∈ Ω and t > 0.

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Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

Cited by 46 publications

References 38 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

Chemical Reactions as $\Gamma$-Limit of Diffusion

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