Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations

Bodineau, Thierry; Lebowitz, Joel L.; Mouhot, Clément; Villani, Cédric

doi:10.1088/0951-7715/27/9/2111

Cited by 30 publications

(44 citation statements)

References 29 publications

(54 reference statements)

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“…and we will show that it is the same function as appears in (4). Note indeed that by replacing here ψ via (52) we find a convex…”

Section: A From Detailed Balance To Gradient Flowmentioning

confidence: 52%

“…Here we assume the constraint (44) for some operator D as appears in (1)-(4) (or, J = 0 in (43)). We will see how under detailed balance conditions the time-symmetric part of the Lagrangian, L(−j; z) + L(j; z), determines the zero-cost flow j z for given entropy S, and in such a way that the autonomous evolution is a gradient flow with respect to the entropy S, as in (4) or as in the examples of Section II A.…”

Section: A From Detailed Balance To Gradient Flowmentioning

confidence: 99%

“…The point is that the entropy is the large deviation rate function for macroscopic fluctuations and is therefore always increasing along the first-order evolution equation for the macroscopic variable. That result has been made precise and illustrated in various contexts [3][4][5].…”

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confidence: 91%

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Deriving GENERIC from a Generalized Fluctuation Symmetry

et al. 2017

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Much of the structure of macroscopic evolution equations for relaxation to equilibrium can be derived from symmetries in the dynamical fluctuations around the most typical trajectory. For example, detailed balance as expressed in terms of the Lagrangian for the path-space action leads to gradient zero-cost flow. We find a new such fluctuation symmetry that implies GENERIC, an extension of gradient flow where a Hamiltonian part is added to the dissipative term in such a way as to retain the free energy as Lyapunov function. I. HISTORY AND OUTLINE OF THE PAPERWhile macroscopic equations describing the return to equilibrium have been conceived and applied even before the atomistic picture of matter was widely accepted, their derivation shows important mathematical and conceptual difficulties. After all, hydrodynamic and thermodynamic behavior is described autonomously in only a few macroscopic variables and it must be understood how these variables get effectively decoupled from the many microscopic degrees of freedom. Moreover, in modifying the scale of description, the character of the dynamics could drastically change, from a unitary or a Hamiltonian to a dissipative evolution as possibly one of the most remarkable features 1 .One of the very first and still much studied examples in transiting from microscopic 1 Perhaps the earliest example where that question became manifest is through d'Alembert's paradox (1752) for reconciling, what we call today, Euler's equation with that of Navier-Stokes. As d'Alembert wrote indeed, "It seems to me that the theory (potential flow), developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance, a singular paradox which I leave to future Geometers to elucidate."

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“…and we will show that it is the same function as appears in (4). Note indeed that by replacing here ψ via (52) we find a convex…”

Section: A From Detailed Balance To Gradient Flowmentioning

confidence: 52%

Section: A From Detailed Balance To Gradient Flowmentioning

confidence: 99%

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Deriving GENERIC from a Generalized Fluctuation Symmetry

et al. 2017

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“…For more details, see Sections 4 and 5 of [16] or also the last example at the end of Subsection 2.4. When dealing with strictly positive Dirichlet data, an entropy method similar to [16] has been developed in [8].…”

Section: (Rcdp)mentioning

confidence: 99%

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Bonforte

Figalli

2020

Comm Pure Appl Math

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We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation u t = ∆u m , posed in a smooth bounded domain Ω ⊂ R N , in the exponent range m s = (N −2) + /(N +2) < m < 1. It is known that bounded positive solutions extinguish in a finite time T > 0, and also that they approach a separate variable solution u(t, x) ∼ (T − t) 1/(1−m) S(x), as t → T − . It has been shown recently that v(x, t) = u(t, x) (T − t) −1/(1−m) tends to S(x) as t → T − , uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincaré inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of "almost orthogonality", which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincaré inequalities and sharp convergence rates for linear flows. Consider the Cauchy-Dirichlet problem for the Fast diffusion Equation (FDE)where m ∈ (0, 1), u 0 ≥ 0, and Ω ⊂ R N is a smooth bounded domain of class C 2,α . The main goal of this paper is to study the fine asymptotic behaviour of nonnegative solutions to this problem.This problem has been addressed for the first time in the '80s by Berryman and Holland in their pioneering work [6]. However, the results of [6] were not conclusive and many question were left open, due to the several difficulties hidden in this apparently simple problem. After that work, only a few relevant improvements appeared, and many basic questions are still open in many relevant aspects. We will give an account of the previous results concerning the problem under consideration, starting by a L ∞ (Ω) ≤ c ′ 0 e −t for all t ≫ 1

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“…However, it is unclear whether J h can be proven to be the rate functional of some microscopic stochastic process. In a recent paper [4], the authors introduce a stochastic particle system, the Ginzburg-Landau dynamics, and show that the porous medium equation is the hydrodynamic limit of the system. Moreover, the m-relative entropy is the rate functional of the invariant measures associated to the system.…”

Section: Discussionmentioning

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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Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations

Cited by 30 publications

References 29 publications

Deriving GENERIC from a Generalized Fluctuation Symmetry

Deriving GENERIC from a Generalized Fluctuation Symmetry

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

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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