Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

Lisini, Stefano; Matthes, Daniel; Savaré, Giuseppe

doi:10.1016/j.jde.2012.04.004

Cited by 68 publications

(91 citation statements)

References 22 publications

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“…has to be equipped with the weighted Wasserstein metric corresponding to the concave mobility η. We refer to [18] for the description of the corresponding metric and to [29] for the rigorous recovery of (22) by a gradient flow approach. The difference between the non-local model (17) and the local one (22) can also be seen as follows.…”

Section: 3mentioning

confidence: 99%

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

Cancès

Matthes

Nabet

2019

Arch Rational Mech Anal

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We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. In difference to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes -but not necessarily the fluxes themselves -annihilate each other. Our main result is a rigorous proof of existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn-Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.∇ · J tot = 0 where J tot = J 1 + J 2 1991 Mathematics Subject Classification. 35K65, 35K41, 49J40, 76T99.

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Section: 3mentioning

confidence: 99%

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

Cancès

Matthes

Nabet

2019

Arch Rational Mech Anal

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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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“…the fourth-order problems studied in [19][20][21]. Possible applications to viscoelasticity are discussed in [22].…”

Section: E(γ (S θmentioning

confidence: 99%

Gradient structures and geodesic convexity for reaction–diffusion systems

Liero

Mielke

2013

Phil. Trans. R. Soc. A.

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We consider systems of reaction–diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic λ -convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift–diffusion system, provide a survey on the applicability of the theory.

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Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

Cited by 68 publications

References 22 publications

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

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

Gradient structures and geodesic convexity for reaction–diffusion systems

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