A new class of transport distances between measures

Dolbeault, Jean; Nazaret, Bruno; Savaré, Giuseppe

doi:10.1007/s00526-008-0182-5

Cited by 170 publications

(282 citation statements)

References 26 publications

(23 reference statements)

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“…The correct way to look at a scalar conservation law as a gradient flow is replacing the classical 2-Wasserstein distance with an alternative transport distance with a nonlinear mobility, see [15,24,42]. We shall describe such approach here.…”

Section: The Nonlinear Mobility Approach: a Microscopic Viewpointmentioning

confidence: 99%

“…Here X = X ρ according to the notation established in section 2. The class of distances (50) has been studied in [15,24,42], together with their relation with certain applied PDEs in which a nonlinear mobility effect is relevant in the dynamics, see e. g. [13] for chemotaxis movements. Let us now consider the functional F : M → R…”

Section: A Generalised Gradient Flow Structure For Scalar Conservatiomentioning

confidence: 99%

“…• To bypass such a problem, in section 6 we introduce a new approach of a gradient flow in the generalised Wasserstein space with a nonlinear mobility, see [42]. We show a rigorous gradient flow structure at a microscopic Lagrangian particle level, and we conjecture a variational structure based on results in [39].…”

Section: Introductionmentioning

confidence: 99%

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Scalar conservation laws seen as gradient flows: known results and new perspectives

Francesco

2016

ESAIM: ProcS

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Abstract. We review some results in the literature which attempted (only partly successfully) at linking the theory of scalar conservation laws with the Wasserstein gradient flow theory. In particular, we consider the problem of writing a scalar conservation law within the Wasserstein gradient flow theory. As a related problem, we also review results on contraction properties of scalar conservation laws in the p-Wasserstein distances. Moreover, we provide a particle-based approach to view a scalar conservation law as a gradient flow in a nonlinear-mobility sense. Finally, we propose a semi-implicit particle method based on the standard 2-Wasserstein distance.

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Section: The Nonlinear Mobility Approach: a Microscopic Viewpointmentioning

confidence: 99%

Section: A Generalised Gradient Flow Structure For Scalar Conservatiomentioning

confidence: 99%

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Scalar conservation laws seen as gradient flows: known results and new perspectives

Francesco

2016

ESAIM: ProcS

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“…A very nice interpretation in terms of a gradient flow was done first in the limit case of the heat equation in [33], and then in the porous media case in [38]. See also [1,25,36,39,28] for further results based on Wasserstein's distance and mass transportation theory. Concerning interpolation inequalities, weights and asymptotic behavior, we also have to quote [13,22,12] for some recent results.…”

Section: Bakry-emery Criterion For Diffusionsmentioning

confidence: 99%

On the Bakry-Emery criterion for linear diffusions and weighted porous media equations

Dolbeault¹,

Nazaret²,

Savaré³

2008

Communications in Mathematical Sciences

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Abstract. The goal of this paper is to give a non-local sufficient condition for generalized Poincaré inequalities which extends the well-known Bakry-Emery condition. Such generalized Poincaré inequalities have been introduced by W. Beckner in the Gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the L 2 norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schrödinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential.Key words. parabolic equations, diffusion, Ornstein-Uhlenbeck operator, porous media, Poincaré inequality, logarithmic Sobolev inequality, convex Sobolev inequality, interpolation, decay rate, entropy, free energy, Fisher information AMS subject classifications. 35B40; 35K55; 39B62; 35J10; 35K20; 35K65The Bakry-Emery method [9] has been extremely successful in establishing logarithmic Sobolev inequalities [31]. It is also known to apply very well to the proof of Poincaré inequalities and inequalities which interpolate between Poincaré and logarithmic Sobolev inequalities [11,6,2,21], the so-called "generalized Poincaré inequalities." The first paper on such inequalities has been written by W. Beckner in [11] in the case of a Gaussian measure. In [6], the inequalities have been slightly generalized by taking into account general x-dependent diffusions and by considering "convex entropies" based on a convex function ψ satisfying the additional admissibility condition:The proof is based on the entropy-entropy production method and the Bakry-Emery condition, [9]. The result has been slightly improved in [5], thus providing "refined inequalities" in the case ψ(s) = (s p − 1 − p(s − 1))/(p − 1), p ∈ (1,2). Other considerations on "generalized Poincaré inequalities" can be found in [2,35,21,3].The Bakry-Emery condition is a sufficient, local condition, which relies on a uniform strict log-concavity of the measure. Using perturbation techniques, it is possible to relax such a strict assumption to some extent, see [32,21]. When Poincaré and logarithmic Sobolev inequalities are known to hold simultaneously, further interpolation inequalities can also be established: see for instance [35,15,10,3] BAKRY-EMERY CRITERION FOR DIFFUSIONSIn case of generalized Poincaré inequalities, many techniques which are available for Poincaré inequalities and spectral gap approaches can be adapted, and more flexib...

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“…Recently, Dolbeault, Nazaret and Savaré introduced in [17] new classes of distances over P(R d ) based on the minimization of the functional (where λ is a given reference measure on R d and ρ and q are identified with their densities w.r.t. λ)…”

Section: Introductionmentioning

confidence: 99%

A Benamou–Brenier Approach to Branched Transport

Brasco¹,

Buttazzo²,

Santambrogio³

2011

SIAM J. Math. Anal.

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The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length ℓ covered by a mass m is proportional to m α ℓ with 0 < α < 1. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks. . . Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path ρ t of probabilities, connecting an initial state µ 0 to a final state µ 1 , satisfying the continuity equation ∂ t ρ + div x q = 0 together with a velocity field v (with q = ρv being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures:

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A new class of transport distances between measures

Cited by 170 publications

References 26 publications

Scalar conservation laws seen as gradient flows: known results and new perspectives

Scalar conservation laws seen as gradient flows: known results and new perspectives

On the Bakry-Emery criterion for linear diffusions and weighted porous media equations

A Benamou–Brenier Approach to Branched Transport

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