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Generalization Of An Inequality By Talagrand, And Links With The Logarithmic Sobolev Inequality
 J. Funct. Anal
, 2000
"... . We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, ..."
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Cited by 244 (12 self)
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. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial dierential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 11 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 30 7. Linearizations 31 Appendix A. A nonlinear approximation argument 35 References 36 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance d(x; y) = inf 8 < : s Z 1 0 j _ w(t)j 2 dt; w 2 C 1 ((0; 1); M); w(0) = x; w(1) = y 9 ...
Best constants for GagliardoNirenberg inequalities and applications to nonlinear diffusions
, 2001
"... ..."
Global stability of vortex solutions of the twodimensional NavierStokes equation
 Comm. Math. Phys
"... NavierStokes equation ..."
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General relative entropy inequality: an illustration on growth models
 J. Math. Pures Appl
"... We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not concervation laws, the notion of relative entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biol ..."
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Cited by 61 (9 self)
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We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not concervation laws, the notion of relative entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biological applications where birth and death can be described by zeroth order terms. But the concept also has applications to more general growth models as the fragmentation equations. We give several types of applications of the General Relative Entropy Inequality: a priori estimates and existence of solution, long time asymptotic to a steady state, attraction to periodic solutions. Keywords: Relative entropy, fragmentation equations, cell division, selfsimilar solutions, periodic solutions, long time asymptotic. AMS class. No: 35B40, 35B10, 82C21, 92B05, 92D25 1 Introduction: Hyperbolic
LongTime Asymptotics of Kinetic Models of Granular Flows
 Arch. Rational Mech. Anal
, 2003
"... We analyze the longtime asymptotics of certain onedimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coe#cient of restitution. These nonlinear ..."
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Cited by 46 (6 self)
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We analyze the longtime asymptotics of certain onedimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coe#cient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending whether their similarity solutions (homogeneous cooling state) extinguish or not in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the similarity solution extinguishes in finite time, we prove that any other solution with initially bounded support extinguishes in finite time, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support.
F.: LongTime Asymptotics of a Multiscale Model for Polymeric Fluid Flows
 Arch. Rational Mech. Anal
"... Communicated by the Editors In this paper, we investigate the longtime behavior of some micromacro models for polymeric fluids (Hookean model and FENE model), in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity, general bounded d ..."
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Cited by 36 (3 self)
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Communicated by the Editors In this paper, we investigate the longtime behavior of some micromacro models for polymeric fluids (Hookean model and FENE model), in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity, general bounded domain with nonhomogeneous Dirichlet boundary conditions on the velocity). We use both probabilistic approaches (coupling methods) and analytic approaches (entropy methods). 1.
On generalized CsiszárKullback inequalities
, 2000
"... The classical CsiszárKullback inequality bounds the L 1 distance of two probability densities in terms of their relative (convex) entropies. Here we generalize such inequalities to not necessarily normalized and possibly nonpositive L 1 functions. Also, our generalized CsiszárKullback inequali ..."
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Cited by 33 (1 self)
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The classical CsiszárKullback inequality bounds the L 1 distance of two probability densities in terms of their relative (convex) entropies. Here we generalize such inequalities to not necessarily normalized and possibly nonpositive L 1 functions. Also, our generalized CsiszárKullback inequalities are in many important cases sharper than the classical ones (in terms of the functional dependence of the L 1 bound on the relative entropy). Moreover our construction of these bounds is rather elementary.
The Derrida–Lebowitz–SpeerSpohn equation: existence, nonuniqueness, and decay rates of the solutions
 SIAM J. Math. Anal
"... Abstract. The logarithmic fourthorder equation ..."
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Inequalities for generalized entropy and optimal transportation
 IN PROCEEDINGS OF THE WORKSHOP: MASS TRANSPORTATION METHODS IN KINETIC THEORY AND HYDRODYNAMICS
, 2003
"... A new concept of Fisherinformation is introduced through a cost function. That concept is used to obtain extensions and variants of transport and logarithmic Sobolev inequalities for general entropy functionals and transport costs. Our proofs rely on optimal mass transport from the MongeKantorovic ..."
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Cited by 31 (4 self)
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A new concept of Fisherinformation is introduced through a cost function. That concept is used to obtain extensions and variants of transport and logarithmic Sobolev inequalities for general entropy functionals and transport costs. Our proofs rely on optimal mass transport from the MongeKantorovich theory. They express the convexity of entropy functionals with respect to suitably chosen paths on the set of probability measures.
A new class of transport distances between measures
 Calc. Var. Partial Differential Equations
"... Abstract We introduce a new class of distances between nonnegative Radon measures in Rd. They are modeled on the dynamical characterization of the KantorovichRubinsteinWasserstein distances proposed by BENAMOUBRENIER [7] and provide a wide family inSobolev distances. From the point of view of o ..."
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Cited by 31 (7 self)
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Abstract We introduce a new class of distances between nonnegative Radon measures in Rd. They are modeled on the dynamical characterization of the KantorovichRubinsteinWasserstein distances proposed by BENAMOUBRENIER [7] and provide a wide family inSobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established KantorovichRubinsteinWasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. terpolating between the Wasserstein and the homogeneous W −1,p γ