Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean; Grillo, Gabriele; Vázquez, Juan Luís

doi:10.1007/s00205-008-0155-z

Cited by 133 publications

(334 citation statements)

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Section: Bakry-emery Criterion For Diffusionsmentioning

confidence: 99%

“…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.In connection with probability theory, entropy -entropy production methods have been successfully applied in a quite general framework of Riemanian manifolds: see [7,24]. However, our approach is more related to a series of attempts which have been made to establish rates of decay for solutions of diffusion equations of second and higher order: see [27,16,34].…”

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confidence: 99%

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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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Section: Bakry-emery Criterion For Diffusionsmentioning

confidence: 99%

mentioning

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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“…In the present work we establish some results in this direction by exploring in detail how 'mass' initially concentrated at spatial infinity spreads over the entire space for the super-fast diffusion equation (1.1) with m < 0. More precisely, we shall be concerned with the Cauchy problem 2) in the strongly degenerate regime m < 0, assuming the initial data v 0 ∈ C 0 (R n ) are positive, and such that v 0 (x) → +∞ as |x| → ∞ (1.3)…”

Section: Introductionmentioning

confidence: 99%

Slow growth of solutions of superfast diffusion equations with unbounded initial data

Фила

Winkler

2017

J. London Math. Soc.

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“…The exponent m * plays a very important role in the results of [2][3][4]. The proofs of convergence with rates are based on the study of the decay in time of a certain relative entropy and a careful analysis of the linearized problem which leads to certain functional inequalities of the Hardy-Poincaré type.…”

Section: Introductionmentioning

confidence: 99%