Hardy–Poincaré inequalities and applications to nonlinear diffusions

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

doi:10.1016/j.crma.2007.01.011

Cited by 54 publications

(125 citation statements)

References 14 publications

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“…Actually they only prove the L ∞ convergence in case (ii) and the L1 ∩ L ∞ convergence in case (i). Our proof was obtained independently and announced in [7]. It is based on entropy estimates and paves the way to the sharper results on convergence with rates, which are the main purpose of the present paper.…”

Section: Introductionmentioning

confidence: 96%

“…In this appendix, we state and prove a result on inequalities which we have already been partially studied in [7]. Here we give more details and a few improvements.…”

Section: Appendix A: Hardy-poincaré Inequalitiesmentioning

confidence: 99%

“…Hence, it makes sense to require that R d g V 2−m D * dx = 0 and use the spectral gap estimate, see [7] and Theorem A.1.…”

Section: Linearizationmentioning

confidence: 99%

“…Appendix A is devoted to the proofs of spectral gap estimates, that is, of weighted Poincaré-Hardy inequalities, which have already been studied in [7], and in [14,27] in the special case m > m c . We consider the family of weights of the form V D or V 2−m D , D > 0, that are obtained from the linearization of the relative entropy.…”

Section: Introductionmentioning

confidence: 99%

“…Analyzing the relationship between the optimal constant C m,d = m/λ m,d in the corresponding functional inequality and the asymptotic rates of the fast diffusion equation is the leitmotiv of this paper. The case m = m * has to be excluded for reasons which are deeply related to Hardy's inequality, see [7]. …”

Section: Introductionmentioning

confidence: 99%

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Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Blanchet

Bonforte

Dolbeault

et al. 2008

Arch Rational Mech Anal

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Abstract. We consider non-negative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d , d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results.

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Section: Introductionmentioning

confidence: 96%

“…In this appendix, we state and prove a result on inequalities which we have already been partially studied in [7]. Here we give more details and a few improvements.…”

Section: Appendix A: Hardy-poincaré Inequalitiesmentioning

confidence: 99%

“…Hence, it makes sense to require that R d g V 2−m D * dx = 0 and use the spectral gap estimate, see [7] and Theorem A.1.…”

Section: Linearizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Blanchet

Bonforte

Dolbeault

et al. 2008

Arch Rational Mech Anal

Self Cite

129

324

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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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Functional Aspects of the Hardy Inequality: Appearance of a Hidden Energy

Vázquez

Zographopoulos

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Starting with a functional difficulty appeared in the paper [14] by Vázquez and Zuazua, we obtain new insights into the Hardy Inequality and the evolution problem associated to it by means of a reformulation of the problem. Surprisingly, the connection of the energy of the new formulation with the standard Hardy functional is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. The problem arises when the equation is posed in a bounded domain, and also when posed in the whole space.We also consider an equivalent problem with inverse square potential on an exterior domain. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.

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Hardy–Poincaré inequalities and applications to nonlinear diffusions

Cited by 54 publications

References 14 publications

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Functional Aspects of the Hardy Inequality: Appearance of a Hidden Energy

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