Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

Carrillo, José A.; Jüngel, Ansgar; Markowich, Peter A.; Toscani, Giuseppe; Unterreiter, Andreas

doi:10.1007/s006050170032

Cited by 343 publications

(381 citation statements)

References 48 publications

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“…[25,13,15] (optimal rates) for m ∈ [m 1 , 1), m 1 = (d − 1)/d, and [14,16] for m ∈ (m c , m 1 ), which are weaker than the ones of Theorem 1.2. See [27] for the detailed analysis of the spectrum of the linearized operator in the range m > m c .…”

Section: Introductionmentioning

confidence: 96%

“…Both questions strongly depend on m. Let us emphasize for instance that the Barenblatt solution U D,T is integrable in y for m > m c , while the pseudo-Barenblatt solution corresponding to m ≤ m c is not integrable. Since much is known in the case m > m c , see for instance [16,25] and [10,11,13,14,15,27,36,45] for more complete results, the main novelty of our paper is concerned with the lower range m ≤ m c , which has several interesting new features. For instance, in the analysis in high space dimensions, that is d > 4, another critical exponent appears,…”

Section: Introductionmentioning

confidence: 99%

“…This has been first done in the range of exponents corresponding to the porous medium equation, with 1 < m < 2, and in the range where standard Gagliardo-Nirenberg inequalities apply, m 1 ≤ m < 1, see [25,13,15]. The class of non-negative, finite mass solutions has to be narrowed to the smaller set of functions with finite free energy, or to be precise, with finite entropy and finite potential energy.…”

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

132

324

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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%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Blanchet

Bonforte

Dolbeault

et al. 2008

Arch Rational Mech Anal

132

324

View full text Add to dashboard Cite

show abstract

“…Thus, |y| ≥ |∇ϕ ε (a+ε)| or equivalently |(∇ϕ ε ) −1 (y)| ≥ a+ε since ∇ϕ ε is moreover radial. Therefore, we conclude from (8) and the definition of ψ ε that…”

mentioning

confidence: 79%

Nonlinear Diffusion: Geodesic Convexity is Equivalent to Wasserstein Contraction

Bolley

Carrillo

2014

Communications in Partial Differential Equations

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Abstract. It is well known that nonlinear diffusion equations can be interpreted as a gradient flow in the space of probability measures equipped with the Euclidean Wasserstein distance. Under suitable convexity conditions on the nonlinearity, due to R. J. McCann [12], the associated entropy is geodesically convex, which implies a contraction type property between all solutions with respect to this distance. In this note, we give a simple straightforward proof of the equivalence between this contraction type property and this convexity condition, without even resorting to the entropy and the gradient flow structure.We consider the nonlinear diffusion equationwhere f (r) is an increasing continuous function on r ∈ [0, +∞) and C 2 smooth for r > 0 such that , with u t ≥ 0 and mass u t (x)dx = M for all t, such thatii) u weakly satisfies equation (1), i.e., it satisfies the identitythere exists a unique such solution to (1) with initial datum u 0 , see [16, Sect. 9.8]. Let us normalize the mass M to unity in the rest of the introduction for convenience. Equation (1) admits the mapas a Liapunov functional. Here the map U ∈ C([0, +∞)) ∩ C 3 ((0, +∞)) is defined in a unique way by the relations f (r) = rU ′ (r)−U (r) on (0, +∞) and U (0) = U ′ (1) = 0. The map f being increasing is equivalent to U being strictly convex, since f ′ (r) = rU ′′ (r), and f being positive on (0, +∞) is equivalent to ψ :

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“…The techniques in [27] and [52] were both (more or less directly) exploiting the Wasserstein gradient flow structure of the PME. Connections with functional inequalities arising in probability theory, such as the Log-Sobolev inequality, the Talagrad inequality, and the Csiszar-Kullback inequality, were established in [3,23,27,52,53] to mention a few. The two papers [25,26] extended the theory to the full class of equations (3) including also nonlocal potentials W .…”

Section: Introductionmentioning

confidence: 99%

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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Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

Cited by 343 publications

References 48 publications

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

Nonlinear Diffusion: Geodesic Convexity is Equivalent to Wasserstein Contraction

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

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