The Porous Medium Equation. New Contractivity Results

Vázquez, Juan Luís

doi:10.1007/3-7643-7384-9_42

Cited by 8 publications

(5 citation statements)

References 59 publications

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“…It is essentially one-dimensional and it was established in the PME case by Vázquez [34] and it proved useful in studies of free boundary location or asymptotic behaviour. It is related to mass transport and Wasserstein distances, [45,36]. It will be crucial in some proofs below, like the proof of the existence for general initial data and the asymptotic behaviour.…”

Section: Shifting Comparisonmentioning

confidence: 99%

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

Vázquez

2016

J. Evol. Equ.

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We consider nonlinear parabolic equations involving fractional diffusion of the form ∂ t u + (−∆) s Φ(u) = 0, with 0 < s < 1, and solve an open problem concerning the existence of solutions for very singular nonlinearities Φ in power form, precisely Φ ′ (u) = c u −(n+1) for some 0 < n < 1. We also include the logarithmic diffusion equation ∂ t u + (−∆) s log(u) = 0, which appears as the case n = 0. We consider the Cauchy problem with nonnegative and integrable data u 0 (x) in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The limit solutions we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, that are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as t → ∞) of the solutions with general integrable data. A new comparison principle is introduced.2000 Mathematics Subject Classification. 35K55, 35K65, 35S10, 26A33, 76S05

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Section: Shifting Comparisonmentioning

confidence: 99%

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

Vázquez

2016

J. Evol. Equ.

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“…, for example, [3,5], the W 2 -contractivity in arbitrary space dimensions being established in [4]. In higher space dimensions, W p -contractivity with p large does not hold [27]. For these reasons, the non-uniform continuity property established in Theorem 1.1 for the semiflow map associated to (1.1) and (1.2) is surprising, the more because for the linear correspondent of (1.1), that is, the heat equation, the semiflow map is a contraction at any fixed positive time, cf.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Non‐uniform continuity of the semiflow map associated to the porous medium equation

Matioc

2014

Bulletin of London Math Soc

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Abstract. We prove that the semiflow map associated to the evolution problem for the porous medium equation (PME) is real-analytic as a function of the initial data in H s (S), s > 7/2, at any fixed positive time, but it is not uniformly continuous. More precisely, we construct two sequences of exact positive solutions of the PME which at initial time converge to zero in H s (S), but such that the limit inferior of the difference of the two sequences is bounded away from zero in H s (S) at any later time.

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“…In particular, contraction in the d ∞ norm allows for fine estimates of the support propagation. Unfortunately, the PME semigroup is not contractive in the distance d p for p large if n > 1, [42]. Actually, a similar negative result happens for the usual L p norms in all space dimensions, see [44].…”

Section: Some Lines Of Researchmentioning

confidence: 94%

Perspectives in nonlinear diffusion: between analysis, physics and geometry

Vázquez¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined. Mathematics Subject Classification (2000). Primary 35K55; Secondary 35K65.

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The Porous Medium Equation. New Contractivity Results

Cited by 8 publications

References 59 publications

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

Non‐uniform continuity of the semiflow map associated to the porous medium equation

Perspectives in nonlinear diffusion: between analysis, physics and geometry

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