Asymptotic L1-decay of solutions of the porous medium equation to self-similarity

By José Antonio Carrillo, Giuseppe Toscani. 2000

325 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/dcadc7036e1614994a4edfd2d539621b78786fd2

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆ (2002, unknown): “The entropy-entropy dissipation method has been used for the heat equation in [42,43,1] and generalized to nonlinear diffusions in [17,12] (also see [36] by F.”
THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION (2001, unknown): “29 have been extended to the case m ≠ = 1 by Carrillo & Toscani [15] (for m > 1) and Dolbeault & del Pino [18] (for m < 1).”
Asymptotics of the Fast Diffusion Equation via Entropy Estimates (2007, methodology): “This has been flrst done in the range of exponents corresponding to the porous medium equation, with 1 15 ].”
Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory (2002, methodology): “I also warmly thank L. Arkeryd, E. Caglioti, C. Cercignani, L. Desvillettes, X. Lu, F. Malrieu, N. Masmoudi, M. Pulvirenti, G. Toscani for providing remarks and corrections, and J.-F. Coulombel for his job of tracking misprints.”
Fine Asymptotics for Fast Diffusion Equations (2003, methodology): “…j 1Þ ¼ Z Rd f ð Þ ð 1Þ 0ð 1Þ ð 1Þgð yÞ dy: ð3:4Þ The definition of relative entropy has been used as the difference between the values of the entropy Eð Þ Eð 1Þ with Eð 1Þ < 1 in Carrillo and Toscani (2000), Carrillo et al. (2001), Dolbeault and del Pino (2002), Otto (2001), equivalent, of course,…”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “It works for solutions with a finite second moment, ∫ u0(x) |x|2 dx, [79] and uses the powerful machinery of entropy methods, that become subsequently very popular in studies of nonlinear diffusion.”
Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the p-Laplacian (2002, methodology): “Recently, an approach based on an entropy-entropy production method [1,3,11,2] has been developed, but it has apparently not been sufficient to catch the intermediate asymptotics, i.”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “the classical works of Vazquez on the self-similar behavior of the porous medium equation, which are nicely collected in the book [53], as well as the papers [45, 19].”
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources (2009, background): “Assuming that ∫ R2 (1 + |v|2) g0(v)dv +∞, J.A. Carrillo and G. Toscani [7] proved that g(t, v) behaves asymptotically in a self-similar way like the Barenblatt-Pattle solution, as t → +∞.”
L1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping (2011, methodology): “One thus expects slower decay rates in our case than in [4].”
Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates (2003, methodology): “1 is by direct calculation in the same spirit used by Arnold, Carrillo, Jüngel, Markowich, Toscani and Unterreiter in [1, 12, 9]; it only relies on differential calculus and integration by parts.”
Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation (2002, methodology): “Following [ 13 , 14] the natural entropy to study the asymptotic behavior for (1.10) is given by”