[Solutions auto-similaires pour une équation des milieux poreux non locale]
Cette Note est consacrée à lʼétude dʼune généralisation non locale de lʼéquation des milieux poreux. Plus précisément, on obtient des formules explicites de solutions auto-similaires à support compact qui ressemblent fortement aux solutions de type Barenblatt. On donne aussi un argument formel qui permet dʼobtenir des estimations des solutions faibles du problème de Cauchy.
We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the decay of weak solutions of the corresponding Cauchy problem.
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@article{CRMATH_2011__349_11-12_641_0,
author = {Biler, Piotr and Imbert, Cyril and Karch, Grzegorz},
title = {Barenblatt profiles for a nonlocal porous medium equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {641--645},
publisher = {Elsevier},
volume = {349},
number = {11-12},
year = {2011},
doi = {10.1016/j.crma.2011.06.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.06.003/}
}
TY - JOUR AU - Biler, Piotr AU - Imbert, Cyril AU - Karch, Grzegorz TI - Barenblatt profiles for a nonlocal porous medium equation JO - Comptes Rendus. Mathématique PY - 2011 SP - 641 EP - 645 VL - 349 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.06.003/ DO - 10.1016/j.crma.2011.06.003 LA - en ID - CRMATH_2011__349_11-12_641_0 ER -
%0 Journal Article %A Biler, Piotr %A Imbert, Cyril %A Karch, Grzegorz %T Barenblatt profiles for a nonlocal porous medium equation %J Comptes Rendus. Mathématique %D 2011 %P 641-645 %V 349 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.06.003/ %R 10.1016/j.crma.2011.06.003 %G en %F CRMATH_2011__349_11-12_641_0
Biler, Piotr; Imbert, Cyril; Karch, Grzegorz. Barenblatt profiles for a nonlocal porous medium equation. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 641-645. doi : 10.1016/j.crma.2011.06.003. http://www.numdam.org/articles/10.1016/j.crma.2011.06.003/
[1] A nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., Volume 294 (2010), pp. 145-168
[2] P. Biler, C. Imbert, G. Karch, Nonlocal porous medium equation: Barenblatt profiles and other weak solutions, in preparation, 2011.
[3] L. Caffarelli, J.L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, preprint , Arch. Rational Mech. Anal., , in press. | arXiv | DOI
[4] Asymptotic behavior of a porous medium equation with fractional diffusion, Discrete Contin. Dynam. Systems, Volume 29 (2011), pp. 1393-1404
[5] Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., Volume 133 (2001), pp. 1-82
[6] First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., Volume 101 (1961), pp. 75-90
[7] C. Imbert, A. Mellet, A higher order non-local equation appearing in crack dynamics, preprint , 2010. | arXiv
[8] On the convergence of solutions of fractal Burgers equation toward rarefaction waves, SIAM J. Math. Anal., Volume 39 (2008), pp. 1536-1549
[9] Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer-Verlag, Berlin–Heidelberg–New York, 1972
[10] Some problems on Markov semigroups, Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top., vol. 11, Akademie Verlag, Berlin, 1996, pp. 163-217
[11] Formulas and Theorems for the Special Functions of Mathematical Physics, Die Grundlehren der mathematischen Wissenschaften, vol. 52, Springer-Verlag New York, Inc., New York, 1966
[12] Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970
[13] The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, Oxford Science Publications, Oxford University Press, Oxford, 2007
[14] Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and Its Applications, vol. 33, Oxford University Press, Oxford, 2006
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