Asymptotic profiles of solutions to convection–diffusion equations

Saı̈d Benachour; Grzegorz Karch; Philippe Laurençot

doi:10.1016/j.crma.2004.01.001

Partial Differential Equations

Asymptotic profiles of solutions to convection–diffusion equations
[Comportement asymptotique des solutions d'équations de convection–diffusion]

Présenté par : Pierre-Louis Lions

Saı̈d Benachour ¹ ; Grzegorz Karch ² ; Philippe Laurençot ³

¹ Institut Elie Cartan-Nancy, Université Henri Poincaré, BP 239, 54506 Vandœuvre lès Nancy cedex, France
² Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, and Institute of Mathematics, Polish Academy of Sciences, Warsaw (2002-2003), Poland
³ Mathématiques pour l'industrie et la physique, CNRS UMR 5640, Université Paul Sabatier-Toulouse 3, 118, route de Narbonne, 31062 Toulouse cedex 4, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 369-374.

Résumé
Abstract

Le comportement asymptotique des solutions de masse nulle du problème de Cauchy pour l'équation de convection–diffusion $u_{t} - u_{xx} + {(| u |}^{q})_{x} = 0, u (x, 0) = u_{0} (x)$ est étudié lorsque q>1 et la donnée initiale u₀ appartient à $L^{1} (ℝ, (1 + | x |) d x)$ et satisfait $\int_{ℝ} u_{0} (x) d x = 0$ . Nous donnons des conditions sur l'amplitude et la forme de la donnée initiale u₀ et sur l'exposant q>1 sous lesquelles le comportement asymptotique des solutions est décrit par la dérivée première du noyau de Gauss–Weierstrass, ou par une solution auto-similaire de l'équation, ou par une N-onde hyperbolique.

The large time behavior of zero-mass solutions to the Cauchy problem for the convection–diffusion equation $u_{t} - u_{xx} + {(| u |}^{q})_{x} = 0, u (x, 0) = u_{0} (x)$ is studied when q>1 and the initial datum u₀ belongs to $L^{1} (ℝ, (1 + | x |) d x)$ and satisfies $\int_{ℝ} u_{0} (x) d x = 0$ . We provide conditions on the size and shape of the initial datum u₀ as well as on the exponent q>1 such that the large time asymptotics of solutions is given either by the derivative of the Gauss–Weierstrass kernel, or by a self-similar solution of the equation, or by hyperbolic N-waves.

Reçu le : 2003-10-23
Accepté le : 2004-01-07
Publié le : 2004-02-06

DOI : 10.1016/j.crma.2004.01.001

Affiliations des auteurs :

Saı̈d Benachour ¹ ; Grzegorz Karch ² ; Philippe Laurençot ³

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     title = {Asymptotic profiles of solutions to convection{\textendash}diffusion equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {369--374},
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     year = {2004},
     doi = {10.1016/j.crma.2004.01.001},
     language = {en},
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Saı̈d Benachour; Grzegorz Karch; Philippe Laurençot. Asymptotic profiles of solutions to convection–diffusion equations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 369-374. doi : 10.1016/j.crma.2004.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.001/

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