The large time behavior of zero-mass solutions to the Cauchy problem for the convection–diffusion equation is studied when q>1 and the initial datum u0 belongs to and satisfies . 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.
Le comportement asymptotique des solutions de masse nulle du problème de Cauchy pour l'équation de convection–diffusion est étudié lorsque q>1 et la donnée initiale u0 appartient à et satisfait . 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.
Accepted:
Published online:
Saı̈d Benachour 1; Grzegorz Karch 2; Philippe Laurençot 3
@article{CRMATH_2004__338_5_369_0, author = {Sa{\i}\ensuremath{\ddot{}}d Benachour and Grzegorz Karch and Philippe Lauren\c{c}ot}, title = {Asymptotic profiles of solutions to convection{\textendash}diffusion equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {369--374}, publisher = {Elsevier}, volume = {338}, number = {5}, year = {2004}, doi = {10.1016/j.crma.2004.01.001}, language = {en}, }
TY - JOUR AU - Saı̈d Benachour AU - Grzegorz Karch AU - Philippe Laurençot TI - Asymptotic profiles of solutions to convection–diffusion equations JO - Comptes Rendus. Mathématique PY - 2004 SP - 369 EP - 374 VL - 338 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2004.01.001 LA - en ID - CRMATH_2004__338_5_369_0 ER -
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/
[1] S. Benachour, G. Karch, Ph. Laurençot, Asymptotic profiles of solutions to viscous Hamilton–Jacobi equations, submitted for publication
[2] S. Benachour, H. Koch, Ph. Laurençot, Very singular solutions to a nonlinear parabolic equation with absorption. II – Uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, in press
[3] Global solutions to viscous Hamilton–Jacobi equations with irregular initial data, Comm. Partial Differential Equations, Volume 24 (1999), pp. 1999-2021
[4] Very singular solutions to a nonlinear parabolic equation with absorption I. Existence, Proc. Roy. Soc. Edinburgh Sect. A, Volume 131 (2001), pp. 27-44
[5] Decay of mass for a semilinear parabolic equation, Comm. Partial Differential Equations, Volume 24 (1999), pp. 869-881
[6] The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces, J. Math. Pures Appl., Volume 81 (2002), pp. 343-378
[7] Asymptotic behavior and source-type solutions for a diffusion–convection equation, Arch. Rational Mech. Anal., Volume 124 (1993), pp. 43-65
[8] Large time behavior for convection–diffusion equations in , J. Funct. Anal., Volume 100 (1991), pp. 119-161
[9] The L1-stability of constant states of degenerate convection–diffusion equations, Asymptotic Anal., Volume 19 (1999), pp. 267-288
[10] The Cauchy problem for ut=Δu+|∇u|q, J. Math. Anal. Appl., Volume 284 (2003), pp. 733-755
[11] On zero mass solutions of viscous conservation laws, Comm. Partial Differential Equations, Volume 27 (2002), pp. 2071-2100
[12] Y.J. Kim, An Oleinik type estimate for a convection–diffusion equation and convergence to N-waves, J. Differential Equations, in press
[13] On the growth of mass for a viscous Hamilton–Jacobi equation, J. Anal. Math., Volume 89 (2003), pp. 367-383
[14] Shock Waves and Reaction–Diffusion Equations, Grundlehren Math. Wiss., vol. 258, Springer-Verlag, New York, 1983
Cited by Sources:
Comments - Policy