Long time behavior for a dissipative shallow water model

Sciacca, V.; Schonbek, M.E.; Sammartino, M.

doi:10.1016/j.anihpc.2016.05.003

Sciacca, V. ; Schonbek, M.E. ; Sammartino, M.

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 731-757.

Résumé
Abstract

Nous considérons le modèle d'eau peu profonde à deux dimensions dérivé dans [29], décrivant le mouvement d'un fluide incompressible, confinèe dans un bassin peu profond, avec topographie du fond variable. Nous construisons des variétés inertielles approximatives pour le système dynamique associé et nous estimons son ordre. Finalement, pour le espace $R^{2}$ avec des conditions appropriées pour la force, nous prouvons la $L^{2}$ décroissance asymptotique des solutions faibles.

We consider the two-dimensional shallow water model derived in [29], describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, working in the whole space $R^{2}$ , under suitable conditions on the time dependent forcing term, we prove the $L^{2}$ asymptotic decay of the weak solutions.

MR Zbl

DOI : 10.1016/j.anihpc.2016.05.003

Mots clés : Inertial manifolds, Attractors, Time decay, Fourier splitting method, Incompressible viscous fluids, Navier–Stokes equations

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     title = {Long time behavior for a dissipative shallow water model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {731--757},
     publisher = {Elsevier},
     volume = {34},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.05.003/}
}

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Sciacca, V.; Schonbek, M.E.; Sammartino, M. Long time behavior for a dissipative shallow water model. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 731-757. doi : 10.1016/j.anihpc.2016.05.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.05.003/

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