Homogenization and correctors for some hyperbolic problems
Identifieur interne : 000852 ( France/Analysis ); précédent : 000851; suivant : 000853Homogenization and correctors for some hyperbolic problems
Auteurs : Florian Gaveau [France]Source :
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Abstract
In this thesis, we are concerned with some convergence and corrector results for hyperbolics problems in heterogeneous media with mixed boundary conditions. This kind of problems models waves spread in heterogeneous media. In the first Chapter, we recall some results of the asymptotic study of problems posed in a heterogeneous media. In the second Chapter, we consider the wave equation in a perforated domain not necessarily periodic. We suppose a hypothesis of H^0-convergence of the elliptic part of the operator. This notion has been introduced by M. Briane, A Damlamian et P. Donato and it generalizes the notion of H-convergence introduced some years before by F. Murat and L.Tartar to perforated domains. We prove two main results, the first one is a homogenization result and the second one is a corrector result which improves the convergence of the solution of the problem under some more restrictive assumptions on the data. To do so, we use the corrector introduced by G. Cardone, P. Donato et A. Gaudiello and we show some of his properties. In the third Chapter, we consider a nonlinear wave equation posed in a periodic perforated domain where the nonlinearity concerns the time derivative of the solution. We suppose that the nonlinearity is bounded above by a monotonous polynomial function which exponent permits to have a suitable Sobolev injection. We study first the existence and the uniqueness for the solution with a Galerkin method, then we check a homogenization result for this problem. In the fourth Chapter, we study the wave equation in a non perforated domain. In a first time, we find again the classical result of homogenization using the method of periodical unfolding introduced by D. Cioranescu, A. Damlamian et G. Griso. Then, under more stronger assumptions on the data, we show a corrector result involving the averaging operator which is the adjoint of the unfolding operator.
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This kind of problems models waves spread in heterogeneous media. In the first Chapter, we recall some results of the asymptotic study of problems posed in a heterogeneous media. In the second Chapter, we consider the wave equation in a perforated domain not necessarily periodic. We suppose a hypothesis of H^0-convergence of the elliptic part of the operator. This notion has been introduced by M. Briane, A Damlamian et P. Donato and it generalizes the notion of H-convergence introduced some years before by F. Murat and L.Tartar to perforated domains. We prove two main results, the first one is a homogenization result and the second one is a corrector result which improves the convergence of the solution of the problem under some more restrictive assumptions on the data. To do so, we use the corrector introduced by G. Cardone, P. Donato et A. Gaudiello and we show some of his properties. In the third Chapter, we consider a nonlinear wave equation posed in a periodic perforated domain where the nonlinearity concerns the time derivative of the solution. We suppose that the nonlinearity is bounded above by a monotonous polynomial function which exponent permits to have a suitable Sobolev injection. We study first the existence and the uniqueness for the solution with a Galerkin method, then we check a homogenization result for this problem. In the fourth Chapter, we study the wave equation in a non perforated domain. In a first time, we find again the classical result of homogenization using the method of periodical unfolding introduced by D. Cioranescu, A. Damlamian et G. Griso. Then, under more stronger assumptions on the data, we show a corrector result involving the averaging operator which is the adjoint of the unfolding operator.</div></front></TEI><affiliations><list><country><li>France</li></country><region><li>Haute-Normandie</li><li>Région Normandie</li></region><settlement><li>Le Havre</li></settlement><orgName><li>Université du Havre</li></orgName></list><tree><country name="France"><region name="Région Normandie"><name sortKey="Gaveau, Florian" sort="Gaveau, Florian" uniqKey="Gaveau F" first="Florian" last="Gaveau">Florian Gaveau</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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