[Couches limites visqueuses pour des systèmes hyperboliques-paraboliques avec condition aux limites de Neumann]
Nous initions l'étude des couches limites non caractéristiques de systèmes hyperboliques-paraboliques avec condition aux limites de Neumann. Plus généralement, nous étudions les couches limites avec condition aux limites de type mixte Dirichlet-Neumann, lorsque le nombre de conditions aux limites de Dirichlet est inférieur au nombre de modes caractéristiques rentrant dans le domaine, pour l'opérateur hyperbolique.
Dans le cas des systèmes linéaires à coefficients constants, nous obtenons un système hyperbolique limite avec des conditions aux limites de type Neumann ou Dirichlet-Neumann. Sous de bonnes hypothèses nous construisons des développements en couches limites BKW à tout ordre.
Dans le cas extrême où tous les modes caractéristiques sont rentrants et avec des conditions de Neumann, nous traitons complètement le cas quasilinéaire, prouvant la convergence vers un problème hyperbolique limite avec des conditions de Neumann au bord. Les estimations maximales de stabilité obtenues pour les problèmes linéarisés sont plus faibles que celles typiques correspondant à des conditions de type Dirichlet.
We initiate the study of noncharacteristic boundary layers in hyperbolic-parabolic problems with Neumann boundary conditions. More generally, we study boundary layers with mixed Dirichlet-Neumann boundary conditions where the number of Dirichlet conditions is fewer than the number of hyperbolic characteristic modes entering the domain, that is, the number of boundary conditions needed to specify an outer hyperbolic solution. We have shown previously that this situation prevents the usual WKB approximation involving an outer solution with pure Dirichlet conditions. It also rules out the usual maximal estimates for the linearization of the hyperbolic-parabolic problem about the boundary layer.
Here we show that for linear, constant-coefficient, hyperbolic-parabolic problems one obtains a reduced hyperbolic problem satisfying Neumann or mixed Dirichlet-Neumann rather than Dirichlet boundary conditions. When this hyperbolic problem can be solved, a unique formal boundary-layer expansion can be constructed. In the extreme case of pure Neumann conditions and totally incoming characteristics, we carry out a full analysis of the quasilinear case, obtaining a boundary-layer approximation to all orders with a rigorous error analysis. As a corollary we characterize the small viscosity limit for this problem. The analysis shows that although the associated linearized hyperbolic and hyperbolic-parabolic problems do not satisfy the usual maximal estimates for Dirichlet conditions, they do satisfy analogous versions with losses.
DOI : 10.24033/asens.2213
Keywords: Boundary layers, mixed Dirichlet-Neumann conditions, Evans-Lopatinski condition.
Mot clés : Couches limites, conditions mixtes Dirichlet-Neumann, condition Evans-Lopatinski.
@article{ASENS_2014__47_1_181_0,
author = {Gues, Olivier and M\'etivier, Guy and Williams, Mark and Zumbrun, Kevin},
title = {Viscous boundary layers in hyperbolic-parabolic systems with {Neumann} boundary conditions},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {181--243},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 47},
number = {1},
year = {2014},
doi = {10.24033/asens.2213},
mrnumber = {3205604},
zbl = {1295.35035},
language = {en},
url = {http://www.numdam.org/articles/10.24033/asens.2213/}
}
TY - JOUR AU - Gues, Olivier AU - Métivier, Guy AU - Williams, Mark AU - Zumbrun, Kevin TI - Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions JO - Annales scientifiques de l'École Normale Supérieure PY - 2014 SP - 181 EP - 243 VL - 47 IS - 1 PB - Société Mathématique de France. Tous droits réservés UR - http://www.numdam.org/articles/10.24033/asens.2213/ DO - 10.24033/asens.2213 LA - en ID - ASENS_2014__47_1_181_0 ER -
%0 Journal Article %A Gues, Olivier %A Métivier, Guy %A Williams, Mark %A Zumbrun, Kevin %T Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions %J Annales scientifiques de l'École Normale Supérieure %D 2014 %P 181-243 %V 47 %N 1 %I Société Mathématique de France. Tous droits réservés %U http://www.numdam.org/articles/10.24033/asens.2213/ %R 10.24033/asens.2213 %G en %F ASENS_2014__47_1_181_0
Gues, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin. Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 1, pp. 181-243. doi : 10.24033/asens.2213. http://www.numdam.org/articles/10.24033/asens.2213/
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