On considère dans cet exposé le modèle asymptotique “shallow-water/shallow-water" obtenu dans [3] à partir du système d’Euler à deux couches avec fond plat et toit rigide pour décrire la propagation d’ondes internes de grande amplitude. En dimension d’espace un, ce système est de type hyperbolique et la théorie locale du problème de Cauchy ne pose pas de difficultés majeures, même si d’autres questions (explosion en temps fini, perte d’hyperbolicité) s’avèrent délicates. En dimension deux d’espace par contre, le système est non local. On montre qu’il conserve cependant des propriétés “de type hyperbolique" et que le problème de Cauchy associé est localement bien posé sous des conditions convenables sur les conditions initiales.
We consider in this talk the “shallow-water/shallow-water" asymptotic model obtained in [3] from the two-layer system with rigid lid, for the description of large amplitude internal waves. For one-dimensional interfaces, this system is of hyperbolic type and its local wellposedness does not raise serious difficulties, though other issues (blow -up, loss of hyperbolicity,...) turn out to be delicate. For two- dimensional interfaces, the system turns out to be nonlocal. We prove that it conserves some properties of “hyperbolic type" and prove that the associated Cauchy problem is locally well posed in suitable Sobolev classes provided some natural restrictions are imposed on the data.
@article{SEDP_2009-2010____A19_0,
author = {Saut, Jean-Claude},
title = {Un mod\`ele asymptotique pour les ondes internes de grande amplitude},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:19},
pages = {1--14},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2009-2010},
language = {fr},
url = {http://archive.numdam.org/item/SEDP_2009-2010____A19_0/}
}
TY - JOUR AU - Saut, Jean-Claude TI - Un modèle asymptotique pour les ondes internes de grande amplitude JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:19 PY - 2009-2010 SP - 1 EP - 14 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_2009-2010____A19_0/ LA - fr ID - SEDP_2009-2010____A19_0 ER -
%0 Journal Article %A Saut, Jean-Claude %T Un modèle asymptotique pour les ondes internes de grande amplitude %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:19 %D 2009-2010 %P 1-14 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_2009-2010____A19_0/ %G fr %F SEDP_2009-2010____A19_0
Saut, Jean-Claude. Un modèle asymptotique pour les ondes internes de grande amplitude. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 19, 14 p. http://archive.numdam.org/item/SEDP_2009-2010____A19_0/
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