On the existence and qualitative theory of stratified solitary water waves

Robin Ming Chen; Samuel Walsh; Miles H. Wheeler

doi:10.1016/j.crma.2016.03.004

Partial differential equations/Mathematical problems in mechanics

On the existence and qualitative theory of stratified solitary water waves
[Sur l'existence et la théorie qualitative des ondes d'eau stratifiées solitaires]

Présenté par : Haïm Brézis

Robin Ming Chen ¹ ; Samuel Walsh ² ; Miles H. Wheeler ³

¹ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
² Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
³ Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 601-605.

Résumé
Abstract

Dans cette note, nous annonçons de nouveaux résultats sur l'existence des ondes de gravité solitaires en deux dimensions se déplaçant à travers un plan d'eau stratifié et situé sous l'air. Le domaine de fluide est limité vers le bas par un fond imperméable, tandis que l'interface entre l'eau et l'air constitue une frontière libre où la pression est constante. Nous montrons que, pour tout choix de profil de vitesse et de distribution de densité en amont, il existe une courbe continue de ces solutions qui comprend les ondes de surface de grande amplitude, qui sont arbitrairement près d'avoir un point de stagnation horizontale. En outre, nous fournissons plusieurs résultats concernant les caractéristiques qualitatives des ondes solitaires stratifiées, notamment des estimations sur le nombre de Froude, la vitesse et la pression, dont certaines sont nouvelles, même pour le cas où la densité constante, une preuve de la non-existence des mascarets monotones dans ce régime, ainsi qu'un théorème énonçant la parité des ondes stratifiées supercritiques d'élévation.

In this note, we announce new results on the existence of two-dimensional solitary waves moving through a body of density stratified water lying beneath air. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the air and water is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity and density distribution, there exists a continuous curve of such solutions that includes large-amplitude waves that come arbitrarily close to having a (horizontal) stagnation point. Additionally, we provide several results characterizing the qualitative features of solitary stratified waves. In part, these include: estimates on the Froude number, velocity, and pressure, some of which are new, even for the constant density case; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical stratified solitary waves of elevation have an axis of even symmetry.

Reçu le : 2015-12-06
Accepté le : 2016-03-11
Publié le : 2016-04-14

DOI : 10.1016/j.crma.2016.03.004

Affiliations des auteurs :

Robin Ming Chen ¹ ; Samuel Walsh ² ; Miles H. Wheeler ³

¹ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
² Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
³ Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

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Robin Ming Chen; Samuel Walsh; Miles H. Wheeler. On the existence and qualitative theory of stratified solitary water waves. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 601-605. doi : 10.1016/j.crma.2016.03.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.004/

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