[Sur l'existence et la théorie qualitative des ondes d'eau stratifiées solitaires]
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.
Accepté le :
Publié le :
@article{CRMATH_2016__354_6_601_0,
author = {Robin Ming Chen and Samuel Walsh and Miles H. Wheeler},
title = {On the existence and qualitative theory of stratified solitary water waves},
journal = {Comptes Rendus. Math\'ematique},
pages = {601--605},
publisher = {Elsevier},
volume = {354},
number = {6},
year = {2016},
doi = {10.1016/j.crma.2016.03.004},
language = {en},
}
TY - JOUR AU - Robin Ming Chen AU - Samuel Walsh AU - Miles H. Wheeler TI - On the existence and qualitative theory of stratified solitary water waves JO - Comptes Rendus. Mathématique PY - 2016 SP - 601 EP - 605 VL - 354 IS - 6 PB - Elsevier DO - 10.1016/j.crma.2016.03.004 LA - en ID - CRMATH_2016__354_6_601_0 ER -
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/
[1] Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 11 (1984), pp. 441-499
[2] On the Stokes conjecture for the wave of extreme form, Acta Math., Volume 148 (1982), pp. 193-214
[3] A global theory of internal solitary waves in two-fluid systems, Trans. Amer. Math. Soc., Volume 298 (1986), pp. 431-484
[4] Analytic Theory of Global Bifurcation: An Introduction, Princeton University Press, Princeton, NJ, USA, 2003
[5] Symmetry of solitary waves, Commun. Partial Differ. Equ., Volume 13 (1988), pp. 603-633
[6] Bifurcation theory for analytic operators, Proc. Lond. Math. Soc., Volume 26 (1973), pp. 359-384
[7] Symmetry and related properties via the maximum principle, Commun. Math. Phys., Volume 68 (1979), pp. 209-243
[8] Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity, Phys. D, Volume 237 (2008), pp. 1530-1538
[9] Bounds for surface solitary waves, Proc. Camb. Philos. Soc., Volume 76 (1974), pp. 345-358
[10] Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. Partial Differ. Equ., Volume 16 (1991), pp. 585-615
[11] Symmetry of internal waves, Nonlinear Anal., Theory Methods Appl., Volume 28 (1997), pp. 87-102
[12] Momentum and energy integrals for gravity waves of finite height, J. Mar. Res., Volume 6 (1947), pp. 175-193
[13] On the theory of oscillatory waves, Mathematical and Physical Papers, vol. 1, 1880, pp. 197-229 (314–326)
[14] Smooth bores in a two-layer fluid with a free surface, Prikl. Mekh. Tekhn. Fiz., Volume 38 (1997), pp. 87-92
[15] Broadening of interfacial solitary waves, Phys. Fluids, Volume 31 (1988), pp. 2486-2490
[16] On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differ. Equ., Volume 246 (2009), pp. 4043-4076
[17] Some criteria for the symmetry of stratified water waves, Wave Motion, Volume 46 (2009), pp. 350-362
[18] Stratified and steady periodic water waves, SIAM J. Math. Anal., Volume 41 (2009), pp. 1054-1105
[19] Large-amplitude solitary water waves with vorticity, SIAM J. Math. Anal., Volume 45 (2013), pp. 2937-2994
[20] The Froude number for solitary water waves with vorticity, J. Fluid Mech., Volume 768 (2015), pp. 91-112
[21] Solitary water waves of large amplitude generated by surface pressure, Arch. Ration. Mech. Anal., Volume 218 (2015), pp. 1131-1187
Cité par Sources :
Commentaires - Politique
Large-amplitude internal fronts in two-fluid systems
Robin Ming Chen; Samuel Walsh; Miles H. Wheeler
C. R. Math (2020)
Didier Bresch; Pascal Noble; Jean-Paul Vila
C. R. Math (2016)