Catalogue et commandes en ligne (paiement sécurisé, VISA ou MASTERCARD uniquement)

Revues disponibles par abonnement

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Séries de livres

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Série Chaire Jean Morlet

SMF/AMS Texts and Monographs

La Série T

Fascicules « Journée Annuelle »

Autres livres

Donald E. Knuth - traductions françaises

Rééditions du Séminaire Nicolas Bourbaki

Rééditions des Œuvres de Jean Leray

Revue de l'Institut Elie Cartan

Editions électroniques

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

Plus d'information / Abonnement

Publications grand public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

Pour les auteurs

Soumission des manuscrits

Formats et documentation

Plus d'info

Liste de diffusion électronique (smf.emath.fr)

Information pour les libraires et diffuseurs (smf.emath.fr)

Publications de la SMF
fr en
Votre numéro IP : 54.224.235.183
Accès aux édit. élec. : SémCong

Bulletin de la SMF

Présentation de la publication

Parutions

Dernières parutions

Comité de rédaction / Secrétariat

Volume :

Faire une recherche


Catalogue & commande

Bulletin de la SMF - Parutions - 145 (2017) 643-710

Parutions < 145

A Paradifferential Reduction for the Gravity-capillary Waves System at Low Regularity and Applications
Thibault de Poyferré, Quang-Huy Nguyen
Bulletin de la SMF 145, fascicule 4 (2017), 643-710

Télécharger cet article : Fichier PDF

Résumé :
Une réduction paradifférentielle du système des vagues de gravité-capillarité à basse régularité et applications
Dans cet article, nous étudions le système des vagues de gravité-capillarité en toutes dimensions, dans la formulation de Zakharov, Craig et Sulem. À l'aide d'une approche paradifférentielle introduite par Alazard, Burq et Zuily, nous symétrisons ce système en une équation dispersive quasilinéaire dont le terme principal est d'ordre 32. La principale nouveauté par rapport aux études précédentes est que cette réduction est effectuée au niveau de régularité des EDPs quasilinéaires : H^s(^d) avec s>32 +d2, d étant la dimension de la surface libre. À partir de cette réduction, nous déduisons un critère d'explosion n'impliquant que la norme Lipschitz de la trace de la vitesse et la norme C^52+ de la surface libre. En outre, nous obtenons une estimation a priori de la norme H^s et la contraction de l'application solution dans la norme H^s-32 , en utilisant le contrôle d'une norme de Strichartz. Ces résultats ont été utilisés pour développer une théorie de Cauchy locale pour des vitesses initiales non Lipschitz, dans le papier compagnon [22].

Mots-clefs : Vagues de gravité-capillarité, réduction paradifférentielle, critère d'explosion, estimations a priori, contraction de l'application solution.

Abstract:
We consider in this article the system of gravity-capillary waves in all dimensions and under the Zakharov/Craig-Sulem formulation. Using a paradifferential approach introduced by Alazard-Burq-Zuily, we symmetrize this system into a quasilinear dispersive equation whose principal part is of order 32. The main novelty, compared to earlier studies, is that this reduction is performed at the Sobolev regularity of quasilinear pdes: H^s(^d) with s>32 +d2, d being the dimension of the free surface. From this reduction, we deduce a blow-up criterion involving solely the Lipschitz norm of the velocity trace and the C^52+-norm of the free surface. Moreover, we obtain an a priori estimate in the H^s-norm and the contraction of the solution map in the H^s-32 -norm using the control of a Strichartz norm. These results have been applied in establishing a local well-posedness theory for non-Lipschitz initial velocity in our companion paper [22].

Keywords: Gravity-capillary waves, paradifferential reduction, blow-up criterion, a priori estimate, contraction of the solution map.

Class. math. : 35Q35, 35A01, 35B45, 35B65, 76B15.


ISSN : 0037-9484
DOI : 10.24033/bsmf.2750
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Alazard, Thomas and Burq, Nicolas and Zuily, Claude
On the water-wave equations with surface tension
Duke Math. J. 158 (2011) 413–499
Math Reviews MR2805065
2
Alazard, Thomas and Burq, Nicolas and Zuily, Claude
Strichartz estimates for water waves
Ann. Sci. Éc. Norm. Supér. 44 (2011) 855–903
Math Reviews MR2931520
3
Alazard, Thomas and Burq, Nicolas and Zuily, Claude
On the Cauchy problem for gravity water waves
Invent. math. 198 (2014) 71–163
Math Reviews MR3260858
4
5
Alazard, Thomas and Métivier, Guy
Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves
Comm. Partial Differential Equations 34 (2009) 1632–1704
Math Reviews MR2581986
6
Alinhac, S.
Paracomposition et opérateurs paradifférentiels
Comm. Partial Differential Equations 11 (1986) 87–121
Math Reviews MR814548
7
Alinhac, S.
Paracomposition et application aux équations non-linéaires
in Bony-Sjöstrand-Meyer seminar, 1984–1985
(1985) Exp.No.11, 11
Math Reviews MR819777
8
Bahouri, Hajer and Chemin, Jean-Yves
Équations d'ondes quasilinéaires et estimations de Strichartz
Amer. J. Math. 121 (1999) 1337–1377
Math Reviews MR1719798
9
Bahouri, Hajer and Chemin, Jean-Yves and Danchin, Raphaël
Fourier analysis and nonlinear partial differential equations
Springer, Heidelberg, 2011
Math Reviews MR2768550
10
Beale, J. T. and Kato, T. and Majda, A.
Remarks on the breakdown of smooth solutions for the 3-D Euler equations
Comm. Math. Phys. 94 (1984) 61–66
Math Reviews MR763762
11
Blair, Matthew
Strichartz estimates for wave equations with coefficients of Sobolev regularity
Comm. Partial Differential Equations 31 (2006) 649–688
Math Reviews MR2233036
12
Bony, Jean-Michel
Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires
Ann. Sci. École Norm. Sup. 14 (1981) 209–246
Math Reviews MR631751
13
Burq, Nicolas and Gérard, P. and Tzvetkov, N.
Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds
Amer. J. Math. 126 (2004) 569–605
Math Reviews MR2058384
14
Christianson, Hans and Hur, Vera Mikyoung and Staffilani, Gigliola
Strichartz estimates for the water-wave problem with surface tension
Comm. Partial Differential Equations 35 (2010) 2195–2252
Math Reviews MR2763354
15
Christodoulou, Demetrios and Lindblad, Hans
On the motion of the free surface of a liquid
Comm. Pure Appl. Math. 53 (2000) 1536–1602
Math Reviews MR1780703
16
Córdoba, Antonio and Córdoba, Diego and Gancedo, Francisco
Interface evolution: water waves in 2-D
Adv. Math. 223 (2010) 120–173
Math Reviews MR2563213
17
Coutand, Daniel and Shkoller, Steve
Well-posedness of the free-surface incompressible Euler equations with or without surface tension
J. Amer. Math. Soc. 20 (2007) 829–930
Math Reviews MR2291920
18
Craig, Walter
An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits
Comm. Partial Differential Equations 10 (1985) 787–1003
Math Reviews MR795808
19
Craig, Walter and Sulem, C.
Numerical simulation of gravity waves
J. Comput. Phys. 108 (1993) 73–83
Math Reviews MR1239970
20
Kre, V. and Ve, K. E.
Mathematical aspects of surface waves on water
Uspekhi Mat. Nauk 62 (2007) 95–116
Math Reviews MR2355420
21
22
de Poyferré, Thibault and Nguyen, Quang-Huy
Strichartz estimates and local existence for the gravity-capillary waves with non-Lipschitz initial velocity
J. Differential Equations 261 (2016) 396–438
Math Reviews MR3487264
23
Ferrari, Andrew B.
On the blow-up of solutions of the 3-D Euler equations in a bounded domain
Comm. Math. Phys. 155 (1993) 277–294
Math Reviews MR1230029
24
Germain, Pierre and Masmoudi, Nader and Shatah, Jalal
Global solutions for the gravity water waves equation in dimension 3
Ann. of Math. 175 (2012) 691–754
Math Reviews MR2993751
25
Germain, Pierre and Masmoudi, Nader and Shatah, Jalal
Global existence for capillary water waves
Comm. Pure Appl. Math. 68 (2015) 625–687
Math Reviews MR3318019
26
Hörmander, Lars
Lectures on nonlinear hyperbolic differential equations
Springer, Berlin, 1997
Math Reviews MR1466700
27
28
Ifrim, Mihaela and Tataru, Daniel
Two dimensional water waves in holomorphic coordinates II: Global solutions
Bull. Soc. Math. France 144 (2016) 369–394
Math Reviews MR3499085
29
Ifrim, Mihaela and Tataru, Daniel
The lifespan of small data solutions in two dimensional capillary water waves
Arch. Ration. Mech. Anal. 225 (2017) 1279–1346
Math Reviews MR3667289
30
Ionescu, Alexandru D. and Pusateri, Fabio
Global solutions for the gravity water waves system in 2d
Invent. math. 199 (2015) 653–804
Math Reviews MR3314514
31
32
Lannes, David
Well-posedness of the water-waves equations
J. Amer. Math. Soc. 18 (2005) 605–654
Math Reviews MR2138139
33
Lannes, David
The water waves problem
Amer. Math. Soc., Providence, RI, 2013
Math Reviews MR3060183
34
Lindblad, Hans
Well-posedness for the motion of an incompressible liquid with free surface boundary
Ann. of Math. 162 (2005) 109–194
Math Reviews MR2178961
35
Lions, J.-L. and Magenes, E.
Problèmes aux limites non homogènes et applications. Vol. 1
Dunod, Paris, 1968
Math Reviews MR0247243
36
Métivier, Guy
Para-differential calculus and applications to the Cauchy problem for nonlinear systems
Edizioni della Normale, Pisa, 2008
Math Reviews MR2418072
37
Ming, Mei and Zhang, Zhifei
Well-posedness of the water-wave problem with surface tension
J. Math. Pures Appl. 92 (2009) 429–455
Math Reviews MR2558419
38
Nalimov, V. I.
The Cauchy-Poisson problem
Dinamika Splošn. Sredy 18 (1974) 104–210, 254
Math Reviews MR0609882
39
Shatah, Jalal and Zeng, Chongchun
Geometry and a priori estimates for free boundary problems of the Euler equation
Comm. Pure Appl. Math. 61 (2008) 698–744
Math Reviews MR2388661
40
Shatah, Jalal and Zeng, Chongchun
A priori estimates for fluid interface problems
Comm. Pure Appl. Math. 61 (2008) 848–876
Math Reviews MR2400608
41
Shatah, Jalal and Zeng, Chongchun
Local well-posedness for fluid interface problems
Arch. Ration. Mech. Anal. 199 (2011) 653–705
Math Reviews MR2763036
42
Smith, Hart F.
A parametrix construction for wave equations with C^1,1 coefficients
Ann. Inst. Fourier (Grenoble) 48 (1998) 797–835
Math Reviews MR1644105
43
Staffilani, Gigliola and Tataru, Daniel
Strichartz estimates for a Schrödinger operator with nonsmooth coefficients
Comm. Partial Differential Equations 27 (2002) 1337–1372
Math Reviews MR1924470
44
Tataru, Daniel
Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation
Amer. J. Math. 122 (2000) 349–376
Math Reviews MR1749052
45
Tataru, Daniel
Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II
Amer. J. Math. 123 (2001) 385–423
Math Reviews MR1833146
46
Taylor, Michael E.
Pseudodifferential operators and nonlinear PDE
Birkhäuser, 1991
Math Reviews MR1121019
47
Wang, Chao and Zhang, ZhiFei
Break-down criterion for the water-wave equation
Sci. China Math. 60 (2017) 21–58
Math Reviews MR3585049
48
49
Wu, Sijue
Well-posedness in Sobolev spaces of the full water wave problem in 2-D
Invent. math. 130 (1997) 39–72
Math Reviews MR1471885
50
Wu, Sijue
Well-posedness in Sobolev spaces of the full water wave problem in 3-D
J. Amer. Math. Soc. 12 (1999) 445–495
Math Reviews MR1641609
51
Wu, Sijue
Almost global wellposedness of the 2-D full water wave problem
Invent. math. 177 (2009) 45–135
Math Reviews MR2507638
52
Wu, Sijue
Global wellposedness of the 3-D full water wave problem
Invent. math. 184 (2011) 125–220
Math Reviews MR2782254
53
Kinsey, Rafe Hand
A Priori Estimates for Two-Dimensional Water Waves with Angled Crests
Ph.D. Thesis, University of Michigan (2014)
54
55
Yosihara, Hideaki
Gravity waves on the free surface of an incompressible perfect fluid of finite depth
Publ. Res. Inst. Math. Sci. 18 (1982) 49–96
Math Reviews MR660822
56
Yosihara, Hideaki
Capillary-gravity waves for an incompressible ideal fluid
J. Math. Kyoto Univ. 23 (1983) 649–694
Math Reviews MR728155
57
Zakhariv, Vladimir E.
Stability of periodic waves of finite amplitude on the surface of a deep fluid
Journal of Applied Mechanics and Technical Physics 9 (1968) 190–194
58
Zakharov, V. E.
Weakly nonlinear waves on the surface of an ideal finite depth fluid
in Nonlinear waves and weak turbulence
Amer. Math. Soc. Transl. Ser. 2 182 (1998) 167–197
Math Reviews MR1618515