Vorticity internal transition layers for the Navier-Stokes equations
Journées équations aux dérivées partielles (2008), article no. 8, 15 p.

We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show -thanks to an asymptotic expansion- that there is a sharp but smooth variation of the fluid vorticity into a internal layer moving with the flow of the Euler equations; as long as this later exists and as t<<1/ν, where ν is the viscosity coefficient.

DOI : 10.5802/jedp.52
Sueur, Franck 1

1 Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie - Paris 6; 175 Rue du Chevaleret 75013 Paris, FRANCE
@article{JEDP_2008____A8_0,
     author = {Sueur, Franck},
     title = {Vorticity internal transition layers for the {Navier-Stokes} equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {8},
     pages = {1--15},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2008},
     doi = {10.5802/jedp.52},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.52/}
}
TY  - JOUR
AU  - Sueur, Franck
TI  - Vorticity internal transition layers for the Navier-Stokes equations
JO  - Journées équations aux dérivées partielles
PY  - 2008
SP  - 1
EP  - 15
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.52/
DO  - 10.5802/jedp.52
LA  - en
ID  - JEDP_2008____A8_0
ER  - 
%0 Journal Article
%A Sueur, Franck
%T Vorticity internal transition layers for the Navier-Stokes equations
%J Journées équations aux dérivées partielles
%D 2008
%P 1-15
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.52/
%R 10.5802/jedp.52
%G en
%F JEDP_2008____A8_0
Sueur, Franck. Vorticity internal transition layers for the Navier-Stokes equations. Journées équations aux dérivées partielles (2008), article  no. 8, 15 p. doi : 10.5802/jedp.52. http://www.numdam.org/articles/10.5802/jedp.52/

[1] P. Brenner. The Cauchy problem for symmetric hyperbolic systems in L p . Math. Scand., 19:27–37, 1966. | MR | Zbl

[2] J.-Y. Chemin. Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel. Invent. Math., 103(3):599–629, 1991. | MR | Zbl

[3] J.-Y. Chemin. Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. (4), 26(4):517–542, 1993. | Numdam | MR | Zbl

[4] J.-Y. Chemin. Two-dimensional Euler system and the vortex patches problem. In Handbook of mathematical fluid dynamics. Vol. III, pages 83–160. North-Holland, Amsterdam, 2004. | MR

[5] N. Depauw. Poche de tourbillon pour Euler 2D dans un ouvert à bord. J. Math. Pures Appl. (9), 78(3):313–351, 1999. | MR | Zbl

[6] A. Dutrifoy. On 3-D vortex patches in bounded domains. Comm. Partial Differential Equations, 28(7-8):1237–1263, 2003. | MR | Zbl

[7] K. O. Friedrichs. The identity of weak and strong extensions of differential operators. Trans. Amer. Math. Soc., 55:132–151, 1944. | MR | Zbl

[8] P. Gamblin and X. Saint Raymond. On three-dimensional vortex patches. Bull. Soc. Math. France, 123(3):375–424, 1995. | Numdam | MR | Zbl

[9] O. Guès, G. Métivier, M. Williams, and K. Zumbrun. Navier-Stokes regularization of multidimensional Euler shocks. Ann. Sci. École Norm. Sup. (4), 39(1):75–175, 2006. | Numdam | MR | Zbl

[10] C. Huang. Remarks on regularity of non-constant vortex patches. Commun. Appl. Anal., 3(4):449–459, 1999. | MR | Zbl

[11] C. Huang. Singular integral system approach to regularity of 3D vortex patches. Indiana Univ. Math. J., 50(1):509–552, 2001. | MR | Zbl

[12] A. Majda. Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math., 39(S, suppl.):S187–S220, 1986. Frontiers of the mathematical sciences: 1985 (New York, 1985). | MR | Zbl

[13] W. Rankine. On the thermodynamic theory of waves of finite longitudinal disturbance. Philos. Trans. Royal Soc. London, 160:277–288, 1870.

[14] J. Rauch. Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Amer. Math. Soc., 291(1):167–187, 1985. | MR | Zbl

[15] P. Serfati. Une preuve directe d’existence globale des vortex patches 2D. C. R. Acad. Sci. Paris Sér. I Math., 318(6):515–518, 1994. | MR | Zbl

[16] F. Sueur. Vorticity internal transition layers for the Navier-Stokes equations. Preprint, available on arXiv.

[17] P. Zhang and Q. J. Qiu. Propagation of higher-order regularities of the boundaries of 3-D vortex patches. Chinese Ann. Math. Ser. A, 18(3):381–390, 1997. | MR | Zbl

Cité par Sources :