no. 11-12
Partial Differential Equations
On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity
[Sur la régularité des solutions des équations de Navier–Stokes 3D : une remarque sur le rôle de l'élicité]

Présenté par : Haïm Brezis

Berselli, Luigi C.1 ; Córdoba, Diego2
1 Dipartimento di Matematica Applicata “U. Dini,” Università di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy
2 Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain
Comptes Rendus. Mathématique, Tome 347 (2009) no. 11-12, pp. 613-618.

Nous démontrons que si la vélocité et le rotationnel sont perpendiculaires partout (avec une borne sur la vitesse avec laquelle ils deviennent perpendiculaires) alors les solutions des équations de Navier–Stokes 3D sont régulières. Cette condition implique que l'élicité est nulle et, dans un certain sens, que le flux ressemble à la situation géométrique bidimensionnelle.

We show that if velocity and vorticity are orthogonal at each point (and they become orthogonal fast enough) then solutions of the 3D Navier–Stokes equations are smooth. This condition implies that the helicity is identically zero and, in a certain sense, the flow resembles the 2D geometric situation.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.03.003
Berselli, Luigi C. 1 ; Córdoba, Diego 2

1 Dipartimento di Matematica Applicata “U. Dini,” Università di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy
2 Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain 
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Berselli, Luigi C.; Córdoba, Diego. On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity. Comptes Rendus. Mathématique, Tome 347 (2009) no. 11-12, pp. 613-618. doi : 10.1016/j.crma.2009.03.003. http://www.numdam.org/articles/10.1016/j.crma.2009.03.003/

[1] Beirão da Veiga, H.; Berselli, L.C. On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations, Volume 15 (2002) no. 3, pp. 345-356

[2] Beirão da Veiga, H.; Berselli, L.C. Navier–Stokes equations: Green matrices, vorticity direction, and regularity up to the boundary, J. Differential Equations, Volume 246 (2009) no. 2, pp. 597-628

[3] Berker, R. Intégration des équations du mouvement d'un fluide visqueux incompressible, Handbuch der Physik, Bd. VIII/2, Springer, Berlin, 1963, pp. 1-384

[4] L.C. Berselli, D. Córdoba, Geometric conditions related to the helicity and the problem of regularity for the Navier–Stokes equations, in preparation

[5] Constantin, P. Geometric statistics in turbulence, SIAM Rev., Volume 36 (1994) no. 1, pp. 73-98

[6] Constantin, P.; Fefferman, C. Direction of vorticity and the problem of global regularity for the Navier–Stokes equations, Indiana Univ. Math. J., Volume 42 (1993) no. 3, pp. 775-789

[7] Constantin, P.; Foias, C. Navier–Stokes Equations, University of Chicago Press, Chicago, IL, 1988

[8] Foias, C.; Hoang, L.; Nicolaenko, B. On the helicity in 3D-periodic Navier–Stokes equations. I. The non-statistical case, Proc. London Math. Soc., Volume 94 (2007) no. 1, pp. 53-90

[9] Galdi, G.P. An Introduction to the Navier–Stokes Initial-Boundary Value Problem, Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000 (pp. 1–70)

[10] Ladyžhenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, vol. 2, Gordon and Breach Sci. Publ., New York, 1969

[11] Ladyžhenskaya, O.A. Solution “in the large” to the boundary-value problem for the Navier–Stokes equations in two space variables, Soviet Phys. Dokl., Volume 123 (1958) no. 3, pp. 1128-1131 (427–429 Dokl. Akad. Nauk SSSR)

[12] Ladyžhenskaya, O.A. Unique global solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) (1968), pp. 155-177

[13] Leray, J. Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1934) no. 1, pp. 193-248

[14] Mahalov, A.; Titi, E.S.; Leibovich, S. Invariant helical subspaces for the Navier–Stokes equations, Arch. Rational Mech. Anal., Volume 112 (1990) no. 3, pp. 193-222

[15] Moffatt, H.K. The degree of knottedness of tangled vortex lines, J. Fluid Mech., Volume 35 (1969), pp. 117-129

[16] Ross Ethier, C.; Steinman, D.A. Exact fully 3D Navier–Stokes solutions for benchmarking, Int. J. Numer. Methods Fluids, Volume 19 (1994), pp. 369-375

[17] Taylor, G.I. On decay of vortices in a viscous fluid, Philos. Mag., Volume 46 (1923) no. 6, pp. 671-674

[18] Ukhovskii, M.R.; Iudovich, V.I. Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., Volume 32 (1968), pp. 52-61

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