Partial Differential Equations/Mathematical Problems in Mechanics
On the regularity up to the boundary in the theory of the Navier–Stokes equations with generalized impermeability conditions
Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 31-36.

In this Note, we extend sufficient conditions for regularity we described in our previous works so that they are valid not only in the interior, but up to the boundary of a flow field. The conditions are based on the integrability properties of either one of the eigenvalues of the rate of deformation tensor or one component of velocity.

Dans cette Note, nous étendons jusqu'à la frontière du domaine spatial, les conditions suffisantes pour la régularité des solutions faibles des équations de Navier–Stokes que nous avions obtenues dans nos travaux précédents. Ces conditions sont basées sur les propriétés d'intégrabilité ou bien d'une des valeurs propres du tenseur de déformation, ou bien d'une des composantes du champ de vitesse.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.10.016
Neustupa, Jiří 1; Penel, Patrick 2

1 Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
2 Université du Sud, Toulon-Var, département de mathématique, BP 20132, 83957 La Garde, France
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Neustupa, Jiří; Penel, Patrick. On the regularity up to the boundary in the theory of the Navier–Stokes equations with generalized impermeability conditions. Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 31-36. doi : 10.1016/j.crma.2004.10.016. http://www.numdam.org/articles/10.1016/j.crma.2004.10.016/

[1] Bellout, H.; Neustupa, J.; Penel, P. On the Navier–Stokes equation with boundary conditions based on vorticity, Math. Nachr., Volume 269/270 (2004), pp. 59-72

[2] Foias, C.; Temam, R. Some analytic and geometric properties of the solutions of the evolution Navier–Stokes equations, J. Math. Pures Appl., Volume 58 (1979), pp. 339-368

[3] Galdi, G.P. An Introduction to the Navier–Stokes initial-boundary value problem (Galdi, G.P.; Heywood, J.; Rannacher, R., eds.), Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000, pp. 1-98

[4] Neustupa, J.; Novotný, A.; Penel, P. An interior regularity of a weak solution to the Navier–Stokes equations in dependence on one component of velocity (Galdi, G.P.; Rannacher, R., eds.), Topics in Mathematical Fluid Mechanics, Quaderni di Matematica (Napoli), vol. 10, 2003, pp. 163-183

[5] Neustupa, J.; Penel, P. The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier–Stokes equations, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 10, pp. 805-810

[6] Neustupa, J.; Penel, P. Regularity of a weak solution to the Navier–Stokes equation in dependence on eigenvalues and eigenvectors of the rate of deformation tensor (Rodrigues, J.F.; Seregin, G.; Urbano, J.M., eds.), Progr. Nonlinear Differential Equations Appl., vol. 61, Birkhäuser, Basel, 2004, pp. 197-213

[7] J. Neustupa, P. Penel, Estimates up to the boundary of a weak solution to the Navier–Stokes equation in a cube in dependence on eigenvalues of the rate of deformation tensor, Banach Center Publ., 2004, submitted for publication

[8] J. Neustupa, P. Penel, Two results on regularity up to the boundary of a weak solution to the Navier–Stokes equation in a bounded smooth domain, in preparation

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