Régularité conditionnelle des équations de Navier–Stokes

Abstract

In this Note, we give sufficient conditions for the regularity of Leray–Hopf weak solutions to the Navier–Stokes equation. We prove that, if one of three conditions (i) ∂ u / ∂ x 3 ∈ L t s 0 L x r 0 where 2 / s 0 + 3 / r 0 ⩽ 2 and 9 / 4 ⩽ r 0 ⩽ 3 , (ii) ∇ u 3 ∈ L t s 1 L x r 1 where 2 / s 1 + 3 / r 1 ⩽ 11 / 6 and 54 / 23 ⩽ r 0 ⩽ 18 / 5 , or (iii) u 3 ∈ L t s 0 L x r 0 where 2 / s 0 + 3 / r 0 ⩽ 5 / 8 and 24 / 5 ⩽ r 0 ⩽ ∞ , is satisfied, then the solution is regular. These conditions improve earlier results on the conditional regularity of the Navier–Stokes equations. To cite this article: I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 343 (2006).