An estimate for the vorticity of the Navier-Stokes equation

Let over(u, →) (ṡ, t) be a strong solution of the Navier-Stokes equation on 3-dimensional torus T3, and over(ω, →) (ṡ, t) = ∇ × over(u, →) (ṡ, t) be the vorticity. In this Note we show that{norm of matrix} over(ω, →) (ṡ, t) {norm of matrix}1 + frac(sqrt(2), 4 ν) {norm of matrix} over(u, →) (ṡ, t) {norm of matrix}22 is decreasing in t as long as the solution over(u, →) (ṡ, t) exists, where ν > 0 is the viscosity constant and {norm of matrix} ṡ {norm of matrix}q denotes the Lq-norm. To cite this article: Z. Qian, C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2008 Académie des sciences.