# On Periodic 3D Navier-Stokes Equations when the initial velocity is in $L^2$ and the initial vorticity is in $L^1$.

On Periodic 3D Navier-Stokes Equations when the initial velocity is in $L^2$ and the initial vorticity is in $L^1$. - Download this document for free, or read online. Document in PDF available to download.

1 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : This paper is devoted to the 3D Navier-Stokes equations in a periodic case. Assuming that the initial data $u 0$ is in $L^2 x$ while the initial vorticity $\omega 0 = abla \times u 0$ is in $L^1 x$, we prove the existence of a distributional solution $u,p$ to the Navier-Stokes equations such that $u\in L^2 tH^1 x\cap L^\infty tL^2 x\cap L^p tW^{2,p} x\forall p< 5-4$, and $\omega = abla \times u\in L^\infty tL^1 x\cap L^p tW^{1,p} x\forall p< 5-4, p \in L^{5-4} tW^{1,5-4} x$. The main remark of the paper is that the equation for the vorticity can be considered as a parabolic equation with a right hand side in $L^1 {t,x}$. Thus one can use tools of the renormalization theory. Studying approximations deduced from a Large Eddy Simulations model, we focus our attention in passing to the limit in the equation for the vorticity. Finally, we look for sufficient conditions yielding uniqueness of the limit.

Mots-clés : inégalités d-etropie simulation des grandes échelles vorticité equations de Navier-Stokes

Author: ** Roger Lewandowski - **

Source: https://hal.archives-ouvertes.fr/