On Periodic 3D Navier-Stokes Equations when the initial velocity is in $L^2$ and the initial vorticity is in $L^1$.
Résumé
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 $\omega_0 = \nabla \times u_0$ is in $L^1_x$, we prove the existence of a distributional solution $(u,p)$ to the Navier-Stokes equations such that $u\in L^2_t(H^1_x)\cap L^\infty_t(L^2_x)\cap L^p_t(W^{2,p}_x)\forall p< 5/4$, and $\omega = \nabla \times u\in L^\infty_t(L^1_x)\cap L^p_t(W^{1,p}_x)\forall p< 5/4, p \in L^{5/4}_t(W^{1,5/4}_x)$. 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.
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