On Periodic 3D Navier-Stokes Equations when the initial velocity is in L2 and the initial vorticity is in L1.
Résumé
This paper is devoted to the 3D Navier-Stokes equations in a periodic case. Assuming that the initial data u0 is in L2x while the initial vorticity ω0=∇×u0 is in L1x, we prove the existence of a distributional solution (u,p) to the Navier-Stokes equations such that u∈L2t(H1x)∩L∞t(L2x)∩Lpt(W2,px)∀p<5/4, and ω=∇×u∈L∞t(L1x)∩Lpt(W1,px)∀p<5/4,p∈L5/4t(W1,5/4x). 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 L1t,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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