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Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces | | D. Q. Khai
; N. M. Tri
; | | Date: |
8 Jan 2016 | | Abstract: | In this paper, we study local well-posedness for the Navier-Stokes equations
(NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces
$dot{H}^s_{L^{q, r}}(mathbb{R}^d):= (-Delta)^{-s/2}L^{q,r}$ for $d geq 2, q
> 1, s geq 0$, $1 leq r leq infty$, and $ frac{d}{q}-1 leq s <
frac{d}{q}$, this result improves the known results for $q > d,r=q, s = 0$
(see M. Cannone (1995) and M. Cannone and Y. Meyer (1995)) and for $q =r= 2,
frac{d}{2} - 1 < s < frac{d}{2}$ (see M. Cannone (1995, J. M. Chemin (1992)).
In the case of critical indexes ($s=frac{d}{q}-1$), we prove global
well-posedness for NSE provided the norm of the initial value is small enough.
The result that is a generalization of the result of M. Cannone (1997) for $q =
r=d, s=0$. | | Source: | arXiv, 1601.1742 | | Services: | Forum | Review | PDF | Favorites | |
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