Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Well-posedness for the Navier-Stokes equations with data in homogeneous
  Sobolev-Lorentz spaces   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 $q =
r=d, s=0$.

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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas