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28 March 2024
 
  » arxiv » 1603.4219

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On the initial value problem for the Navier-Stokes equations with the initial datum in the Sobolev spaces
D. Q. Khai ; V. T. T. Duong ;
Date 14 Mar 2016
AbstractIn this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces $dot{H}^s_p(mathbb{R}^d)$ for $d geq 2, p > frac{d}{2}, { m and} frac{d}{p} - 1 leq s < frac{d}{2p}$. The obtained result improves the known ones for $p > d$ and $s = 0$ (see M. Cannone (1995), M. Cannone and Y. Meyer (1995)). In the case of critical indexes $s=frac{d}{p}-1$, we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the ones in Cannone (1999) and P. G. Lemarie-Rieusset (2002) in which $(p = d, s = 0)$ and $(p > d, s = frac{d}{p} - 1)$, respectively.
Source arXiv, 1603.4219
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