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arXiv · 1603.04219

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
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Abstract

In 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 $d \geq 2, p > \frac{d}{2},\ {\rm and}\ \frac{d}{p} - 1 \leq s < \frac{d}{2p}$. 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.

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D. Q. Khai, V. T. T. Duong. 2016-03-14. On the initial value problem for the Navier-Stokes equations with the initial datum in the Sobolev spaces. https://arxiv.org/abs/1603.04219

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