On the initial value problem for the Navier-Stokes equations with the initial datum in the Sobolev spaces

TL;DR

This paper establishes local well-posedness results for the Navier-Stokes equations with initial data in Sobolev spaces, extending previous results and proving global well-posedness for small initial data at critical indexes.

Contribution

It generalizes existing well-posedness results for Navier-Stokes equations to broader Sobolev space settings, including critical cases.

Findings

Improved local well-posedness results for $p > d$ and $s=0$.

Proved global well-posedness for small initial data at critical indexes $s = d/p - 1$.

Extended previous results to more general Sobolev spaces.

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.