Well-posedness for the Navier-Stokes equations with datum in the Sobolev spaces

TL;DR

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

Contribution

It generalizes existing well-posedness results for Navier-Stokes equations to broader Sobolev spaces and includes a global well-posedness result at critical indices.

Findings

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

Global well-posedness for small initial data at critical indices

Extension of previous work to more general Sobolev spaces

Abstract

In this paper, we study local well-posedness for the Navier-Stokes \linebreak 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 $s = 0$ 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 one in M. Cannone (1997) in which $p = d$ and $s = 0$.