Self-similar solutions in weak $L^p$-spaces of the Navier-Stokes equations

Abstract

The most important result stated in this paper is a theorem on the existence of global solutions for the Navier-Stokes equations in ${\mathbb{R}}^n$ when the initial velocity belongs to the space weak $L^n({\mathbb{R}}^n)$ with a sufficiently small norm. Furthermore, this fact leads us to obtain self-similar solutions if the initial velocity is, besides, an homogeneous function of degree -1. Partial uniqueness is also discussed.