Weak Solutions for the Navier-Stokes Equations for $B_{\infty \infty }^{-1(ln)}+B_{{\dot{X}}_r}^{-1+r,\frac{2}{1-r}}+L^2$ Initial Data

Abstract

In 1934 Leray proved that the Navier-Stokes equations have global weak solutions for initial data in $L^2({\mathbb{R}}^N)$. In 1990 Calderón extended this result to the initial value spaces $L^p({\mathbb{R}}^N)$ ($2\le p<\infty$). In the book "{\em Recent developments in the Navier-Stokes problems}" (2002), Lemarié-Rieusset extended this result of Calderón to the space $B_{{\tilde{X}}_r}^{-1+r,\frac{2}{1-r}}({\mathbb{R}}^N)+L^2({\mathbb{R}}^N)$ ($0<r<1$), where $X_r$ is the space of functions whose pointwise products with $H^r$ functions belong to $L^2$, ${\tilde{X}}_r$ denotes the closure of $C_0^{\infty }({\mathbb{R}}^N)$ in $X_r$, and $B_{{\tilde{X}}_r}^{-1+r,\frac{2}{1-r}}({\mathbb{R}}^N)$ is the Besov space over ${\tilde{X}}_r$. In this paper we further extend this result of Lemarié-Rieusset to the larger initial value space $B_{\infty \infty }^{-1(ln)}({\mathbb{R}}^N)+B_{{\tilde{\dot{X}}}_r}^{-1+r,\frac{2}{1-r}}({\mathbb{R}}^N)+L^2({\mathbb{R}}^N)$ ($0<r<1$).

24 pages. This new version of the manuscript repairs some mistakes contained in the previous version of this manuscript

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