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

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