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In 1934 Leray proved that the Navier-Stokes equations have global weaksolutions for initial data in $L^2(\mathbb{R}^N)$. In 1990 Calder\'on extendedthis 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 productswith $H^r$ 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 spaceover $\widetilde{X}_r$. In this paper we further extend this result ofLemari\'e-Rieusset to the larger initial value space${B}^{-1(ln)}_{\infty\infty}(\mathbb{R}^N)+{B}_{\widetilde{\dot{X}}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)+L^2(\mathbb{R}^N)$($0<r<1$).

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