Existence of local suitable weak solutions to the Navier-Stokes equations for initial data in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2({\mathbb{R}}^3)$

We consider the Navier-Stokes equations in ${\mathbb{R}}^3$ subject to the initial condition with initial velocity field in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2({\mathbb{R}}^3)$ such that ${\mathrm{lim\hspace{0.17em}sup}}_{R\to +\mathrm{\infty }}{R}^{-1}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2(B(R))}<+\mathrm{\infty }$. Our aim is to show the local existence of a weak solution, global existence of weak solution if $C=0$ and the partial regularity in the sense of Caffarelli-Kohn-Nirenberg.