  We study the existence of strong solutions to the initial value problem for
the incompressible Navier-Stokes equations in $\mathbb{R}^N, N\geq 3$. Our
investigation shows that local in-time classical solutions do not develop
singularity as long as the initial velocity lies in $(L^2(\mathbb{R}^N))^N\cap
(L^\infty(\mathbb{R}^N))^N$.
