  In this paper we study existence of solutions of the initial-boundary value
problems of the Navier-Stokes equations with a periodic boundary value
condition for initial data in the Sobolev spaces
$\mathcal{H}^{s}(\mathbb{T}^N)$ with a negative order $-1<s<0$, where $N=2, 3$.
By using the randomization approach of N. Burq and N. Tzvetkov, we prove that
for almost all $\omega\in\Omega$, where $\Omega$ is the sample space of a
probability space $(\Omega,\mathcal{A},p)$, for the randomized initial data
$\vec{f}^\omega\in\mathcal{H}_{\sigma}^{s}(\mathbb{T}^N)$ with $-1<s<0$, such a
problem has a unique local solution.
