Random-data Cauchy Problem for the Periodic Navier-Stokes Equations with Initial Data in Negative-order Sobolev Spaces

Abstract

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.