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

TL;DR

This paper demonstrates that for the Navier-Stokes equations with initial data in negative-order Sobolev spaces, a randomized approach ensures almost sure local existence and uniqueness of solutions.

Contribution

It introduces a probabilistic method to establish local solutions for Navier-Stokes with rough initial data in negative Sobolev spaces, extending previous deterministic results.

Findings

Almost sure local existence of solutions for randomized initial data.

Unique solutions are obtained for initial data in negative-order Sobolev spaces.

The approach applies to periodic boundary conditions in 2D and 3D.

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.