Remarks on the almost sure well-posedness for incompressible Navier-Stokes equations with arbitrary regularity

TL;DR

This paper investigates the almost sure well-posedness of incompressible Navier-Stokes equations with data of arbitrary regularity, utilizing a specific randomization technique to analyze solution existence and uniqueness.

Contribution

It introduces a new approach to study the Navier-Stokes equations with irregular data using randomization, extending previous results to arbitrary regularity levels.

Findings

Established almost sure well-posedness for a broader class of initial data.

Applied randomization techniques to analyze solution behavior.

Extended prior work to include irregular initial conditions.

Abstract

In this note, we study the random data problem for incompressible Navier-Stokes equations in Euclidean space with arbitrary regularity, using the randomization introduced in \cite{Shen-Soffer-Wu-2021-1}.