On existence of local and global strong solutions for the stochastic tamed Navier-Stokes equations on $\mathbb{R}^3$

TL;DR

This paper establishes the existence and uniqueness of local and global strong solutions for stochastic tamed Navier-Stokes equations in three-dimensional space, considering multiplicative noise and initial data in specific function spaces.

Contribution

It proves the existence of local and global strong solutions for the stochastic tamed Navier-Stokes equations with multiplicative noise in -dimensional space, addressing the non-local pressure issue.

Findings

Existence of a unique maximal local strong solution for initial data in L^p spaces.

Global strong solutions are established under additional initial data regularity.

The results handle multiplicative Wiener and Lévy jump noise in -space.

Abstract

We study the theory of local and global strong solution for the stochastic tamed Navier--Stokes equations with multiplicative Wiener and L\'evy jump noise in the whole space $\R^3$. More specifically, we first prove the existence of a pathwise unique maximal local strong solution for $\mathcal{F}_0$-measurable initial data in $L^p(\Omega; L^p(\mathbb{R}^3;\mathbb{R}^3)$ for $p > 3$. Furthermore, by assuming initial data in $L^p(\Omega; L^p(\mathbb{R}^3;\mathbb{R}^3)) \cap L^2(\Omega; H^1(\mathbb{R}^3;\mathbb{R}^3))$, we overcome the non-local pressure obstruction to establish the existence of a unique global strong solution.