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

We study the existence of local and global strong solutions for the stochastic tamed Navier–Stokes equations on the whole space ${\mathbb{R}}^3$, driven by multiplicative Wiener noise and compensated Lévy jump noise. For $p>3$, we first prove the existence of a pathwise unique maximal local $L^p$-strong solution for divergence-free, ${\mathcal{F}}_0$-measurable initial data in $L^p(\mathrm{\Omega };L^p({\mathbb{R}}^3;{\mathbb{R}}^3))$. For initial data additionally belonging to $L^2(\mathrm{\Omega };H^1({\mathbb{R}}^3;{\mathbb{R}}^3))$, we overcome the non-local pressure obstruction inherent to the whole space, to establish the existence of a pathwise unique global strong solution.