Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on ${\mathbb{R}}^3$. II. Global $L^p$ Well-Posedness

We establish intrinsic continuation criteria and finite-energy global solvability for the stochastic tamed Navier–Stokes equations on ${\mathbb{R}}^3$ driven simultaneously by multiplicative cylindrical Wiener and compensated Poisson noise. Under local coefficient hypotheses, the maximal local $L^p$ solution, $p>3$, satisfies a blow-up alternative independent of auxiliary cutoffs and a Serrin criterion with time exponent $2p\mathrm{/}(p-3)$. Additional coercivity of the taming term and compatible $L^2$ and gradient noise bounds yield the global well-posedness theory for divergence-free initial data $u_0\in L^p(\mathrm{\Omega };L^p)\cap L^2(\mathrm{\Omega };L^2)$. No smallness or initial $H^1$ regularity is required. The solution has càdlàg $L^p\cap L^2$ paths and gains $H^1$ regularity at positive times, with time-weighted $H^1$ and $H^2$ estimates. The proof combines finite-energy persistence with endpoint completion that retains terminal Poisson jumps and permits restart in the original uniqueness class.