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

Abstract

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 $2p/(p-3)$. 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 $u_0\in L^p(\Omega ;L^p)\cap L^2(\Omega ;L^2)$. 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.

75 pages, This is the Second part of our previous arXiv manuscript I. Maximal $L^p$ well-posedness