Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on $\mathbb R^3$. I. Maximal Local $L^p$-Well-Posedness

We establish maximal local $L^p$-well-posedness, for $p>3$, for the stochastic tamed Navier--Stokes equations on $\mathbb R^3$ driven simultaneously by multiplicative cylindrical Wiener noise and a compensated Poisson random measure. For divergence-free initial data $ u_0\in L^p\bigl(Ω,\mathcal F_0;