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

Abstract

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; L^p(\mathbb R^3;\mathbb R^3)\bigr), $ we prove existence and pathwise uniqueness of a local strong solution with $L^p$-valued càdlàg trajectories and the local $L^p$-energy regularity. For solutions driven by the same noises, we establish localized Lipschitz dependence on the initial datum in the path supremum and space--time norms. The discontinuous forcing makes the whole-space Gaussian construction non formal as stopping levels may be overshot by jumps, while convergence of the compensated-Poisson term requires simultaneous control of its quadratic and $p$th integrability modes. We resolve these difficulties by a jump-compatible localization based on strict pre-exit bounds and predictable left limits. We further establish bounded-time restart and stochastic pasting on the prescribed stochastic basis. The resulting family of attainable lifetimes is upward directed and yields a unique maximal local strong solution, which inherits the localized dependence estimate. Continuation criteria, blow-up alternatives, and global well-posedness under stronger finite-energy hypotheses are treated in the companion Part~II.