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 $u_0\in L^p(\Omega ,{\mathcal{F}}_0;L^p({\mathbb{R}}^3;{\mathbb{R}}^3)),$ 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.

Substantially revised and reorganized. The original manuscript has been split into two papers. This version develops the maximal local $L^p$-strong well-posedness theory for stochastic tamed Navier--Stokes equations on ${\mathbb{R}}^3$ with Wiener and jump noise. The global results in v1--v2 have been removed and will appear separately with additional results