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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(\Omega,\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.

Submission history

From: Bikram Podder [view email]
[v1] Tue, 5 May 2026 13:22:29 UTC (105 KB)
[v2] Mon, 1 Jun 2026 17:06:45 UTC (88 KB)
[v3] Wed, 16 Sep 2026 13:12:52 UTC (69 KB)