Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on ℝ³. I. Maximal Local L^p-Well-Posedness

@misc{podder2026stochastic,
  title = {{Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on $\mathbb R^3$. I. Maximal Local $L^p$-Well-Posedness}},
  author = {Bikram Podder and Surendra Kumar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.03734},
  eprint = {2605.03734},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.03734},
}

TY  - UNPB
TI  - Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on ℝ^3. I. Maximal Local L^p-Well-Posedness
AU  - Bikram Podder
AU  - Surendra Kumar
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2605.03734
UR  - https://arxiv.org/abs/2605.03734
AB  - We establish maximal local L^p-well-posedness, for p>3, for the stochastic tamed Navier--Stokes equations on 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 pth 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.
ER  -