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PreprintCC BY 4.0HAL

Well-posedness of a stochastic Navier-Stokes system with dynamically coupled subgrid scales

  • Arnaud Debussche
  • Etienne Mémin
  • Sébastien Moskowitz
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Abstract

We study a stochastic Navier-Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier-Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent s ≥ 1, and pathwise uniqueness for s > 1 under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for s > 3/2. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.

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Cite

BibTeXRIS
BibTeX
@misc{debussche2026wellposedness,
  title = {{Well-posedness of a stochastic Navier-Stokes system with dynamically coupled subgrid scales}},
  author = {Arnaud Debussche and Etienne Mémin and Sébastien Moskowitz},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24396},
  eprint = {2609.24396},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05759193v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Well-posedness of a stochastic Navier-Stokes system with dynamically coupled subgrid scales
AU  - Arnaud Debussche
AU  - Etienne Mémin
AU  - Sébastien Moskowitz
PY  - 2026
DA  - 2026/09/21
PB  - HAL
DO  - 10.48550/arxiv.2609.24396
UR  - https://hal.science/hal-05759193v1
AB  - We study a stochastic Navier-Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier-Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent s ≥ 1, and pathwise uniqueness for s > 1 under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for s > 3/2. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.
ER  -