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Existence of ergodic invariant measures for 3D Navier-Stokes eqs with superlinear noise

  • Zdzislaw Brzezniak
  • Benedetta Ferrario
  • Margherita Zanella
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Abstract

We consider the Navier-Stokes equations in a smooth bounded domain D⊂ ℝ3. It has been proved in arXiv:2609.08284 that these equations have a unique solution - globally defined in time - when a superlinear noise is introduced and the initial velocity belongs to the space V of divergence-free H10(D;ℝ3)-vector fields. Here we show that the same result holds when there is also an additive noise and a deterministic forcing term. Moreover, when the deterministic forcing term is time-independent we prove the existence of at least one invariant measure for this stochastic equation. This is obtained by the Krylov-Bogoliubov method, based on tightness and Feller property in V. Finally, we show that the set of invariant measures contains one ergodic invariant measure.

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Cite

BibTeXRIS
BibTeX
@misc{brzezniak2026existence,
  title = {{Existence of ergodic invariant measures for 3D Navier-Stokes eqs with superlinear noise}},
  author = {Zdzislaw Brzezniak and Benedetta Ferrario and Margherita Zanella},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13266},
  eprint = {2610.13266},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13266},
}
RIS
TY  - UNPB
TI  - Existence of ergodic invariant measures for 3D Navier-Stokes eqs with superlinear noise
AU  - Zdzislaw Brzezniak
AU  - Benedetta Ferrario
AU  - Margherita Zanella
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13266
UR  - https://arxiv.org/abs/2610.13266
AB  - We consider the Navier-Stokes equations in a smooth bounded domain D⊂ ℝ^3. It has been proved in arXiv:2609.08284 that these equations have a unique solution - globally defined in time - when a superlinear noise is introduced and the initial velocity belongs to the space V of divergence-free H^1_0(D;ℝ^3)-vector fields. Here we show that the same result holds when there is also an additive noise and a deterministic forcing term. Moreover, when the deterministic forcing term is time-independent we prove the existence of at least one invariant measure for this stochastic equation. This is obtained by the Krylov-Bogoliubov method, based on tightness and Feller property in V. Finally, we show that the set of invariant measures contains one ergodic invariant measure.
ER  -