Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918161035558912 |
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| author | Coe, James Hairer, Martin Tolomeo, Leonardo |
| author_facet | Coe, James Hairer, Martin Tolomeo, Leonardo |
| contents | We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process.
Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13460 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations Coe, James Hairer, Martin Tolomeo, Leonardo Probability Analysis of PDEs 60H15, 35Q30 We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process. Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations. |
| title | Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations |
| topic | Probability Analysis of PDEs 60H15, 35Q30 |
| url | https://arxiv.org/abs/2510.13460 |