Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913238501818368 |
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| author | Lü, Huaxiang Zhu, Xiangchan |
| author_facet | Lü, Huaxiang Zhu, Xiangchan |
| contents | We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+÷(u\otimes u)\dif t+\nabla p\dif t&=Δu\dif t + (-Δ)^{\fa/2}\dif B_t,\ \ \ \ ÷u=0,\notag \end{align*} driven by derivative of space-time white noise, where $\fa\in[0,\frac13)$. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution $N(0,\frac12(-Δ)^{\fa-1})$. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13743 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise Lü, Huaxiang Zhu, Xiangchan Probability Analysis of PDEs We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+÷(u\otimes u)\dif t+\nabla p\dif t&=Δu\dif t + (-Δ)^{\fa/2}\dif B_t,\ \ \ \ ÷u=0,\notag \end{align*} driven by derivative of space-time white noise, where $\fa\in[0,\frac13)$. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution $N(0,\frac12(-Δ)^{\fa-1})$. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces. |
| title | Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2402.13743 |