Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909327134031872 |
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| author | Brzeźniak, Zdzisław Ferrari, Matteo |
| author_facet | Brzeźniak, Zdzisław Ferrari, Matteo |
| contents | We prove the existence and some moment estimates for an invariant measure $μ$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain $\mathbb R^2$ and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous $2$D Navier-Stokes equations with kinematic viscosity $ν>0$ and an additive stochastic noise scaling as $\sqrt ν$. We then study the inviscid limit and prove that, as $ν$ tends to $0$, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure $μ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$ Brzeźniak, Zdzisław Ferrari, Matteo Probability Analysis of PDEs We prove the existence and some moment estimates for an invariant measure $μ$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain $\mathbb R^2$ and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous $2$D Navier-Stokes equations with kinematic viscosity $ν>0$ and an additive stochastic noise scaling as $\sqrt ν$. We then study the inviscid limit and prove that, as $ν$ tends to $0$, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure $μ$. |
| title | Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$ |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2409.17697 |