Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912046816165888 |
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| author | Tahraoui, Yassine Cipriano, Fernanda |
| author_facet | Tahraoui, Yassine Cipriano, Fernanda |
| contents | This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stabilty result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using "Krylov-Bogoliubov Theorem" we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subset of $L^2$, which leads to the existence of an ergodic invariant measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains Tahraoui, Yassine Cipriano, Fernanda Probability Analysis of PDEs 35Q35, 37L40, 60H15, 60J25, 76A05 This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stabilty result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using "Krylov-Bogoliubov Theorem" we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subset of $L^2$, which leads to the existence of an ergodic invariant measure. |
| title | Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains |
| topic | Probability Analysis of PDEs 35Q35, 37L40, 60H15, 60J25, 76A05 |
| url | https://arxiv.org/abs/2401.04566 |