Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911969926184960 |
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| author | Coghi, Michele Maurelli, Mario |
| author_facet | Coghi, Michele Maurelli, Mario |
| contents | We consider the 2D Euler equations on $\R^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $α\in (0,1)$.
We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-α})$ regularity.
For every $p>3/2$ and for certain regularity indices $α\in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_03216 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity Coghi, Michele Maurelli, Mario Probability Analysis of PDEs 35Q31, 60H15, 60H50, 35Q35 We consider the 2D Euler equations on $\R^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $α\in (0,1)$. We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-α})$ regularity. For every $p>3/2$ and for certain regularity indices $α\in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise. |
| title | Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity |
| topic | Probability Analysis of PDEs 35Q31, 60H15, 60H50, 35Q35 |
| url | https://arxiv.org/abs/2308.03216 |