Regularization by rough Kraichnan noise for the generalised SQG equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909554526126080 |
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| author | Bagnara, Marco Galeati, Lucio Maurelli, Mario |
| author_facet | Bagnara, Marco Galeati, Lucio Maurelli, Mario |
| contents | We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $β\in (0,1)$, an active scalar model interpolating between SQG ($β=1$) and the 2D Euler equations ($β=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $θ_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $α$ of the noise and the singularity degree $β$ of the velocity field; in particular, we can cover any $β\in (0,1)$ for suitable $α$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularization by rough Kraichnan noise for the generalised SQG equations Bagnara, Marco Galeati, Lucio Maurelli, Mario Probability Analysis of PDEs 35Q35, 60H15, 60H50 We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $β\in (0,1)$, an active scalar model interpolating between SQG ($β=1$) and the 2D Euler equations ($β=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $θ_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $α$ of the noise and the singularity degree $β$ of the velocity field; in particular, we can cover any $β\in (0,1)$ for suitable $α$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise. |
| title | Regularization by rough Kraichnan noise for the generalised SQG equations |
| topic | Probability Analysis of PDEs 35Q35, 60H15, 60H50 |
| url | https://arxiv.org/abs/2405.12181 |