Weak well-posedness by transport noise for a class of 2D fluid dynamics equations
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| Format: | Preprint |
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2023
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| author | Galeati, Lucio Luo, Dejun |
| author_facet | Galeati, Lucio Luo, Dejun |
| contents | A fundamental open problem in fluid dynamics is whether solutions to $2$D Euler equations with $(L^1_x\cap L^p_x)$-valued vorticity are unique, for some $p\in [1,\infty)$. A related question, more probabilistic in flavour, is whether one can find a physically relevant noise regularizing the PDE. We present some substantial advances towards a resolution of the latter, by establishing well-posedness in law for solutions with $(L^1_x\cap L^2_x)$-valued vorticity and finite kinetic energy, for a general class of stochastic 2D fluid dynamical equations; the noise is spatially rough and of Kraichnan type and we allow the presence of a deterministic forcing $f$. This class includes as primary examples logarithmically regularized 2D Euler and hypodissipative 2D Navier-Stokes equations. In the first case, our result solves the open problem posed by Flandoli. In the latter case, for well-chosen forcing $f$, the corresponding deterministic PDE without noise has recently been shown by Albritton and Colombo to be ill-posed; consequently, the addition of noise truly improves the solution theory for such PDE. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_08761 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak well-posedness by transport noise for a class of 2D fluid dynamics equations Galeati, Lucio Luo, Dejun Probability Analysis of PDEs 60H15, 60H50, 35Q35 A fundamental open problem in fluid dynamics is whether solutions to $2$D Euler equations with $(L^1_x\cap L^p_x)$-valued vorticity are unique, for some $p\in [1,\infty)$. A related question, more probabilistic in flavour, is whether one can find a physically relevant noise regularizing the PDE. We present some substantial advances towards a resolution of the latter, by establishing well-posedness in law for solutions with $(L^1_x\cap L^2_x)$-valued vorticity and finite kinetic energy, for a general class of stochastic 2D fluid dynamical equations; the noise is spatially rough and of Kraichnan type and we allow the presence of a deterministic forcing $f$. This class includes as primary examples logarithmically regularized 2D Euler and hypodissipative 2D Navier-Stokes equations. In the first case, our result solves the open problem posed by Flandoli. In the latter case, for well-chosen forcing $f$, the corresponding deterministic PDE without noise has recently been shown by Albritton and Colombo to be ill-posed; consequently, the addition of noise truly improves the solution theory for such PDE. |
| title | Weak well-posedness by transport noise for a class of 2D fluid dynamics equations |
| topic | Probability Analysis of PDEs 60H15, 60H50, 35Q35 |
| url | https://arxiv.org/abs/2305.08761 |