Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914945162018816 |
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| author | Hofmanová, Martina Zhu, Rongchan Zhu, Xiangchan |
| author_facet | Hofmanová, Martina Zhu, Rongchan Zhu, Xiangchan |
| contents | We consider stochastic forced Navier--Stokes equations on $\mathbb{R}^{3}$ starting from zero initial condition. The noise is linear multiplicative and the equations are perturbed by an additional body force. Based on the ideas of Albritton, Brué and Colombo \cite{ABC22}, we prove non-uniqueness of local-in-time Leray--Hopf solutions as well as joint non-uniqueness in law for solutions on $\mathbb{R}^{+}$. In the deterministic setting, we show that the set of forces, for which Leray--Hopf solutions are non-unique, is dense in $L^{1}_{t}L^{2}_{x}$. In addition, by a simple controllability argument we show that for every divergence-free initial condition in $L^{2}_{x}$ there is a force so that non-uniqueness of Leray--Hopf solutions holds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_03668 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations Hofmanová, Martina Zhu, Rongchan Zhu, Xiangchan Probability Analysis of PDEs We consider stochastic forced Navier--Stokes equations on $\mathbb{R}^{3}$ starting from zero initial condition. The noise is linear multiplicative and the equations are perturbed by an additional body force. Based on the ideas of Albritton, Brué and Colombo \cite{ABC22}, we prove non-uniqueness of local-in-time Leray--Hopf solutions as well as joint non-uniqueness in law for solutions on $\mathbb{R}^{+}$. In the deterministic setting, we show that the set of forces, for which Leray--Hopf solutions are non-unique, is dense in $L^{1}_{t}L^{2}_{x}$. In addition, by a simple controllability argument we show that for every divergence-free initial condition in $L^{2}_{x}$ there is a force so that non-uniqueness of Leray--Hopf solutions holds. |
| title | Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2309.03668 |