Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917987421782016 |
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| author | Chen, Weiquan Dong, Zhao Zheng, Yang |
| author_facet | Chen, Weiquan Dong, Zhao Zheng, Yang |
| contents | We show non-uniqueness of local strong solutions to stochastic fractional Navier-Stokes equations with linear multiplicative noise and some certain deterministic force. Such non-uniqueness holds true even if we perturb such deterministic force in appropriate sense.This is closely related to a critical condition on force under which Leray-Hopf solution to the stochastic equations is locally unique. Meanwhile, by a new idea, we show that for some stochastic force the system admits two different global Leray-Hopf solutions smooth on any compact subset of $(0,\infty) \times \mathbb{R}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index Chen, Weiquan Dong, Zhao Zheng, Yang Probability Analysis of PDEs We show non-uniqueness of local strong solutions to stochastic fractional Navier-Stokes equations with linear multiplicative noise and some certain deterministic force. Such non-uniqueness holds true even if we perturb such deterministic force in appropriate sense.This is closely related to a critical condition on force under which Leray-Hopf solution to the stochastic equations is locally unique. Meanwhile, by a new idea, we show that for some stochastic force the system admits two different global Leray-Hopf solutions smooth on any compact subset of $(0,\infty) \times \mathbb{R}^d$. |
| title | Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2503.18041 |