Global dissipative solutions of the 3D Naiver-Stokes and MHD equations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910863992029184 |
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| author | Cheskidov, Alexey Zeng, Zirong Zhang, Deng |
| author_facet | Cheskidov, Alexey Zeng, Zirong Zhang, Deng |
| contents | For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05692 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global dissipative solutions of the 3D Naiver-Stokes and MHD equations Cheskidov, Alexey Zeng, Zirong Zhang, Deng Analysis of PDEs Fluid Dynamics 35A02, 35D30, 76W05 For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term. |
| title | Global dissipative solutions of the 3D Naiver-Stokes and MHD equations |
| topic | Analysis of PDEs Fluid Dynamics 35A02, 35D30, 76W05 |
| url | https://arxiv.org/abs/2503.05692 |