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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2605.16997 |
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| _version_ | 1866910226447335424 |
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| author | Zhang, Yao Soe, Han Ni Xu, Zhipeng |
| author_facet | Zhang, Yao Soe, Han Ni Xu, Zhipeng |
| contents | We prove existence of a weak solution to the three-dimensional Beris--Edwards system in the whole space under the stable bulk assumption $c>0$. The solution satisfies the natural bounds $Q\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and $u\in L^\infty_tL^2_x\cap L^2_tH^1_x$, the distributional form of the equations, and the expanded Leray--Hopf type energy inequality used in weak--strong uniqueness arguments. The proof does not pass directly to the limit in that expanded inequality, where the non-corotational terms contain products of the form $|Q^n|^4Q^n:\nabla u^n$. It first obtains the physical free-energy inequality through a hyperviscous approximation and a localized tail estimate, and then derives the expanded inequality from a low-order chain rule for the bulk part of the energy. The last section records the elementary uniaxial reduction which explains why the present argument is restricted to stable bulk potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16997 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Leray--Hopf Type Weak Solutions for the Three-Dimensional Beris--Edwards System with Stable Landau--de Gennes Potential Zhang, Yao Soe, Han Ni Xu, Zhipeng Analysis of PDEs We prove existence of a weak solution to the three-dimensional Beris--Edwards system in the whole space under the stable bulk assumption $c>0$. The solution satisfies the natural bounds $Q\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and $u\in L^\infty_tL^2_x\cap L^2_tH^1_x$, the distributional form of the equations, and the expanded Leray--Hopf type energy inequality used in weak--strong uniqueness arguments. The proof does not pass directly to the limit in that expanded inequality, where the non-corotational terms contain products of the form $|Q^n|^4Q^n:\nabla u^n$. It first obtains the physical free-energy inequality through a hyperviscous approximation and a localized tail estimate, and then derives the expanded inequality from a low-order chain rule for the bulk part of the energy. The last section records the elementary uniaxial reduction which explains why the present argument is restricted to stable bulk potentials. |
| title | Leray--Hopf Type Weak Solutions for the Three-Dimensional Beris--Edwards System with Stable Landau--de Gennes Potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.16997 |