Weak-strong uniqueness of the full coupled Navier-Stokes and Q-tensor system in dimension three
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| Format: | Preprint |
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2025
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| _version_ | 1866910255765520384 |
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| author | Yang, Fan Zhou, Junjie |
| author_facet | Yang, Fan Zhou, Junjie |
| contents | In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the Beris-Edwards model of nematic liquid crystals in $\R^3$ with an arbitrary parameter $ξ\in\R$, which measures the ratio of tumbling and alignment effects caused by the flow. This result is obtained by proposing a new uniqueness criterion in terms of $(ΔQ,\nabla u)$ with regularity $L_t^qL_x^p$ for $\frac{2}{q}+\frac{3}{p}=\frac{3}{2}$ and $2\leq p\leq 6$, which enables us to deal with the additional nonlinear difficulties arising from the parameter $ξ$. Compared with the known results, our finding reveals that the criterion of weak-strong uniqueness for $ξ\ne 0$ is a sub-regime of the one for the corotational case. The associated regularity assumption rises with the nonlinearity of the model. Moreover, we establish the global well-posedness of this model for small initial data in $H^s$-framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak-strong uniqueness of the full coupled Navier-Stokes and Q-tensor system in dimension three Yang, Fan Zhou, Junjie Analysis of PDEs In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the Beris-Edwards model of nematic liquid crystals in $\R^3$ with an arbitrary parameter $ξ\in\R$, which measures the ratio of tumbling and alignment effects caused by the flow. This result is obtained by proposing a new uniqueness criterion in terms of $(ΔQ,\nabla u)$ with regularity $L_t^qL_x^p$ for $\frac{2}{q}+\frac{3}{p}=\frac{3}{2}$ and $2\leq p\leq 6$, which enables us to deal with the additional nonlinear difficulties arising from the parameter $ξ$. Compared with the known results, our finding reveals that the criterion of weak-strong uniqueness for $ξ\ne 0$ is a sub-regime of the one for the corotational case. The associated regularity assumption rises with the nonlinearity of the model. Moreover, we establish the global well-posedness of this model for small initial data in $H^s$-framework. |
| title | Weak-strong uniqueness of the full coupled Navier-Stokes and Q-tensor system in dimension three |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.09281 |