The global well-posedness for the Q-tensor model of nematic liquid crystals in the half-space
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| Format: | Preprint |
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2025
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| _version_ | 1866912689724325888 |
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| author | Barbera, Daniele Murata, Miho Shibata, Yoshihiro |
| author_facet | Barbera, Daniele Murata, Miho Shibata, Yoshihiro |
| contents | In this paper, we consider the Q-tensor model of nematic liquid crystals, which couples the Navier-Stokes equations with a parabolic-type equation describing the evolution of the directions of the anisotropic molecules, in the half-space. The aim of this paper is to prove the global well-posedness for the Q-tensor model in the $L_p$-$L_q$ framework. Our proof is based on the Banach fixed point argument. To control the higher-order terms of the solutions, we prove the weighted estimates of the solutions for the linearized problem by the maximal $L_p$-$L_q$ regularity. On the other hand, the estimates for the lower-order terms are obtained by the analytic semigroup theory. Here, the maximal $L_p$-$L_q$ regularity and the generation of an analytic semigroup are provided by the R-solvability for the resolvent problem arising from the Q-tensor model. It seems to be the first result to discuss the unique existence of a global-in-time solution for the Q-tensor model in the half-space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_03309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The global well-posedness for the Q-tensor model of nematic liquid crystals in the half-space Barbera, Daniele Murata, Miho Shibata, Yoshihiro Analysis of PDEs Mathematical Physics 76A15, 35Q35, 35A01 In this paper, we consider the Q-tensor model of nematic liquid crystals, which couples the Navier-Stokes equations with a parabolic-type equation describing the evolution of the directions of the anisotropic molecules, in the half-space. The aim of this paper is to prove the global well-posedness for the Q-tensor model in the $L_p$-$L_q$ framework. Our proof is based on the Banach fixed point argument. To control the higher-order terms of the solutions, we prove the weighted estimates of the solutions for the linearized problem by the maximal $L_p$-$L_q$ regularity. On the other hand, the estimates for the lower-order terms are obtained by the analytic semigroup theory. Here, the maximal $L_p$-$L_q$ regularity and the generation of an analytic semigroup are provided by the R-solvability for the resolvent problem arising from the Q-tensor model. It seems to be the first result to discuss the unique existence of a global-in-time solution for the Q-tensor model in the half-space. |
| title | The global well-posedness for the Q-tensor model of nematic liquid crystals in the half-space |
| topic | Analysis of PDEs Mathematical Physics 76A15, 35Q35, 35A01 |
| url | https://arxiv.org/abs/2511.03309 |