The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913407345623040 |
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| author | Barbera, Daniele Murata, Miho |
| author_facet | Barbera, Daniele Murata, Miho |
| contents | In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields $Q$. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal $L^p$ -$L^q$ regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the $\mathcal R$-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space Barbera, Daniele Murata, Miho Analysis of PDEs 76A15, 35Q30, 35Q35 In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields $Q$. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal $L^p$ -$L^q$ regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the $\mathcal R$-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data. |
| title | The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space |
| topic | Analysis of PDEs 76A15, 35Q30, 35Q35 |
| url | https://arxiv.org/abs/2406.19805 |