Abstract
The three-dimensional nematic liquid crystal flows with damping are considered in this paper. The existence and uniqueness of strong solutions for the 3D nematic liquid crystal flows with damping are proved for betageq4 with any alpha>0.
Highlights
In this paper, we consider the following three-dimensional nematic liquid crystal flows with damping: ⎧ ⎪⎪⎪⎪⎪⎪⎨ ∂t u ∂t d – +ν (u u + (u · ∇)u + α|u|β–1u · ∇)d = γ ( d – f (d)), + ∇p = –λ∇ ·
In [8], the existence and uniqueness of strong solutions for the 3D magneto-micropolar equations were proved for β ≥ 4 with any α > 0
Then we get the above estimate easily forFor I2(t), integrating by parts, applying the Hölder inequality and the Young inequality, we get
Summary
We consider the following three-dimensional nematic liquid crystal flows with damping:. The 3D nematic liquid crystal flows were proposed by Lin ([1, 2]) and have been extensively investigated. When d = 0, the problem (1) reduces to the three-dimensional Navier–Stokes equations with damping. In [3,4,5,6], the well-posedness of the three-dimensional Navier–Stokes equations with damping is proved for β. The global existence of weak solutions of the 3D nematic liquid crystal flow was proved in [7]. In [8], the existence and uniqueness of strong solutions for the 3D magneto-micropolar equations were proved for β ≥ 4 with any α > 0
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