Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect
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Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect Xiaopeng Zhao1 · Mingxuan Zhu2 Received: 1 May 2020 / Revised: 3 September 2020 / Accepted: 11 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, for the incompressible nematic liquid crystal flows with heat effect and density-dependent viscosity coefficient in three-dimensional bounded domain, by using the elliptic regularity result of the Stokes equations and the linearization and iteration method, we investigate the local existence and uniqueness of strong solutions. Keywords Strong solutions · Nematic liquid crystal flows · Non-negative density · Heat effect Mathematics Subject Classification 35Q35 · 76A15 · 35D35
1 Introduction Let ⊂ R3 be a bounded domain with smooth boundary ∂ whose unit outward normal is n, we are concerned with the incompressible nematic liquid crystal flows with heat effect and density-dependent viscosity coefficient in : ⎧ ρt + ∇ · (ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ (ρu)t + ∇ · (ρu ⊗ u) − ∇ · (2μ(ρ)D(u)) + ∇ p = −λ∇ · (∇d ∇d), ⎪ ⎨ dt + u · ∇d = ϑ(d + |∇d|2 d), (1.1) ⎪ ⎪ ⎪ 2 2 2 ⎪ cν [(ρθ )t +∇ · (ρuθ )]−∇ · (κ(ρ)∇θ ) = 2μ(ρ)|D(u)| +ρ|d +|∇d| d| , ⎪ ⎪ ⎩ ∇ · u = 0,
Communicated by Yong Zhou.
B
Mingxuan Zhu [email protected] Xiaopeng Zhao [email protected]
1
School of Science, Northeastern University, Shenyang 110819, China
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
123
X. Zhao, M. Zhu
where ρ, u, θ and p are fluid density, velocity, absolute temperature and the pressure, respectively. The constants λ > 0, cν > 0 and ϑ > 0 are competition between kinetic and potential energy, heat conductivity and microscopic elastic relaxation time, respectively; for convenience, we will set λ = ϑ = 1 in this paper. The force term ∇d ∇d in the equation of the conservation of momentum denotes the 3 × 3 matrix whose i jth entry is given by ‘∇i d · ∇ j d’ for 1 ≤ i, j ≤ 3. We also denote by D(u) the deformation tensor 21 [∇u + (∇u)T ]. The two continuously differentiable functions μ(ρ) and κ(ρ) satisfy μ(ρ), κ(ρ) ∈ C 1 [0, ∞) and 0 < μ ≤ μ, 0 < κ ≤ κ on [0, ∞),
(1.2)
for some positive constants μ and κ. System (1.1) is a simplified version of those proposed in [49, Chapter BIII2], [51, Chapter 3], [40, Section 9] and [21, Section 2]. In this paper, on the basis of physical considerations, we suppose that equations (1.1) is coupled with the following initial conditions (ρ, u, d, θ )|t=0 = (ρ0 , u 0 , d0 , θ0 ) with ∇ · u 0 = 0 and |d0 | = 1, in ,
(1.3)
and boundary value conditions u=
∂θ ∂d = = 0, on ∂, ∂ n ∂ n
(1.4)
where n is the unit outward normal vector to ∂. If ρ ≡ constant > 0 and the heat effect is neglected, the simplified incompressible nematic liquid crystals flow is [12,13,35] ⎧ ⎪ ⎨ u t + ∇ · (u ⊗ u) − u + ∇ p = −∇ · (∇d ∇d), dt + u · ∇d = d + |∇d|2 d, ⎪ ⎩ ∇ · u = 0, |d| = 1.
(1.5)
In order to understand the nematic liquid crystal flows distinctly, Lin
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