A remark on regularity of liquid crystal equations in critical Lorentz spaces
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A remark on regularity of liquid crystal equations in critical Lorentz spaces Xiangao Liu1 · Yueli Liu1 · Zixuan Liu1 Received: 14 August 2020 / Accepted: 9 November 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The regularity for the 3-D nematic liquid crystal equations is considered in this paper, it is proved that the Leray–Hopf weak solutions (u, d) is in fact smooth, if the velocity field u ∈ L∞ (0, T;Lx3,∞ (ℝ3 )) satisfies some addition local small condition }| |{ r−3 | x ∈ Br (x0 ) ∶ |u(x, t0 )| > 𝜀r−1 | ≤ 𝜀, | | which is inspired by the papers [2, 35].
Keywords Weak - L3 space · Backward uniqueness · Nematic liquid crystals equations Mathematics Subject Classification 76A15 · 35B65 · 76D03
1 Introduction The three-dimensional incompressible liquid crystals system are the following coupled equations
⎧ u − Δu + u ⋅ ∇u + ∇p = −div(∇d ⊙ ∇d), ⎪ t ⎨ div u = 0, ⎪ dt + u ⋅ ∇d − Δd = −f (d), ⎩
(1.1)
This work was supported partly by NSFC Grant 11971113, 11631011. * Xiangao Liu [email protected] Yueli Liu [email protected] Zixuan Liu [email protected] 1
School of Mathematic Sciences, Fudan University, Shanghai, China
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Vol.:(0123456789)
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in the domain QT ≡ ℝ3 × (0, T) . Here, the unknowns u = (u1 , u2 , u3 ) is the velocity field, p is the scalar pressure and d = (d1 , d2 , d3 ) is the optical molecule direction after penalization, and f (d) = 𝜎12 (|d|2 − 1)d , ∇d ⊙ ∇d is a symmetric tensor with its component (∇d ⊙ ∇d)ij is given by 𝜕i d ⋅ 𝜕j d . And the initial conditions are
u(x, 0) = u0 (x),
div(u0 ) = 0,
d(x, 0) = d0 (x),
(1.2)
with |d0 | = 1. System (1.1) is the simplified system of the original Ericksen–Leslie system of variable length for the flow of liquid crystals, that is the Ginzburg–Landau energy 2 )2 ∫Ω ( 21 |∇d|2 + (1−|d| ) . For this system, Lin and Liu [19–21] first proved a global existence 2 4𝜎 of weak solutions under L2 data and regularity result of the suitable weak solution under the C-K-N condition. Other results to liquid crystals equations refer to [8–11, 15, 16, 18, 22, 23, 28]. Let us now recall the notion of a suitable weak solution of liquid crystals equations.
Definition 1.1 ([20]) a triple (u, d, p) is called a suitable weak solution of (1.1) in ℝ3 × (0, T) if the following conditions hold: (1). the weak solution (u, d) satisfies system (1.1) in the distribution sense; (2). the solution (u, d) satisfy the energy inequality, i.e.,
‖u‖2L2,∞ (ℝ3 ×(0,T)) + ‖∇d‖2L2,∞ (ℝ3 ×(0,T))
+ ‖∇2 d‖2L2 (ℝ3 ×(0,T))) + ‖∇u‖2L2 (ℝ3 ×(0,T))) ≤ c0 ; 3
2 (ℝ3 × (0, T)); (3). the press p ∈ Lloc (4). the triple (u, d, p) satisfy the modified generalized local energy inequality, for a.e. t ∈ (0, T) and for all 𝜙 ∈ C0∞ (ℝ3 × (0, T)) with 𝜙 ≥ 0,
t
�ℝ3 ×{t}
(|u|2 + |∇d|2 )𝜙dx + 2
�0 �ℝ3
(|∇u|2 + |∇2 d|2 )𝜙
t
≤
�0 �ℝ3
(|u|2 + |∇d|2 )(𝜙t + Δ𝜙)
t
+
�0 �ℝ3
(|u|2 + |∇d|2 + 2p)(u ⋅ ∇𝜙) t
t
+2
�0 �ℝ3
((u ⋅ ∇)d ⊙ ∇d) ⋅ ∇𝜙 − 2
�0 �ℝ3
∇x f (d) ⋅ ∇d𝜙.
Denote for z = (x, t)
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