Two new regularity criteria for nematic liquid crystal flows

Remarks on the regularity for the solutions to liquid crystal flows

“…For example, Zhou and Pokorný [9] have established the following regularity criteria: ∇v 3 ∈ L 𝛼 (0, T; L 𝜁 (R 3 )), 2∕𝛼 + 3∕𝜁 ≤ 19∕12 + 1∕2𝜁, 𝜁 ∈ (30∕19, 3], (1.2) or 2∕𝛼 + 3∕𝜁 ≤ 3∕2 + 3∕4𝜁, 𝜁 ∈ (3, ∞]. Similarly, there are also some interesting results for liquid crystal system; see [3,[16][17][18][19][20][21][22][23][24][25][26][27]. Fan and Guo [19] gave the regularity criterion for the liquid crystal system (1.1), v ∈ L 𝛼 (0, T;…”

“…Wei et al [22] has established the regularity of the weak solutions to 3D liquid crystal equations as follows:…”

“…For example, Zhou and Pokorný [9] have established the following regularity criteria: $$$$ \nabla {v}^3\in {L}^{\alpha}\left(0,T;{L}^{\zeta}\left({\mathrm{\mathbb{R}}}^3\right)\right),2/\alpha +3/\zeta \le 19/12+1/2\zeta, \zeta \in \left(30/19,3\right], $$$$ or $$$ 2/\alpha +3/\zeta \le 3/2+3/4\zeta, \zeta \in \left(3,\infty \right] $$$. Similarly, there are also some interesting results for liquid crystal system; see [3, 16–27]. Fan and Guo [19] gave the regularity criterion for the liquid crystal system (), $$$$ v\in {L}^{\alpha}\left(0,T;{\dot{M}}_{\zeta, \gamma}\left({\mathrm{\mathbb{R}}}^3\right)\right),2/\alpha +3/\zeta =1,\zeta >3,\zeta \ge \gamma \ge 1, $$$$ or …”

“…Wei et al [22] has established the regularity of the weak solutions to 3D liquid crystal equations as follows: $$$$ {v}^3,\nabla n\in {L}^p\left(0,T;{L}^q\left({\mathrm{\mathbb{R}}}^3\right)\right),2/p+3/q\le 3/4+1/2q,q>10/3. $$$$ …”

BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations

“…Under this simplification, Lin and Liu proved in [22] the global existence of a unique strong solution in dimension two and in dimension three under the large viscosity ν, moreover, in [23], they proved the existence of suitable weak solutions and the one-dimensional space-time Hausdorff measure of the singular set of suitable weak solutions is zero, analogous to the celebrated partial regularity theorem by Caffarelli-Kohn-Nirenberg [3] for the three-dimensional incompressible Navier-Stokes equations. Note that the regularity and uniqueness of global weak solutions to the three-dimensional simplified system are still open problems, we refer the readers to see [31], [37], [39] and [40] for some recent results concerning the global existence of strong solutions with small initial data, regularity and uniqueness criteria of weak solutions to the simplified Ericksen-Leslie system.…”