Global regularity of the generalized liquid crystal model with fractional diffusion

Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows

“…here 0 ≤ r ≤ 1, then (u, d) can be extended beyond T smoothly. Tat is to say, if the solution blows up at T, then 17) is an improvement in some sense compared to the condition (12). However,4/3 − 2r � 4/3 when r � 0, hence the condition (12) is better than the condition (17) in the end point.…”

“…Hence much eforts have been paid to study the regularity criteria to extend local solutions. For the regularity criteria readers may refer to [5][6][7][8][9][10][11][12][13] and references therein.…”

Two regularity criteria of solutions to the liquid crystal flows

“…where 12 7 ≤ 𝛼 = 4p p+6 < 4. Combining condition (1.6), the above inequality indicates that by Gronwall inequality,…”

“…It is obvious that system () reduces to the well‐known Navier–Stokes equations when the orientation field $$$ d\equiv 1 $$$. Therefore, in view of the results of regularity criterion for the 3D Navier–Stokes equations, some Prodi–Serrin type regularity criteria of the nematic liquid crystal flows have been obtained based on the velocity field $$$ u $$$, the gradient of the velocity field $$$ \nabla u $$$, and their components; see earlier studies 3–12 and the references therein.…”