Regularity criteria for weak solutions to the 3d co-rotational Beris-Edwards system via the pressure.

We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q -tenser. Our results yield that for any positive constant γ > 0 , if either the negative part of the associated pressure Π satisfies Π − [ ln ⁡ (1 + Π −) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , or the quantity 2 Π + | u | 2 + | ∇ Q | 2 satisfies (2 Π + + | u | 2 + | ∇ Q | 2) [ ln ⁡ (1 + 2 Π + + | u | 2 + | ∇ Q | 2) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , then the weak solution (u , Q) , to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript "−" and "+" denote the negative and the nonnegative part, respectively. L 3 2 , ∞ (R 3) denotes the standard weak Lebesgue space. If Q ≡ 0 , then our results extend some previous known results from the theory of the 3d Navier–Stokes equations. [ABSTRACT FROM AUTHOR]