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

Abstract

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 satisfiesΠ−[ln⁡(1+Π−)]1+γ∈L∞(R+;L32,∞(R3)), or the quantity 2Π+|u|2+|∇Q|2 satisfies(2Π++|u|2+|∇Q|2)[ln⁡(1+2Π++|u|2+|∇Q|2)]1+γ∈L∞(R+;L32,∞(R3)), 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. L32,∞(R3) 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.