Logarithmically improved criteria for the 3D nematic liquid crystal flows in the Morrey–Campanato space

Abstract

a b s t r a c tWe provide a sufficient condition for the regularity of solutions to the 3D nematic liquid crystal flow in the Morrey-Campanato space. More precisely, we prove that if the velocity u satisfies3 r , or the gradient of the velocity ∇u satisfies  T 0 ∥∇u(·, t)∥ 2 2−γ M p, 3 γ 1 + ln(e + ∥u(·, t)∥ L 6 ) dt < ∞ with 0 < γ ≤ 1 and 2 ≤ p ≤ 3 γ , then the solution (u, d) remains smooth on (0, T ].Crown