Endpoint regularity criterion for weak solutions of the 3D incompressible liquid crystals system

Abstract

In this paper, we consider the endpoint case regularity for the 3D liquid crystals system. We prove that if v ∈ L ∞ (0, T; L 3 (R 3 )), then weak solution (v, d) is smooth, and our main observation is that the condition ∇d ∈ L ∞ (0, T; L 3 (R 3 )) is not necessary in this situation. The proof is based on the blow-up analysis and backward uniqueness for the parabolic operator developed by Escauriaza-Seregin-Sverák. KEYWORDSbackward uniqueness, liquid crystals system, regularity criterion 3672