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

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We consider an elliptic equation with unbounded drift in an exterior domain, and obtain quantitative uniqueness estimates at infinity, i.e. the non-trivial solution of −△u1, which is sharp with the help of some counterexamples. These results also generalize the decay theorem by in the whole space. As an application, the asymptotic behavior of an incompressible fluid around a bounded obstacle is also considered. Specially for the two-dimensional case, we can improve the decay rate in [16] to exp(−C|x| log 2 |x|), where the minimal decaying rate of exp(−C|x| 1 2 |x|) and |Dv| ≤ o(|x| − 3 4 (log |x|) 9 8 ) provided that