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In this paper, we establish the local well-posedness and blow-up criteria ofstrong solutions to the Ericksen-Leslie system in $\Bbb R^3$ for the well-knownOseen-Frank model. The local existence of strong solutions to liquid crystalflows is obtained by using the Ginzburg-Landau approximation approach toguarantee the constraint that the direction vector of the fluid is of lengthone. We establish four kinds of blow-up criteria, including (i) the Serrintype; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin typecondition for one field and Beal-Kato-Majda type condition on the other one;(iv) a new one, which characterizes the maximal existence time of the strongsolutions to the Ericksen-Leslie system in terms of Serrin type norms of thestrong solutions to the Ginzburg-Landau approximate system. Furthermore, wealso prove that the strong solutions of the Ginzburg-Landau approximate systemconverge to the strong solution of the Ericksen-Leslie system up to the maximalexistence time.

Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$