Abstract. In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in R 3 for the wellknown Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition 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 strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.
IntroductionThe Ericksen-Leslie theory is successful in describing dynamic flows of liquid crystals in physics, which is based on the fundamental Oseen-Frank model. Mathematically, the static theory of nematic liquid crystals involves a unit vector field u in a region Ω ⊂ R 3 . The Oseen-Frank density W (u, ∇u) is given bywhere k 1 , k 2 , k 3 and k 4 are positive constants. The free energy for a configurationThe Euler-Lagrange system for the Oseen-Frank energy E(u, Ω) is:, where the standard summation convention is adopted. Since the divergence of tr(∇u) 2 −(div u) 2 is free ([5]), one can rewrite the density W (u, ∇u) aswhere 2 . However, the question on the global weak solution on the system in 3D is still unknown. In the study of the Navier-Stokes equations, there are two well-known blow-up criteria for the strong (smooth) solutions: the Serrin (also called Ladyzhenskaya-Prodi-Serrin type) criterion [23] and the Beal-Kato-Majda type criteria [2]. Recently, for the simplified model, i.e. k 1 = k 2 = k 3 , the local strong solutions was obtained by Wen and Ding [14], and the blow up criterions were obtained by Huang and Wang [15], and there have been many new results developed in this direction [16].In this paper, we consider the Cauchy problem to the Ericksen-Leslie system (1.1)-(1.3) for the general Oseen-Frank model in R 3 . Suppose that the initial data is given byThroughout this paper, we always assume that (u 0 , v 0 ) satisfiesfor some constant unit vector b.In order to state our results, we give the definition of strong solutions and introduce some notations. and it satisfies the equation (1.1)-(1.3) a.e. (x, t) ∈ R 3 × (0, T ). The maximal existence time of the strong solution to the approximate system (1.5)-(1.7) can be defined similarly. For T > 0, we denote