Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions

We consider weak solutions to a simplified Ericksen-Leslie system of two-dimensional compressible flow of nematic liquid crystals. An initialboundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity |∇d|d in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible NavierStokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition. Fei Jiang College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China. E-mail: [email protected] Song Jiang Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China. E-mail: [email protected] Dehua Wang Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA. E-mail: [email protected] 2 F. Jiang, S. Jiang, D. Wang