Weak compactness of simplified nematic liquid flows in 2D

For any bounded, smooth domain $Ω\subset \R^2$, %(or $Ω=\R^2$), we will establish the weak compactness property of solutions to the simplified Ericksen-Leslie system for both uniaxial and biaxial nematics, and the convergence of weak solutions of the Ginzburg-Landau type nematic liquid crystal flow to a weak solution of the simplified Ericksen-Leslie system as the parameter tends to zero. This is based on the compensated compactness property of the Ericksen stress tensors, which is obtained by the $L^p$-estimate of the Hopf differential for the Ericksen-Leslie system and the Pohozaev type argument for the Ginzburg-Landau type nematic liquid crystal flow.

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