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SUMMARYIn this paper, we study a simplified system for the flow of nematic liquid crystals in a bounded domain in the three-dimensional space. We derive the basic energy law which enables us to prove the global existence of the weak solutions under the condition that the initial density belongs to L ( ) for any > 3 2 . Especially, we also obtain that the weak solutions satisfy the energy inequality in integral or differential form.

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We consider the compressible Navier-Stokes-Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the optimal L 2 and L p , p 2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L 2 decay rate of the second-order spatial derivatives is obtained. The proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates.

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