Well-posedness of the co-rotational Beris-Edwards system with the Landau-De Gennes energy in $ L^{3}_{uloc}(\mathbb{R}^3) $.

We investigate the well-posedness of the 3d co-rotational Beris-Edwards system for incompressible nematic liquid crystal flows with the Landau-De Gennes bulk potential. The system under consideration consists of the Navier–Stokes equations for the fluid velocity $ \boldsymbol u $, and an evolution equation for the $ Q $-tensor order parameter. We prove the existence of solutions to the Cauchy problem of the system with initial data $ (\boldsymbol u_0,Q_0) $ having small $ L_{uloc}^3(\mathbb{R}^3) $-norm of $ (\boldsymbol u_0,\nabla Q_0) $. Moreover, the uniqueness of $ L^3_{uloc} $-solutions is obtained. [ABSTRACT FROM AUTHOR]