Global Well-Posedness and Analyticity for the Three-Dimensional Incompressible Nematic Liquid Crystal Flows in Scaling Invariant Spaces

Abstract

The Cauchy problem for the three-dimensional incompressible flows of liquid crystals in scaling invariant spaces is considered. In this work, we exhibit three results. First, we prove the global well-posedness of mild solution for the system without the supercritical nonlinearity ${\left|\nabla d\right|}^{2}d$ when the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. Our second result is a proof of the global existence of mild solution in the time dependent spaces for the system including the term ${\left|\nabla d\right|}^{2}d$ for small initial data. Lastly, we also get analyticity of the solution.