Global Solvability of 3D Inhomogeneous Nematic Liquid Crystal Flows With Only Bounded Density in Critical Besov Space

Abstract

This paper is devoted to proving the global existence and uniqueness of weak solutions to the three-dimensional inhomogeneous incompressible nematic liquid crystal flows. The results hold under the conditions that the initial density lies in the bounded function space with a positive lower bound, the initial velocity is sufficiently small in the critical Besov space ${\dot{B}}_{2,1}^{\frac{1}{2}}\left({ℝ}^3\right)$, and the gradient of the initial molecular orientation is also small enough in the same critical Besov space ${\dot{B}}_{2,1}^{\frac{1}{2}}\left({ℝ}^3\right)$. These results align with the Fujita-Kato theorem for the inhomogeneous incompressible Navier-Stokes equations, as established by Zhang. Additionally, our work provides a lower bound for the lifespan of smooth solutions to the three-dimensional inhomogeneous incompressible nematic liquid crystal flows. The findings also address the uniqueness problem raised by De Anna.