Global solution to the 3D inhomogeneous nematic liquid crystal flows with variable density

In this paper, we investigate the global existence and uniqueness of solution to the 3D inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of Besov spaces. It is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data (ρ0−1,u0,n0−e3)∈M(B˙p,13p−1(R3))×B˙p,13p−1(R3)×B˙p,13p(R3) with 1≤p<6, and satisfies‖ρ0−1‖M(B˙p,13p−1)+‖u0‖B˙p,13p−1+‖n0−e3‖B˙p,13p≤cfor some small c>0 depending only on p. Here M(B˙p,13p−1(R3)) is the multiplier space of Besov space B˙p,13p(R3). Using the Lagrangian approach in Danchin and Mucha (2012, 2013) [7], [8] is the key to our results.