Uniform well-posedness and low Mach number limit to the compressible nematic liquid crystal flows in a bounded domain

We prove global-in-time and uniform-in-ϵϵ of the strong solutions to the 3D3D compressible nematic liquid crystal flows in a bounded domain, where ϵϵ is the Mach number. Consequently, we obtain the strong solution of compressible nematic liquid crystal model that converges to that of incompressible nematic liquid crystal model. This is the first result on the low Mach number limit for compressible nematic liquid crystal flows in 3D3D bounded domain. Our proof relies on the dedicated estimates on the solutions and the subtle use of the boundary conditions.