Incompressible Limit of the Compressible Q-tensor System of Liquid Crystals

We study the connection between the compressible Navier-Stokes equations coupled by the Q-tensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.