The Low Mach Number Limit Of Compressible Hydrodynamic Flow Of Liquid Crystals

This thesis is devoted to the low Mach number limit of compressible hydrodynamic flow of liquid crystals. Based on the convergence-stability principle, it is shown that, for the Mach number sufficiently small, the Cauchy problem of compressible liquid crystal flow has a unique smooth solution on the(finite) time interval where the incompressible liquid crystal flow exists. Namely, the compressible model converges to the incompressible model as the low Mach number tends to zero.Moreover, the sharp convergence order is also obtained. The thesis is divided into four chapters.In the first chapter, we first introduce the equations of liquid crystals, and deduce the limit problem formally. Then we briefly give the development of related equations. Finally, the main result is stated.In the second chapter, we bring in the Sobolev space and state the main properties about it as well as some inequalities which will be useful in the following proofs.In the third chapter, based on the continuation principle for hyperbolic-parabolic composite systems, we obtain the existence and uniqueness of the solution to the compressible model. Then,from the error energy estimates, we justify the asymptotic process rigorously by using the energy methods.In the forth chapter, we take on the further prospect on the problem.