Some Properties Of Solution To The Euler Equations With Damping And The Liquid Crystal Model

In this dissertation, the research is mainly divided into two parts:One part is devoted to the comparison of weak solutions of the isentropic compressible Euler equations with damping and its potential flow equations; The other part is concerned with the regularity criterion for the liquid crystal systems in R3. Following is the structure of this thesis.The first chapter is the introduction. On one hand, we introduce some interrelated studies on the isentropic compressible Euler equations with damping, and then give some research situation about the approximate estimates of weak solutions for some equations in hydromechanics and its approximate model. On the other hand, we introduce some research background and research situation for the liquid crystal systems. Finally we give several important definitions and lemmas.The second chapter is concerned with the difference of the weak solutions for the isentropic compressible Euler equations with damping and its approximate model. If the system of isentropic flow is assumed to be irritation, we can get the potential flow equations. As the initial value of the two equations is measurable function and its total variation is sufficiently small, we can prove that in any bounded domain the L1 norm of the difference weak entropy solution between system of isentropic flow and systems of potential flow with the same data can be bounded by the cube of total variations of the initial data.The third chapter is concerned with regularity of solution for the liquid crystal systems with fractional diffusions in R3. The regularity of solution include the regularity of global weak solution and the regularity of local strong solution. In this chapter we consider the regularity of local strong solution for liquid crystal systems. According to the difference value of the real parameter α,β, we establish several different regularity criterions to guarantee smoothness of solutions.