Cauchy Problem For Modified Chaplyain Gas Equations

The isentropic Euler equations for modified Chaplygin gas, whose pressure law is a combination of ideal fluid and the generalized Chaplygin gas, is called the modified Chaplygin gas equations which can be used to understand the mysterious behavior of dark matter and dark energy in the universe. In recent years it has received extensive attention of physics and mathematics. This thesis mainly studies Cauchy problem of the modified Chaplygin gas equations.Firstly, we consider interaction of two rarefaction waves for the one-dimensional mod-ified Chaplygin gas equations. By using the characteristic and phase plane analysis method, the existence and uniqueness of smooth solution for the Goursat problem de-veloped on the penetration domain of two rarefaction waves is obtained, and the collision results are presentSecondly, we study the Cauchy problem of the one-dimensional modified Chaplygin gas equations. Under some reasonable assumptions of initial values, with the aid of characteristic analysis and uniformly a priori estimate, we establish sufficient conditions of the existence and uniqueness of global smooth solution as well as continuous and piecewise smooth solution for the Cauchy problem. We also obtain the corresponding results for ideal fluid and Chaplygin gas, as well as generalized Chaplygin gas pressure law, etc. Finally we study the brow-up solution, and estimate the life span of classical solution.