| In this dissertation, we first systematically study the properties of two-dimensional steady supersonic isentropic and adiabatic Euler Equations. We study the classical Riemann problem, and analyze the properties of the Riemann solutions to the isentropic and adiabatic Euler equations, which are essential for the interaction estimates among the nonlinear waves and for the existence and behavior of entropy solutions of the wedge problem and Lipschitz wall problem in Chapter 2 and 3, respectively.; Then, in Chapter 2, we establish the existence and nonlinear stability of supersonic Euler flows past a Lipschitz wedge when the total variation of the tangent angle functions along the wedge boundaries is suitably small. We develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution. The asymptotic stability of entropy solutions, as well as the strong shock emanated from the vertex, are also established in the flow direction.; In Chapter 3, we are concerned with the stability of supersonic vortex sheets in two-dimensional steady Euler flows over Lipschitz walls under a BV boundary perturbation. By carrying out a similar analysis as in Chapter 2, it is proved in this chapter that steady supersonic vortex sheets, as time-asymptotics, are stable in structure globally, in contrast with the prediction of the instability of supersonic vortex sheets at high Mach number as time evolves. |
