Existence and Stability of Vortex Sheets and Entropy Waves for the Euler Equations for Steady Compressible Flows

In this dissertation, we first give a short introduction to the topic of research in Chapter 1. Then we introduce some background knowledge and necessary tools for our problems in Chapter 2. It mainly includes a brief discussion about the Euler equations of gas dynamics, the Wave front tracking method, preliminary BV theory and approximation of Lipschitz flows. In Chapters 3--6, we establish the global existence in BV and L1-stability of vortex sheets and entropy waves for the Euler Equations for steady supersonic and transonic Euler flows. These results indicate that steady supersonic and transonic vortex sheets/entropy waves, as time-asymptotics, are stable in structure globally, in contrast with the prediction of the instability of supersonic and transonic vortex sheets at high Mach numbers as time evolves.;Particularly, in Chapter 3, we are concerned with the stability of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls under a BV boundary perturbation. In this chapter, we establish the L1 well-posedness for two-dimensional steady supersonic Euler flows containing a strong vortex sheet/entropy wave past a Lipschitz wall whose boundary slope function has small total variation, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is small. In this case, both the Lipschitz wall and incoming flow perturb the background strong vortex sheet/entropy wave, and the waves reflect after interacting with the strong vortex sheet/entropy wave and the wall boundary. We first establish the existence of solutions in BV, when the incoming flow perturbation of the background strong vortex sheet/entropy wave has small total variation by the wave-front tracking method and then establish the L 1-stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated both by the wall boundary and from the incoming flow to develop a Lyapunov functional between two solutions containing strong vortex sheets/entropy waves, which is equivalent to the L1-norm, and prove that the functional decreases in the flow direction. Then the L1-stability is established, so is the uniqueness of the solutions by the wave-front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.;Then, in Chapter 4, we are establishing the stability of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows under a small BV perturbation of the initial data. More specifically, in this chapter, we establish the L 1 well-posedness for two-dimensional steady supersonic Euler flows containing two strong vortex sheets/entropy waves when the total variation of the incoming flow is small. In this situation, a uniform supersonic flow with two straight vortex sheets/entropy waves is perturbed at t=0 and for t>0 weak waves interact and reflect off the two strong vortex sheets/entropy waves. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L1 stability of the solutions with respect to the incoming flows. Specifically, we incorporate the nonlinear waves generated by weak and strong waves interactions to develop a Lyapunov functional between two solutions containing the two strong vortex sheets/entropy waves, which is equivalent to the L 1 norm, and prove that the functional decreases in the flow direction. Then the L1-stability is proved, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also attained. For two-dimensional steady supersonic Euler flows past a convex corner, there may appear a contact discontinuity separating the supersonic flow from gases at rest (hence subsonic flow). In Chapter 5, we prove that such transonic contact discontinuity is structurally stable under small perturbation in the space of bounded variation of the upstream supersonic flow. This is essentially a free boundary problem of a non-strictly hyperbolic system of conservation laws. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e. contact discontinuity) is proved by using the front tracking method. In Chapter 6, we further show such transonic contact discontinuity is unique and the L1-stability with respect to small perturbation of the incoming supersonic flow also holds in the Lagrangian coordinates.