Stability Of Transonic Shocks In Supersonic Flow Past A Wedge

The present Ph.D. dissertation is concerned with stability of transonic shocks in supersonic flow past a wedge.It is well known that when a supersonic flow attacks a given body, a shock will be generated ahead of it. R.Courant and K.O.Friedrichs indicated in their famous book [19] that the generated shock will attach its edge if the given body is a straight wedge with its vertex angle less than the criticle value. The attached shocks can be supersonic( with supersonic flow behind the shock fronts) or transonic( with subsonic flow behind the shock fronts). For a given straight wedge with a vertex angle less than a critical value, these two kinds of shocks are both possible and satisfy the Rankine-Hugoniot conditions and the entropy condition. The weaker one may be supersonic, and the stronger one is transonic. Since any bistable shock does not actually occur, then the study on stability the shocks becomes extremely important. It is known that, supersonic shocks are stable, but only a little is known on stability of transonic shocks. This dissertation is concerned with the stability of transonic shocks in supersonic flow past a wedge. Taking the potential equation and the full Euler system as the model to describe the steady compressible fluids, we prove that transonic shocks are conditionally stable with respect to perturbation of the upstream flow or the wedge boundary, that is, if the upstream flow or the wedge boundary is slightly perturbed, and the perturbation satisfies some control conditions at infinity, then there exists a new position of shock front and downstream flow, which is also a perturbation of the original state.The whole dissertation is organized as follows.Chapter One is Preface. This chapter is devoted to the physical and mathematical background. The main results of this Ph.D. dissertation are also illustrated.