Persistence of Steady Structures under Unsteady Perturbations

The aim of this project is to develop the rigorous mathematical theory on the persistence of steady structures under unsteady perturbations for the compressible irrotational Euler flow. In particular, it will mainly focus on the stability of static solutions near a corner as well as the stability of oblique shock solutions over a wedge under unsteady perturbations, governed by the two dimensional unsteady potential flow equation, and other related problems. As said by Courant-Friedrichs in, "Whether or not a flow compatible with the boundary condition occurs depends moreover on its stability", so the study of the stability of solutions with boundary conditions for the compressible Euler flow is necessary and important.Up to now, there are lots of literatures on the steady compressible Euler flows motivated by the problems introduced in. As said by von Karman in the meeting chaired by von Neumann in, a steady motion "can occur only as a limiting case" of a physical process. Therefore, it is an important topic to study the stability of such steady structures under unsteady perturbations, i.e., to investigate whether the structures will be persistent, at least in a short time, under such perturbations. In this project, we will first try to establish the stability of static solutions near a corner under an unsteady perturbation. Next, we will consider the stability of steady oblique shock solutions over a wedge under an unsteady perturbation and other related problems. Mathematically, these problems can be formulated into an initial boundary value problem on a cornered space domain governed by a nonlinear hyperbolic equation of second order. The crucial difficulty is that the boundary of the space domain is not smooth, especially coupled with other difficulties such as nonlinearity, free boundaries, etc. In fact, Osher gave examples in [38] to show that the initial-boundary value problem governed by hyperbolic equations on a cornered space domain may be ill-posed. On the other hand, there are plenty of problems in gas dynamics which can be formulated as hyperbolic problems in non-smooth domains. While the mathematical theory for such problems is far away from satisfied. We believe the ideas and techniques developed in this project would be useful both to deal with these problems with similar difficulties and to the mathematical theory on the multi-dimensionalconservation laws. Other mathematicians in the fields of partial differential equations and gas dynamics would be interested in this work too. We have been working on the related problems for several years. We proved the dynamic stability of two-dimensional steady normal shock in and the problem for the three-dimensional case in. We also proved the stability of three dimensional weak transonicshock over a wedge in; the stability of three dimensional rarefaction wave with a contact discontinuity in; the stability of supersonic contact discontinuity in a finitely long nozzle in [29]; the stability of transonic contact discontinuity in a finitely long nozzle in [30]. Inthis project, we will mainly focus on a further development of the technics of proving the obtained results, to study the mentioned objectives.