PERSISTENCE OF THE STEADY NORMAL SHOCK STRUCTURE FOR THE UNSTEADY POTENTIAL FLOW

This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-posedness problem of an initial boundary value problem of a nonlinear wave equation in a cornered space domain with a free boundary. The corner singularity is the essential difficulty and there is no result available even for the linear problem without the symmetry assumptions, i.e., "even"or "odd"traces vanish on the solid boundary, which allows extension from the cornered space domain to the half-space domain, as in the previous works. In this paper, we first obtain an existence result for the initial boundary value problem of linear hyperbolic equations of second order in a cornered space domain without such symmetry assumptions. The key idea is based on the construction of a new auxilliary problem, which allows us to reduce the linear problem to a new one that can be even extended to a half-space domain such that the existence of H2η-solutions can be established. However, due to the lack of the symmetry assumptions, the low regularity of the extended coefficients block us from obtaining the higher regularity of the solutions in the extended domain, which is necessary for the iteration to the nonlinear problem. In order to deal with it, new hyperbolic type and elliptic type estimates in the cornered space domain are established carefully. The results on the general linear problems can be applied to the linearized problem that we are concerned with in this paper. Due to the loss of regularity in the estimates of the linearized problem, a modified Nash-Moser iteration is developed.