Stability of contact discontinuities in three-dimensional compressible steady flows

Abstract

In this paper, we study the stability of contact discontinuities in three-dimensional compressible isentropic steady flows. By developing Kreiss, Coulombel and Secchiʼs arguments for a boundary value problem of hyperbolic equations involving poles in coefficients derived from the linearized problem at a supersonic contact discontinuity, we obtain a necessary and sufficient condition for the linear weak stability of contact discontinuities and observe that only weak Kreiss–Lopatinskii condition holds. Both of planar and non-planar contact discontinuities are studied. The energy estimates of solutions to the linearized problem at a planar contact discontinuity are obtained in the weakly stable region, by constructing the Kreiss symmetrizers microlocally away from the poles of the coefficient matrices, and studying the equations directly at each pole. The linear stability of non-planar contact discontinuities is studied by developing the argument of the planar case and using the calculus of paradifferential operators. The failure of the uniform Kreiss–Lopatinskii condition results in a loss of tangential derivatives of solutions to the linearized boundary value problem with respect to the source terms and boundary data.