Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows

In this paper, we study the stability of contact discontinuities that separate a C1 supersonic flow from a static gas, governed by the three-dimensional steady non-isentropic compressible Euler equations. The linear stability problem of this transonic contact discontinuity is formulated as a one-phase free boundary problem for a hyperbolic system with the boundary being characteristics. By calculating the Kreiss–Lopatinskii determinant for this boundary value problem, we conclude that this transonic contact discontinuity is always stable, but only in a weak sense because the Kreiss–Lopatinskii condition fails exactly at the poles of the symbols associated with the linearized hyperbolic operators. Both of the planar and nonplanar contact discontinuities are studied. We establish the energy estimates of solutions to the linearized problem at a contact discontinuity, by constructing the Kreiss symmetrizers microlocally away from the poles of the symbols, and studying the equations directly at each pole. The nonplanar case is studied by using the calculus of para-differential operators. The failure of the uniform Kreiss–Lopatinskii condition leads to a loss of derivatives of solutions in estimates.