Structural Stability of Supersonic Contact Discontinuities in   Three-Dimensional Compressible Steady Flows
  In this paper, we study the structurally nonlinear stability of supersonic contact discontinuities in three-dimensional compressible isentropic steady flows. Based on the weakly linear stability result and the $L^2$-estimates obtained by the authors in J. Diff. Equ. 255(2013), for the linearized problems of three-dimensional compressible isentropic steady equations at a supersonic contact discontinuity satisfying certain stability conditions, we first derive tame estimates of solutions to the linearized problem in higher order norms by exploring the behavior of vorticities. Since the supersonic contact discontinuities are only weakly linearly stable, so the tame estimates of solutions to the linearized problems have loss of regularity with respect to both of background states and initial data, so to use the tame estimates to study the nonlinear problem we adapt the Nash-Moser-H\"ormander iteration scheme to conclude that weakly linearly stable supersonic contact discontinuities in three-dimensional compressible steady flows are also structurally nonlinearly stable. 
Ya-Guang Wang, Fang Yu
math.AP