Structural Stability of Supersonic Contact Discontinuities in Three-Dimensional Compressible Steady Flows

Abstract

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 in [Y.-G. Wang and F. Yu, J. Differential Equations, 255 (2013), pp. 1278--1356] 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 background states and initial data. Thus, to use the tame estimates to study the nonlinear problem we adapt the Nash--Moser--Hörmander iteration scheme to conclude that supersonic contact discontinuities in three-dimensional compressible steady flows satisfying the stability conditions of Wang and Yu are structurally nonlinearly stable at least locally in space.