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

TL;DR

This paper proves the nonlinear structural stability of supersonic contact discontinuities in three-dimensional steady compressible flows, extending linear stability results through advanced iterative techniques.

Contribution

It introduces a novel application of the Nash-Moser-Hörmander iteration scheme to establish nonlinear stability based on previous linear stability and tame estimates.

Findings

Weakly linearly stable discontinuities are also nonlinearly stable.

Derived higher-order tame estimates for linearized solutions.

Extended stability results to three-dimensional 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 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.