Steady Euler flows with contact discontinuities in infinitely long nozzles

Abstract

We investigate two-dimensional steady subsonic Euler flows with a contact discontinuity in infinitely long nozzles. An Euler--Lagrange transformation reduces the problem to a scalar quasilinear elliptic equation in divergence form on a fixed strip, with discontinuous coefficients and the contact discontinuity conditions incorporated into the weak formulation. By solving this elliptic problem, we establish existence, uniqueness, and regularity of the flow on the maximal interval of wall amplitudes containing the straight-nozzle configuration. These results apply to general piecewise smooth subsonic upstream data and dispense entirely with the monotonicity, convexity, and small-perturbation assumptions imposed in earlier studies. In particular, the contact discontinuity curve is of class $C^{2,\alpha }$. We further investigate the downstream asymptotic behavior and show, under the corresponding decay assumptions on the nozzle walls, that the convergence rate of the flow matches that of the walls.