CONTACT DISCONTINUITIES FOR 2-DIMENSIONAL INVISCID COMPRESSIBLE FLOWS IN INFINITELY LONG NOZZLES

Abstract

We prove the existence of a subsonic weak solution (u, rho, p) to a steady Euler system in a two-dimensional infinitely long nozzle when prescribing the value of the entropy (= p/rho gamma) at the entrance by a piecewise C-2 function with a discontinuity at a point. Due to the variable entropy condition with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity and contains a contact discontinuity x(2) = g(D)(x(1)). We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a piecewise smooth subsonic flow with a contact discontinuity and nonzero vorticity. We also analyze the asymptotic behavior of the solution at far field.