Contact discontinuities for 3-D axisymmetric inviscid compressible flows in infinitely long cylinders

Abstract

We prove the existence of a subsonic axisymmetric weak solution (u,ρ,p) with u=uxex+urer+uθeθ to steady Euler system in a three-dimensional infinitely long cylinder N when prescribing the values of the entropy (=pργ) and angular momentum density (=ruθ) at the entrance by piecewise C2 functions with a discontinuity on a curve on the entrance of N. Due to the variable entropy and angular momentum density (=swirl) conditions with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity, nonzero swirl, and contains a contact discontinuity r=gD(x). 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 solution and analyze the asymptotic behavior of the solution at far field.