Sonic-supersonic solutions for the steady Euler equations

Given a smooth curve as a sonic line in the plane, we construct a local smooth supersonic solution on one side of the curve for the steady compressible Euler system of equations in two space dimensions. Our construction hinges on a new set of coordinates introduced here to handle the inherent degeneracy of the system at the sonic curve. We analyze the streamlines of the solutions to illustrate that the shock-free portion of the solutions may be combined with known results of existence of sonic-subsonic solutions of Xie and Xin [33] on the other side of the curve to form shock-free transonic flows in a channel. The existence result is also a partial generalization of the exact solution of Ringleb [28, ZAMM (1940)] toward a flexible existence.