On a Supersonic-Sonic Patch in the Three-Dimensional Steady Axisymmetric Transonic Flows

This paper focuses on the structure of solutions in a supersonic bubble arising from the three-dimensional transonic flows. Given a velocity distribution on a streamline, we construct a small smooth supersonic-sonic patch for the three-dimensional axisymmetric steady isentropic irrotational Euler equations. This patch can be regarded as the region near the upstream vertex of a supersonic bubble. The main difficulty is the coupling of nonhomogeneous terms and sonic degeneracy. To overcome it, we adopt the idea of characteristic decompositions to solve the problem in a partial hodograph coordinate system composed by the Mach angle and the potential function. By converting to the physical plane, a smooth solution for the original problem is established and the uniform regularity of solution up to the sonic curve is also discussed.