Existence and regularity of solutions of a supersonic-sonic patch arising in axisymmetric relativistic transonic flow with general equation of state

In this article, we prove the existence and regularity of a smooth solution for supersonic-sonic patch arising in modified Frankl problem study three-dimensional axisymmetric steady isentropic relativistic transonic flows over symmetric airfoil. We consider general convex equation state which makes complicated as well interesting context theory flows. Such type patches appear many an airfoil flow near nozzle throat. Here main difficulty is coupling nonhomogeneous terms due to axisymmetry sonic degeneracy flow. However, using well-received characteristic decompositions angle variables partial hodograph transformation plane first. Further, by inverse construct physical discuss uniform up associated curve. Finally, also