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

TL;DR

This paper proves the existence and smoothness of solutions for a complex relativistic transonic flow problem involving a supersonic-sonic patch in axisymmetric conditions, addressing challenges of degeneracy and coupling.

Contribution

It introduces a novel approach combining characteristic decompositions and hodograph transformations to establish solution regularity in relativistic transonic flows with a general convex equation of state.

Findings

Existence of smooth solutions for the relativistic transonic flow problem.

Regularity of solutions up to the sonic curve.

Construction of solutions in the physical plane from the hodograph plane.

Abstract

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