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

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

“…Kuz'min discussed the existence and uniqueness of the solution of the modified Frankl problem for a linearized version of the von Karman equation in a finite domain; see [37,38]. Recently, Hu and Li established the existence and regularity of solutions of a sonic-supersonic patch extracted from a modified Frankl problem for 2-D steady isentropic Euler equations and 3-D steady axisymmetric isentropic Euler equations with ideal gas [39,40]. The recent development in the context of transonic flows has motivated us to ask naturally whether such analysis can be performed for more complicated mixed-type systems for a more general equation of state or not.…”

“…To overcome these complexities, we use partial hodograph transformation where the independent variables are Mach angle and the potential function to convert the governing axisymmetric relativistic Euler system into a new degenerate hyperbolic system. The idea of choosing such independent variables is taken from a very recent work of Hu [40]. However, unlike the Mach-flow angle plane, the reduced hyperbolic equations in our case do not form a closed system and additional equations are needed to be added to the system in order to close the system which makes the current problem even more complicated.…”