In this paper, we study the transonic shock problem for the full compressible
Euler system in a general two-dimensional de Laval nozzle as proposed in
Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New
York, 1948): given the appropriately large exit pressure pe(x), if the upstream flow
is still supersonic behind the throat of the nozzle, then at a certain place in the
diverging part of the nozzle, a shock front intervenes and the gas is compressed and
slowed down to subsonic speed so that the position and the strength of the shock
front are automatically adjusted such that the end pressure at the exit becomes pe(x).
We solve this problem completely for a general class of de Laval nozzles whose
divergent parts are small and arbitrary perturbations of divergent angular domains
for the full steady compressible Euler system. The problem can be reduced to solve
a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system.
One of the key ingredients in the analysis is to solve a nonlinear free boundary
value problem in a weighted Hölder space with low regularities for a second order
quasilinear elliptic equation with a free parameter (the position of the shock curve
at one wall of the nozzle) and non-local terms involving the trace on the shock of
the first order derivatives of the unknown function.