Monotonicity and uniqueness of a 3D transonic shock solution in a conic nozzle with variable end pressure

Abstract

We focus on the uniqueness problem of a 3D transonic shock solution in a conic nozzle when the variable end pressure in the diverging part of the nozzle lies in an appropriate scope. By establishing the monotonicity of the position of shock surface relative to the end pressure, we remove the nonphysical assumptions on the transonic shock past a fixed point made in previous studies and further obtain uniqueness.

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler–Poisson system

“…To our best knowledge, all the previous studies on the steady axially symmetric flows mostly concern the case of zero swirl component. We expect not only the result of this work contributes to understand a stabilizing or instabilizing effect of vorticity to three dimensional subsonic flows of Euler-Poisson system, but also the new Helmholtz decomposition introduced in this work may open a new approach to investigate multidimensional transonic shock solutions to Euler-Poisson system or even transonic shock solutions to Euler system, which were previously studied in [7,8,9,13,14,15,20,21,23,24] and in the references therein.…”

“…In particular, the function ψ concerns the swirl (=angular momentum density). There are many other studies of axially symmetric smooth subsonic solutions to the steady compressible Euler equations [2,12,21,30]. To our best knowledge, all the previous studies on the steady axially symmetric flows mostly concern the case of zero swirl component.…”

Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments

“…For steady 2-D flows in an expanding nozzle, with appropriate pressure condition at the exit, one can obtain a shock solution by assuming that the flow depends only on the radius parameter and the position of the shock front can be uniquely determined. The well-posedness of the shock solution has been established, for instance, in [5,6,8,16,17,18]. While for steady 2-D flows in a finite flat nozzle, there exist infinite shock solutions and the position of the shock front can be arbitrary.…”

Transonic shock in finitely long nozzle with porous medium boundary condition

“…Compared with the results in [11][12][13], we do not need to require that the diverging part of the nozzle wall changes slowly. The key ingredient in the analysis of [11][12][13] is to establish the monotonic property of the shock position along the nozzle wall with respect to the exit pressure so that one can avoid the difficulties caused by the unknown position of the shock.…”

“…The key ingredient in the analysis of [11][12][13] is to establish the monotonic property of the shock position along the nozzle wall with respect to the exit pressure so that one can avoid the difficulties caused by the unknown position of the shock. After some modification of the new elaborate scheme developed in [14], we can determine the shock position together with the solution in each iteration step.…”