Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

Abstract

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C 1 smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing the point the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of piecewise constant can be replaced by some more weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions.Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of shock polar applied on the (maybe curved) shock-fronts.

Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations

“…He achieved this by introducing Lagrangian coordinates via conservation of mass and using characteristic decomposition method. This was then generalized by other authors, and paved the way for a quite well developed theory of steady subsonic flows and transonic shocks in two dimensional case (see, for example, [21,5,32,15,18,29] and references therein).…”

On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure

“…233-234], there exists a unique solution P W 2 q ./, up to an additive constant, to the problem (A.13)-(A. 18), provided that the solvability condition (A.7) holds. Furthermore, because it is the first derivatives of that are really needed, the additive constant is not important and can be chosen such that´ h 0.…”

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

“…Then the result was generalized in [46] to finitely long nozzles as small perturbations of a straight one. The uniqueness for the transonic shocks in two-dimensional steady Euler flows in a finitely long duct was considered in [26] for a class of piecewise C 1 smooth functions. For the three-dimensional case we refer to [17,20,31,42,44,45].…”

“…Many works have been done on the compressible Euler flow in nozzles. The problem of the transonic shock in a nozzle, governed by an equation of the hyperbolic-elliptic mixed-type (see [18,34]), has been extensively studied, see [5,8,9,16,17,23,32,33,40,41,53] and their references for the recent progress. The problem of the subsonic flow in nozzles was first studied in [4].…”