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We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the coordinate transformation of Euler-Lagrange type and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the Euler system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for a second-order nonlinear elliptic equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or quivalently the entropy function along the shock-front downstream, all the physical variables can expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at the infinite exit can be uniquely determined with uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the second-order equations, and then update the entropy function to prove that it has a fixed point. Another advantage in our analysis here is in the context of the real full Euler equations so that the solutions do not necessarily obey Bernoullis law with a uniform Bernoulli constant, that is, the Bernoulli constant is allowed to change for different fluid trajectories.
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We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.
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: $(rmnum{1})$ the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; $(rmnum{2})$ the oblique transonic shocks attached to an infinite wedge; $(rmnum{3})$ 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 $L^{infty}$ 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.
This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observation and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.
In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs are based on the study of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function, which is shown to satisfy a semilinear elliptic equation in the whole domain. In exterior or punctured domains, the method of moving planes is applied to some almost circular domains located between some streamlines of the flow, and the radial symmetry of the stream function is shown by a limiting argument. The paper also contains two Serrin-type results in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary and the domains are then proved to be disks or annuli.
This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzles boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution.
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