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A variational inequality formulation for transonic compressible steady potential flows: Radially symmetric transonic shock

We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual Rankine-Hugoniot jump conditions for a shock. By means of example we validate our formulation, and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.


Introduction
In an effort to understand steady transonic potential flow in compressible gas dynamics we introduce a variational inequality formulation. To validate our formulation, we build a model problem with an exact solution, and establish the necessary and sufficient conditions for the existence of transonic shock in our model using the new variational scheme. While we study a simple model system at this point, our variational inequality formulation is applicable to more general multidimensional settings.
It is well known that the steady compressible Euler equation is the governing model system for various applications such as a flow over a profile (a steadily moving aircraft). Consequently understanding boundary value problem in multidimensional gas dynamics has always been of interest to mathematicians and engineers alike. The classical book by Courant and Friedrichs [12] contains many more such applications that involve steady compressible Euler equations in multidimensional settings. The following quoted from Morawetz [24] has been a long standing open problem: In the study of transonic flow, one of the most illuminating theorems to prove would be: Given an airfoil profile and a continuous two dimensional irrotational transonic compressible inviscid flow past it with some given speed at infinity, there does not exist a corresponding flow with a slightly different speed at infinity.
In order to answer Morawetz's question, one must face issues arising from the multi-dimensionality of the boundary value problem. For a given subsonic constant speed at infinity, Shiffman [27] used the direct method of calculus of variations to establish an existence theorem for the subsonic flow in the entire region around a profile. The existence and uniqueness results for this subsonic flow over the profile is completed by Bers [1], a stronger uniqueness result by Finn and Gilbarg [16] and for higher dimensions by Dong and Ou [14]. All these results rely upon the condition that the given speed at infinity must be sufficiently subsonic. To our best knowledge, it is unknown whether a critical point for the variational problem exists for any arbitrary speed at infinity.
In fact subsonic flow at infinity does not necessarily remain subsonic everywhere; it can become transonic and create a shock over a convex profile [12]. For simpler configurations when the supersonic flow at infinity is given, a shock wave is formed in front of the profile. In general the shock location is not known apriori. In other words a steady transonic flow that contains a shock is a free boundary problem.
We propose to tackle the transonic problem using a variational inequality formulation with the shock location represented by a free boundary to be determined as a part of the solution. Brézis and Stampacchina [2] showed that the problem of infinite plane flow around a symmetric convex profile can be reduced to a variational inequality in the hodograph plane, and established the existence of a subsonic flow. Shimborsky [28] utilized the variational inequality in the hodograph plane setting studied in [2] to establish a subsonic symmetric channel flow (Laval nozzles). Both [2,28] results focused on subsonic flow. Later a few attempts have been tried to extend the formulation to cover supersonic flow as well as shock formation [17,18] and the references therein. For example in [18], an artificial entropy condition was imposed on admissible functions so as to ensure the existence of a minimizer to a functional. We note, however, that it is not clear a priori whether the minimizer is a solution of the transonic problem. Furthermore, their variational formulations do not provide any information how to locate a shock, and whether the Rankine-Hugoniot jump conditions are satisfied.
In this paper, we consider a configuration for the given supersonic upstream flow be isentropic. Since the disturbance caused by an immersed body can only propagate downstream, the supersonic flow will not be affected by the formation of a shock. In general the flow will not be isentropic behind the shock (it can be, if the entropy increases by the same amount across the shock). In other words the replacement of the full system governing compressible fluid flow by the transonic equation induces inconsistency. Such an inconsistency usually results in not satisfying all the three Rankine-Hugoniot conditions across a shock.
The main goals of this paper is to establish a correct variational inequality formulation incorporating a transonic free boundary, and find the corresponding critical point which satisfies the Rankine-Hugoniot jump conditions across the transonic shock. More precisely, we consider a configuration for which the supersonic flow upstream of the shock is known, so that one can find a potential function, denoted by ϕ sup , which gives rise to this simple flow. We next consider the shock as a free boundary where the supersonic flow in the direction normal to the shock makes an abrupt change to a subsonic flow. Since the flow behind the shock is assumed to be potential, it suffices to find the corresponding subsonic potential function ϕ sub , satisfying ϕ sub ≥ ϕ sup with ϕ sub = ϕ sup at the shock. This obstacle problem is a free boundary problem that can be formulated in a variational inequality setting. That is, we perform the variational calculus on a closed convex set in a function space rather than on the whole function space. Such a formulation will automatically guarantee the continuity of tangential velocity across a shock, which is a physical requirement (see equation (81.6), p.318, [23]). The remaining hurdle is to ensure the natural boundary condition associated with the variational functional agrees with some (or a linear combination) of the Rankine-Hugoniot conditions. As remarked earlier one cannot expect all such conditions to be satisfied as the flow behind the shock may not be isentropic.
To test the correctness of our variational inequality formulation we consider a model problem that involves only radial flow. This problem admits an exact solution which can be used to validate our formulation. With the actual flow remains potential behind the shock, it is remarkable that our variational formulation captures all the Rankine-Hugoniot conditions in this case so that its critical point, which solves the transonic equation, is also an exact solution to the full system of compressible fluid flow with shock wave. This is only possible since in our formulation we have allowed entropy to change behind the shock. We note, however, that there are many works that assume the same constant entropy in front and behind the shock for simplicity. This automatically eliminates any chance of capturing the exact solution to the full system of compressible fluid flow in variational formulation.
More importantly, the model problem shows clearly that the critical point of our variational inequality formulation corresponds to a shock solution to the transonic equation is a saddle point. Our variational formulation ensures that the saddle point satisfies one Rankine-Hugoniot condition and a linear combination of the remaining two. Thus one can think of it satisfying two such conditions. And if the flow is actually potential behind the shock, like in the model problem, one can have all three jump conditions being satisfied.
The paper is organized as follows. In section 2 we present an overview of the steady flow of compressible system. We pay a particular attention to the physical entropy constant which is typically normalized and simplified elsewhere. Section 3 comprises our model problem and its exact solution, see Subsection 3.1. This is also substantiated by numerical results. In Section 4, we discuss the variational inequality formulation and the free (transonic shock) boundary condition. We show that its critical point satisfies the mass conservation equation and some jump conditions on the free boundary that corresponds to the transonic shock. We establish the existence of the saddle point; the necessary and sufficient conditions of such an existence agree with that obtained in Section 3.1. We conclude the paper by presenting numerical results that demonstrate the transonic solution being the saddle node of the variational function in Section 6.
We believe our results will serve as a vehicle for understanding transonic flows in particular the long standing open problem of the flow over a profile.

Review of governing equations
Conservation laws are governing principles in gas dynamics. While there are many references dedicated to explain compressible flow, we give a short survey on the conservation principles which will be useful for our variational formulation.
The two dimensional steady state compressible Euler system for an ideal gas in conservation form reads Here ρ is the density, p is the pressure, u = (u 1 , u 2 ) is the velocity vector, q = u 2 1 + u 2 2 is the gas speed and ι is the enthalpy per unit mass of the gas.
Entropy of any ideal gas particle remains constant during its motion except when crossing a shock. As a consequence one has p = kρ γ (2.5) for some positive constants 1 < γ < ∞ (typically 1 < γ ≤ 5/3) and k. Physically, the adiabatic exponent γ is the ratio of specific heat capacity per unit mass at constant pressure to that at constant temperature for the gas, and k is a parameter that depends only on the local entropy [12, p.6]. In case when upstream flow is homogeneous, the entropy, and hence k, are the same everywhere in this spatial region. After a gas particle crosses a shock, its entropy increases and so does the parameter k. With (2.5) the enthalpy of an ideal gas reads where c is the (local) sound speed defined by the above equation.
We consider a potential flow that is supersonic far upstream; the potential function is denoted by ϕ sup so that u = −∇ϕ sup . The negative sign in the last equation is convenient in our variational inequality formulation that will be discussed later.
The subsequent steady adiabatic flow satisfies the Bernoulli's law where q is the speed of the flow andq 0 is a known positive constant along a streamline determined by the given upstream flow. In general the Bernoulli's constant 1 2q 2 0 may have different values along different streamlines, and the same is true for its entropy. Since the upstream flow is homogeneous in our case, 1 2q 2 0 is the same on every streamline. Where a stationary shock is formed, one obtains from (2.1)-(2.4) the Rankine-Hugoniot jump conditions Here [·] in the above equations denotes the jump across the shock and v is the velocity component normal to the shock. Thus there is no jump in v 2 /2 + ι across a shock. Since the tangential speed across a shock is continuous (which can be considered as a 4 th Rankine-Hugoniot condition), this implies q 2 /2 + ι is continuous across a shock and hence equals to the same constantq 2 0 /2 everywhere immediate behind a shock.
After crossing the shock, the flow may no longer be potential. As the Bernoulli's equation requires q 2 /2 + ι be constant along each individual stream line for a steady state, one can conclude that (2.7) holds everywhere before and after the shock with the same Bernoulli's constantq 2 0 /2 as in upstream flow. As a consequence, we can replace the energy equation (2.4) and its jump condition (2.10) by (2.7).
We now adopt the usual simplification that the flow behind a shock is potential with its velocity vector u sub = −∇ϕ sub for some velocity potential function ϕ sub to be determined. Even the shock is a strong one, this may still be a good approximation so long as the increase in entropy across the shock is more or less uniform, as illustrated by the model problem below. We like to emphasize that even we use the superscript sub to designate the variables behind a shock, the flow may not be necessarily subsonic everywhere.
From the Bernoulli's law (2.7) and the flow being assumed to be potential, we can now define both c 2 and ρ as functions of the speed |∇ϕ| everywhere and with ϕ being ϕ sup or ϕ sub . The entropy increase across the shock results the changes in k that increases from k sup to a larger value k sub after crossing the shock. It is important to note a major difference between (2.7) and (2.12): the potential flow assumption has been built in the latter equation already. If the flow behind the shock were truly potential, the momentum equations will be automatically satisfied.
Consequently, the continuity equation (2.1) behind the shock can be written in an equation of a potential flow ϕ div(ρ ∇ϕ) = div(ρ(|∇ϕ| 2 ) ∇ϕ) = 0. (2.13) where ρ satisfies (2.12). This continuity equation can be written in a non-divergence form, and denoted by, and thus the operator Q is elliptic whenever c 2 > |∇ϕ| 2 , that is, which is equivalent to subsonic flow. Our model problem, which will be used to validate our variational inequality formulation, concerns a radial fluid flow without angular motion. Let r be the radial distance from the origin and and v denote the radial velocity (positive in the increasing r direction) which is a function of r only. It is convenient to write the Euler system in such a coordinate system: With the assumption of potential flow behind the shock, one only needs to study (2.16) and (2.12).
Since dι dr = 1 ρ dp dr , it is easy to verify the momentum equation (2.17) is an easy consequence of (2.12). Hence the solution to the transonic equation in this case turns out to be exact for the Euler system, because radial flow behind a shock is always potential.

A model problem
Let a, R be given positive constants with a < R. We consider a simple configuration that involves radial flow in the annulus {x ∈ R 2 : a < |x| < R} with a circular shock inside. We study (2.16) and (2.12) under the assumption that the flow is potential behind the shock. Since radial flow is always potential, a solution to these two equations will be the exact solution for the Euler system (2.16)- (2.18). Impose the boundary conditions with both v 0 , v a > 0. The given velocities at both r = a and r = R are negative since they point towards the origin. Because of (2.16) and (2.8), the total flux is a constant everywhere, regardless to be in front or behind the shock. Combining this information with (2.12) and the boundary conditions at r = R, we obtain That is, v and ρ can be written in terms of r and k by using We let the boundary conditions (3.1) and (3.2) satisfy Employing (3.3) and the boundary condition (3.2), we conclude that We note that k a depends only on the given boundary conditions, but not on any other features of the subsonic solution behind the shock. One expects the change in entropy k may, in general, depends on the location of the shock.

Exact solutions to the model system
The radially symmetric model has a stationary shock which satisfies all the Rankine-Hugoniot conditions under appropriate conditions, and the exact solution will be found explicitly; in particular we need to pin down the shock location. Later in Section 5, we obtain a solution from our variational formulation, which is in exact match with the explicit solution obtained in this section. This validates our methodology. For weak shock it is known that the change in entropy is of third order of the pressure change [23, p.323]. This is the rationale behind many transonic potential flow studies for setting the entropy constant k to stay the same after passing the shock. However we test our our variational inequality formulation without the weak shock assumption in the model. To follow the change in k, from (3.3) we have To satisfy the increase of entropy after the shock, we need k a > k 0 . Therefore a necessary condition for a shock wave to exist is While the second inequality is immediate, in order to have a transonic flow, we choose a subsonic v a to be large enough so that k a > k 0 .
In the supersonic region in front of the shock, we have k = k 0 . With any given r > a, one can solve for a unique supersonic v from (3.9) and write −v = V sup (k 0 , r) (recall that v < 0 so that V sup is a positive function). Correspondingly we find ρ = ρ sup (k 0 , r) by substituting k = k 0 and v = −V sup (k 0 , r) in the second equation in (3.3). Similarly in the subsonic region behind the shock, we have k = k a and we can solve for a unique subsonic −v so that −v = V sub (k a , r), and ρ = ρ sub (k a , r) for any r > a.
Across the transonic shock (if it exists), denoted by Σ, the Rankine-Hugoniot conditions take the form of where ρ 1 = ρ sup (k 0 , r s ), −v 1 = V sup (k 0 , r s ) and ρ 2 = ρ sub (k a , r s ), −v 2 = V sub (k a , r s ). Note that (3.12) and (3.14) are satisfied because of the first equation in (3.3) and our choice of Bernoulli's constant being unchanged. If there exists r = r s satisfying (3.13), we have a transonic shock conserving mass, momentum and energy across it. We find the transonic shock that conserves the momentum by including the changes in entropy, that is the entropy constant k is changed across the shock. 1 We observe that the momentum flux can be written as Let us denote Recall that −v = V sup (k 0 , r) in front of the shock and −v = V sub (k a , r) behind the shock. We then consider H as a function of r: where we have, and sometimes will again, suppressed the dependency of H on k for notational simplicity. In the (r, H) coordinates we let point A be (a, H sub (k a , a)) and point C be (R, H sup (k 0 , R)), which can be computed easily using the given boundary conditions at r = a and r = R, respectively. Figure 2 depicts the curves H sup (k 0 , r) and H sub (k a , r) as functions of r. Then there must be the corresponding end points B = (a, H sup (k 0 , a)) and D = (R, H sub (k a , R)), respectively. Thus the shock position r = r s corresponds to a point at which the curves H sup and H sub cross. If they do cross, the intersection point has to be unique. The uniqueness can be easily seen as follows. From (3.3) and p = kρ γ , one easily verifies that which is a decreasing function for 0 ≤ |v| ≤q 2 0 . Since V sup > c * > V sub , we have p sub > p sup . Thus as a consequence of (2.17), we have dH sub /dr = p sub > p sup = dH sup /dr for all r, which leads to a unique crossing point. We now turn to the existence of the intersection of the curves. Let conditions (3.4), (3.5) and (3.11) hold. Since the curve H sub always has a steeper slope than the curve H sup , for given A and C the necessary and sufficient conditions for crossing are: These are equivalent to Further simplifications lead to Thus (3.19) and (3.20) are the necessary and sufficient conditions for the curves H sup (k 0 , r) and H sub (k a , r) to cross at r = r s with a < r s < R. Once we know the position of the shock, the exact solution is completely known and satisfies all the Rankine-Hugoniot conditions and the Euler system (2.16)-(2.18). We point out that conditions (3.4), (3.5), (3.11) do not imply (3.19) and (3.20). In fact using the Rankine-Hugoniot jump conditions, it is shown in [12, (67.03), p.148] that the density compression ratio ρ 2 /ρ 1 due to a shock in an ideal gas is always restricted to  where we denote V 1 = V sup (k 0 , r s ) and V 2 = V sub (k a , r s ), and consequently Employing the Prandtl's relation [12, page 147] at the shock, i.e.
which implies yet another constraint on v a , v a ≥ V 2 > c * γ − 1 γ + 1 := ν. Here we have used |v| being a decreasing function of r in the subsonic regime behind the shock. This can be easily seen from Figure 1 by observing the horizontal line k = k a intersects the curves at successively smaller subsonic |v| with increasing r. There exists v a satisfying (3.4), (3.5) and (3.11), but not (3.22). We emphasize that condition (3.22) is still a necessary condition for the H sup and H sub curves to cross. The necessary and sufficient conditions are (3.19) and (3.20). While data leading to Figure 2 shows that the H-curves cross, Figure 3 depicts a counter example in which v a satisfies all the constraints except for (3.22). More precisely we have v a = √ 0.4 < ν = √ 0.5. As predicted the H-curves do not cross.

Variational inequality formulation
We now introduce a variational inequality formulation to solve the one-dimensional model problem and show that the solution of the variational inequality is the same as the exact solution we found in Subsection 3.1. The formulation is first derived for a general multidimensional case, and then reduced to the model problem of the radial flow.
Since the upstream supersonic flow is potential, there exists a ϕ sup such that the velocity u = −∇ϕ sup . We solve for ϕ sup by using the Euler system with the boundary condition at the outer boundary of the domain ( the entering flow). For example, in the one-dimensional radial flow model, we use the radial supersonic solution that we have found in Subsection 3.1. Hence we have radial velocity −v = ϕ sup r = V sup (k 0 , r), density ρ sup (k 0 , r), pressure p sup = k 0 (ρ sup (k 0 , r)) γ and the total inward flux 2πM 0 = 2πρ 0 v 0 R. Without loss of generality we let ϕ sup = 0 at r = R.
Let the supersonic flow be converging, that is, the flow moves inward toward the origin, starting from r = R hits a smooth inner boundary denoted by Γ, which can be considered as the circle r = a in Section 3. If there is no shock, we assume that V sup (k 0 , r) remains supersonic before it hits Γ.
Let Ω be the domain in R N for an arbitrary N ≥ 1 enclosed by Γ and A R ≡ B R \ Ω. Define the non-empty closed convex set K := {w ∈ W 1,∞ (A R ) : w ≥ ϕ sup , |∇w| ≤q 0 } and c 2 := γ−1 2 (q 2 0 − |∇w| 2 ) as in (2.11). (This is the sound speed as given by the Bernoulli's law if w is the velocity potential for the flow; but at this point w is just any element in K and c 2 becomes a definition.) We introduce the functional I : K → R such that where Ω ≡ {x ∈ A R : w(x) > ϕ sup (x)}, m(s) is a given smooth function on Γ satisfying Γ m(s)ds = 2πM 0 , and k a > k 0 is a positive constant to be specified later. (In the one dimensional case, k a is given by (3.7).) Define the coincidence set as A R \ Ω. Let ϕ be a critical point of the functional I. We now derive the jump condition across Σ, which is the part of ∂(A R \ Ω) that lies in the interior of A R , and study its relation to the Rankine-Hugoniot jump conditions. It is easy to see that there is no jump in the tangential derivatives of ϕ and ϕ sup across Σ. For the one dimensional case we confirm later that the jump conditions imply the rest of the Rankine-Hugoniot conditions as well.
We use ∂/∂n to denote the normal derivative in the increasing r direction on the shock Σ, and in the outward normal direction for the remaining two boundaries ∂B R and Γ. Taking the Fréchet derivative, we have a m(s)δϕ ds .   ∂ϕ sup /∂n−∂ϕ/∂n δϕ, depending on whether we are dealing with Ω or A R \ Ω, we can also derive the same formula for δI. We also refer Courant and Hilbert [13, p.260-262] for variable domains for the calculus of variations.
Assuming a certain smoothness of ϕ and the shock Σ, the first term in the last equation becomes In Ω, we define ρ by which is inspired by (2.11), and p = k a ρ γ ; hence We can now rewrite δI as a m(s)δϕ ds .
Assume there is only one continuous shock that encloses Γ, by integrating div(ρ∇ϕ) = 0 we have Hence the total flux will be conserved. We have now established the following theorem.
Theorem 4.2 Assume the existence and smoothness of a critical point ϕ ∈ K for the functional I in (4.1) and the coincidence set (where ϕ = ϕ sup ) has a smooth boundary Σ. Then the critical point ϕ satisfies where m is a given function that satisfies (4.7).

Characterization of the critical point
We note that for general multi-dimensional configurations, the constant k a may not be determined uniquely, and may be with the same constant k 0 = k a in the entire region when the shock is weak. Moreover we assume that the prescribed v a will make k a satisfy (3.11). We need to ensure that V sup (k 0 , r) remains supersonic for a ≤ r ≤ R. As r decreases from R, ρ|v| has to increase to maintain a constant flux M 0 . In the supersonic regime, |v| has to decrease for ρ|v| to increase. Once |v| reaches the sonic speed c * before reaching r = a, the local flux ρ|v| cannot increase further and there is no supersonic flow afterward. Hence in order that ϕ sup to be defined on the entire region a ≤ r ≤ R, we impose where ρ * is the density at the sonic speed. Since ρ * = ( c 2 * γk0 ) 1/(γ−1) , the above relation is equivalent to which is already satisfied by imposing condition (3.11). If we use dx dy = rdrdθ in the derivation in Section 4, or simply convert the equations to polar coordinates at the end, the governing transonic flow equation for radially symmetric solutions in Theorem 4.2 becomes (ρϕ r r) r = 0 (5.2) in {x ∈ A R : ϕ > ϕ sup , |ϕ r | <q 0 }. Since ϕ r (a) = v a < c * , (5.2) holds in a neighborhood of r = a with ρϕ r r = M 0 . Thus ϕ r > 0 and as r increases, ρϕ r = M 0 /r decreases. It is easy to show that ρϕ r is an increasing function of the speed ϕ r so long as ϕ r < c * , and thus ϕ r decreases with larger r and will never reach c * . Hence the equation ρϕ r = M 0 /r can uniquely be solved to get a subsonic ϕ r in term of M 0 , k a and r.
We now locate the shock by a variational method. Let a ≤ η ≤ R. Define With a given η, the function ϕ ∈ K 1 is uniquely determined. We now consider ϕ ∈ K 1 as a function of η and define J : (a, R) → R such that J(η) = I(ϕ(η)) with ϕ ∈ K 1 .
Since ϕ ∈ C 2 [a, η] satisfies the transonic flow equation and the boundary condition at r = a, (4.4) can be simplified to Recall that δr = δϕ/(ϕ sup r − ϕ r ). Thus we can evaluate J ′ explicitly to If an interior critical point of J exists at η = r s , then ρϕ 2 r + p = ρϕ r ϕ sup r + p sup at r = r s . (5.5) As m is prescribed to ensure (4.7) is satisfied, thus ρϕ r = ρ sup ϕ sup r at r = r s . Combining this equation with equation (5.5), we have Hence all the Rankine-Hugoniot conditions are satisfied at the shock. In fact by using ϕ ∈ K 1 and the same calculations in obtaining (4.7), we always have ρϕ r = ρ sup ϕ sup r at any η, which may not be a critical point of J. Hence with the definition (3.15), we can rewrite (5.4) as We already known that dH sub /dη > dH sup /dη in Section 3. If δϕ ≡ w = 0 at the shock Σ, the term involving AR\Ω in (4.1) does not change and the subsonic domain is fixed. It is known that the solution of the transonic equation in the subsonic case can be obtained as a minimizer of I, see [14] which has essentially the same functional. We verify that the subsonic solution is the minimizer of I for our model.
Recall that dxdy becomes 2πrdr and ds on Γ becomes 2πa, hence with critical point ϕ satisfying (ρϕ r r) r = 0 with boundary conditions ρϕ r a = M 0 at r = a, and ϕ = ϕ sup at r = η. Taking another derivative, which shows that I is strictly convex for subsonic flow. Hence when we varies ϕ with δϕ = 0 at Γ (so that the subsonic domain [a, η] is fixed), I attains its minimum when ϕ satisfies the transonic equation for fixed η. This accounts for our choice of the set K 1 . Thus we expect the critical point that we are looking for may be characterized as a saddle point by sup a<η<R { min ϕ ∈ K, shock at r = η I(ϕ)} = sup a<η<R J(η) .
Because we can solve the 1D variational problem exactly, we have the following saddle node theorem.

Remark 5.2
The functional I is well defined when we restrict ourselves to W 1,∞ functions. To prove existence of a critical point, a cut-off of our functional may be introduced (such as the one in [14]) so that a different function space can be employed as its domain.
The natural question is whether the procedures developed in this paper can be extended to more general setting of the multi-dimensional case. We leave the following open question for future work.
Given any curve C in the interior of the flow domain A R ., we find a minimizer ϕ C of the "subsonic" problem. Let J(C) = I(ϕ C ). Now identify a C = Σ such that J attains either its maximum or a saddle point.

Numerical results for variational formulation of the model problem
This section comprises the numerical results that validate our variational formulation and the model problem. In all computations, we use γ = 3, c 2 * = 1, M 0 = 1, R = 6, and a = 5. The figures are generated by MATLAB. As in the last section we let J(η) = min ϕ ∈ K 1 , shock at r = η I(ϕ) .
For clarity, we use η s instead of η to denote the arbitrarily assigned shock position; the computed value of the true position of the shock is denoted by r s .
Since γ = 3, from the Bernoulli's law and the mass conservation we write V (which we only consider positive) explicitly to V±(k,r) + V ± (k, r)); H + = H sup and H − = H sub are computed by substituting V by V + (k 0 , r) and V − (k a , r), respectively. As expected there exists a unique intersection point for the graphs of H sup and H sub at η s = r s ≈ 5.260220746. The resulting velocity V = V sub/sup and the corresponding density ρ = ρ sub/sup are depicted in the right figure in Figure 5. Such a solution satisfies the physical entropy condition, namely density increases across the shock in the direction of flow.
Many studies make an assumption that the entropy is with the same constant, for example k = k a = k 0 in the entire region. This may lead to an erroneous conclusion when the shock is strong. It can be easily shown that both H sup (·, r) and H sub (·, r) are decreasing functions of k. That explains why the graph of H sub when V is replaced by V − (k 0 , r), represented by the solid line labeled as H − (k 0 , r) in the left figure of Figure 5, is located well above the correct H sub . This curve does not intersect H sup and give the incorrect conclusion that no shock develops.
Two figures in Figure 6 confirm that the critical point of I is a saddle node. The left figure in Figure 6 is the graph of J/2π with respect to η s , where     Figure 6 is the graph of I/2π with respect to a perturbation parameter x. Precisely let ∆V = x(r − a) be the perturbation in speed in the subsonic region while keeping the assigned shock location η s fixed. The given speed at the boundary r = a is unchanged. One can also assume we have the same ϕ at η s , as the functional depends on the derivative of ϕ. Regarding I as a function of x, we have   When a shock is not formed in the interior of the flow domain, the maximum of J is attained either at r = a or r = R depending on the boundary data. The corresponding solution ϕ ∈ K 1 does not satisfies the boundary conditions at either r = R or r = a, since it remains either supersonic (when J attains its maximum at r = a as in Figure 8) or subsonic (maximum at r = R as in Figure 7) in the entire region.  Figure 9 where H curves are intersecting) whereas the shock position r s = 5.260220746 obtained in the example in Figure 2.
Finally we compare the necessary and sufficient conditions for the transonic shock and the first variation by evaluating them explicitly. More precisely, we use the same data as in Figure 2 and We conclude the paper with the following remark. Our variational inequality formulation holds for general multidimensional setting. The boundary of its coincidence set forms a shock, across which two out of three Rankine-Hugoniot jump conditions are satisfied. A good estimate of the (average) change in the entropy constant in the configuration would be useful especially when a strong shock develops. However even if we set k a = k 0 the formulation still stands provided the existence of a critical point. We leave the further study on general multidimensional transonic problems in variational inequality formulations for future work.