Incompressible Limit of Solutions of Multidimensional Steady Compressible Euler Equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompresibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.


Introduction
We are concerned with the incompressible limit of solutions of multidimensional steady compressible Euler equations. The steady compressible full Euler equations take the form: where x := (x 1 , · · · , x n ) ∈ R n with n ≥ 2, u := (u 1 , · · · , u n ) ∈ R n is the flow velocity, is the flow speed, ρ, p, and E represent the density, pressure, and total energy respectively, and u ⊗ u := (u i u j ) n×n is an n × n matrix. For the full Euler case, the total energy is with adiabatic exponent γ > 1, the local sonic speed is c = γp ρ , (1.5) and the Mach number is For the homentropic case, the pressure-density relation is The local sonic speed is (1.11) However, the rigorous justification of this limit for weak solutions has been a challenging mathematical problem, since it is a singular limit for which singular phenomena usually occur in the limit process. In particular, both the uniform estimates and the convergence of the nonlinear terms in the incompressible models are usually difficult to obtain. Moreover, tracing the boundary conditions of the solutions in the limit process is a tricky problem. Generally speaking, there are two processes for the incompressible limit: The adiabatic exponent γ tending to infinity, and the Mach number M tending to zero [22,23]. The latter is also called the low Mach number limit. A general framework for the low Mach number limit for local smooth solutions for compressible flow was established in Klainerman-Majda [16,17]. In particular, the incompressible limit of local smooth solutions of the Euler equations for compressible fluids was established with well-prepared initial data i.e., the limiting velocity satisfies the incompressible condition initially, in the whole space or torus. Indeed, by analyzing the rescaled linear group generated by the penalty operator (cf. [27]), the low Mach number limit can also be verified for the case of general data, for which the velocity in the incompressible fluid is the limit of the Leray projection of the velocity in the compressible fluids. This method also applies to global weak solutions of the isentropic Navier-Stokes equations with general initial data and various boundary conditions [10,11,20]. In particular, in [20], the incompressible limit on the stationary Navier-Stokes equations with the Dirichlet boundary condition was also shown, in which the gradient estimate on the velocity played the major role. For the one-dimensional Euler equations, the low Mach number limit has been proved by using the BV space in [3]. For the limit γ → ∞, it was shown in [21] that the compressible isentropic Navier-Stokes flow would converge to the homogeneous incompressible Navier-Stokes flow. Later, the similar limit from the Korteweg barotropic Navier-Stokes model to the homogeneous incompressible Navier-Stokes model was also considered in [18].
For the steady flow, the uniqueness of weak solutions of the steady incompressible Euler equations is still an open issue. Thus, the incompressible limit of the steady Euler equations becomes more fundamental mathematically; it may serve as a selection principle of physical relevant solutions for the steady incompressible Euler equations since a weak solution should not be regarded as the compressible perturbation of the steady incompressible Euler flow in general. Furthermore, for the general domain, it is quite challenging to obtain directly a uniform estimate for the Leray projection of the velocity in the compressible fluids.
In this paper, we formulate a suitable compactness framework for weak solutions with weak uniform bounds with respect to the adiabatic exponent γ by employing the weak convergence argument. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity, which was introduced in [7,15]. Another observation is that the incompressibility of the limit for the homentropic Euler flow follows directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. Finally, we find a suitable framework to satisfy the boundary condition without the strong gradient estimates on the velocity. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles. As a consequence, we can establish the new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.
The rest of this paper is organized as follows. In §2, we establish the compactness framework for the incompressible limit of approximate solutions of the steady full Euler equations and the homentropic Euler equations in R n with n ≥ 2. In §3, we give a direct application of the compactness framework to the full Euler flow through infinitely long nozzles in R 2 . In §4, the incompressible limit of homentropic Euler flows in the three-dimensional infinitely long axisymmetric nozzle is established.
Proof. We divide the proof into four steps.

Now we show that ρ
Since γ → ∞, for given q ≥ 1, we may assume γ > q. Then we find by Jensen's inequality that where |Ω| is the Lebesgue measure of Ω. Then, for (p (γ) ) 1 γ , we have On the other hand, since ln y is concave with respect to y, we have which implies from the Hölder inequality that which, together with (2.7) and (2.9), gives (2.11) Note that both the left and right sides of the above inequality tend to |Ω| 1 q as γ → ∞, owing to condition (H). Then we have In particular, taking q = 1 and q = 2 respectively, we have This implies that ρ (γ) are uniformly bounded in L 2 (Ω). Then there exists a subsequence of ρ (γ) (still denoted by ρ (γ) ) such that ρ (γ) weakly converges toρ in L 2 (Ω). By a simple computation, we obtain from (2.13) that That is, ρ (γ) converges to 1 a.e. in x ∈ Ω, as γ → ∞.
3. By the div-curl lemma of Murat [24] and Tartar [26], the Young measure representation theorem for a uniformly bounded sequence of functions in L p (cf. Tartar [26]; also see Ball [1]), we use (2.1) 1 and (A.3) to obtain the following commutation identity: where we have used that ν(ρ, u) is the associated Young measure (a probability measure) for the sequence (ρ (γ) , u (γ) )(x). Then the main point in the compensated compactness framework is to prove that ν(ρ, u) is in fact a Dirac measure, which in turn implies the compactness of the sequence (ρ (γ) , u (γ) )(x). On the other hand, from lim we see that ν(ρ, u) = δ 1 (ρ) ⊗ ν(u), where δ 1 (ρ) is the Delta mass concentrated at ρ = 1.
Combining both sides of (2.14) together, we have Exchanging indices (1) and (2), we obtain the following symmetric commutation identity: which immediately implies that, i.e., ν(u) concentrates on a single point. If this would not be the case, we could suppose that there are two different pointsú andù in the support of ν. Then (ú,ú), (ú,ù), (ù,ú), and (ù,ù) would be in the support of ν ⊗ ν, which contradicts with u (1) = u (2) . Therefore, the Young measure ν is a Dirac measure, which implies the strong convergence of u (γ) . This completes the proof.
Remark 2.6. The main difference between Propositions 2.3 and 2.4 is that, when γ → ∞, the compressible homentropic Euler equations converge to the homogeneous incompressible Euler equations with the unknown variables (u, p), while the full Euler equations converge to the inhomogeneous incompressible Euler equations with the unknown variables (ρ, u, p). Furthermore, the incompressibility of the limit for the homentropic case follows directly from the approximate continuity equation (2.1) 1 , while the incompressibility for the full Euler case is from a combination of all the equations in (2.17).
There are various ways to construct approximate solutions by either numerical methods or analytical methods such as numerical/analytical vanishing viscosity methods. As direct applications of the compactness framework, we now present two examples in §3- §4 for establishing the incompressible limit for the multidimensional steady compressible Euler flows through infinitely long nozzles.

Incompressible Limit for Two-Dimensional Steady Full Euler Flows in an Infinitely Long Nozzle
In this section, as a direct application of the compactness framework established in Theorem 2.2, we establish the incompressible limit of steady subsonic full Euler flows in a twodimensional, infinitely long nozzle.
The infinitely long nozzle is defined as with the nozzle walls ∂Ω := W 1 ∪ W 2 , where Suppose that W 1 and W 2 satisfy and there exists α > 0 such that for some positive constant C. It follows that Ω satisfies the uniform exterior sphere condition with some uniform radius r > 0. See Fig 3.1. holds for some constant m, which is the mass flux, where Σ is any curve transversal to the x 1 -direction, and l is the normal of Σ in the positive x 1 -axis direction. We assume that the upstream entropy function is given, i.e., ρ p 1/γ −→ S − (x 2 ) as x 1 → −∞, (3.5) and the upstream Bernoulli function is given, i.e., For this problem, the following theorem has been established in Chen-Deng-Xiang [5]. (ii) The flow satisfies the following asymptotic behavior in the far field: As uniformly for , the constant p − and function u − (x 2 ) can be determined by m, S − (x 2 ), and B − (x 2 ) uniquely.
Next, we take the incompressible limit of the full Euler flows.
Proof. We divide the proof into four steps.

(3.11)
From the far-field behavior of the Euler flows, we can define Since both the upstream Bernoulli and entropy functions are given, B (γ) and S (γ) have the following expressions: , with uniformly upper and lower bounds with respect to γ.
Since the flow is subsonic so that the Mach number M (γ) ≤ 1, then we have and (3.13) Since |u (γ) | 2 and p (γ) are uniformly bounded, we conclude that |u (γ) | 2 and p (γ) are uniformly bounded in L 1 loc (Ω). Thus, conditions (A.1)-(A.2) are satisfied. It is observed that, even though the lower bound of pressure p (γ) may tend to zero as γ → ∞ with polynomial rate, so that (F.1) holds for any bounded domain.

2.
For fixed x 1 , (ψ γ − ) −1 (ψ γ (·)) can be regarded as a backward characteristic map with The uniform boundedness and positivity of p 1 implies that the map is not degenerate. Then we have (3.14) Thus, S (γ) is uniformly bounded in BV , which implies its strong convergence. Then condition (F.2) follows.
Remark 3.1. In the two-dimensional homentropic case, the subsonic results in [2,28] can also be extended to the incompressible limit by using Proposition 2.3.

Incompressible Limit for the Three-Dimensional Homentropic Euler Flows in an Infinitely Long Axisymmetric Nozzle
We consider Euler flows through an infinitely long axisymmetric nozzle in R 3 given by  In Du-Duan [13], axisymmetric flows without swirl are considered for the fluid density ρ = ρ(x 1 , r) and the velocity in the cylindrical coordinates, where u 1 , u 2 , and u 3 are the axial velocity, radial velocity, and swirl velocity, respectively, and r = x 2 2 + x 2 3 . Then, instead of (1.2), we have Rewrite the axisymmetric nozzle as with the boundary of the nozzle: where Σ is any curve transversal to the x 1 -axis direction, andl is the unit normal of Σ.
Notice that the quantity For the homentropic Euler flows in the axisymmetric nozzle, we assume that the upstream Bernoulli is given, that is, We denote the above problem as Problem 2(m, γ). It is shown in [13] that 0 ,m), there exists a global C 1 -solution (i.e. a homentropic Euler flow) (ρ, U, V ) ∈ C 1 (Ω) through the nozzle with mass flux condition (4.7) and the upstream asymptotic condition (4.8). Moreover, the flow is uniformly subsonic, and the axial velocity is always positive, i.e., sup in Ω. (4.10) (ii) The subsonic flow satisfies the following properties: As uniformly for r ∈ K 1 (0, 1), where ρ − is a positive constant, and ρ − and U − (r) can be determined by m and B(r) uniquely.
As above, we have the following incompressible limit theorem for this case.
Similar to the previous case, the flow is subsonic so that the Mach number M (γ) ≤ 1, On the other hand, the vorticity ω (γ) has the following expressions: ). (4.18) A direct calculation yields which implies that ω (γ) is uniformly bounded in the bounded measure space and (A.3) is satisfied.