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 incompressibility 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: ⎧ ⎪ ⎨ ⎪ ⎩ div (ρu) = 0, div (ρu ⊗ u) + ∇p = 0, div (ρuE + up) = 0, (1.1) while the steady homentropic Euler equations have the form: div (ρu) = 0, div (ρu ⊗ u) + ∇p = 0, (1.2) 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 (cf. [8]). 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 , (1.8) and the Mach number is defined as (1.9) The incompressible limit is one of the fundamental fluid dynamic limits in fluid mechanics. Formally, the steady compressible full Euler equations (1.1) converge to the steady inhomogeneous incompressible Euler equations: ⎧ ⎪ ⎨ ⎪ ⎩ div u = 0, div (ρu) = 0, div (ρu ⊗ u) + ∇p = 0, (1.10) while the homentropic Euler equations (1.2) converge to the steady homogeneous incompressible Euler equations: div u = 0, div (u ⊗ u) + ∇p = 0. (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 [19,20]. 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 [14,15]. 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. [23]), 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 [9,10,17]. In particular, in [17], 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 [18] 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 [16]. 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 [4,7,13]. 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 Sect. 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 Sect. 3, we give a direct application of the compactness framework to the full Euler flow through infinitely long nozzles in R 2 . In Sect. 4, the incompressible limit of homentropic Euler flows in the three-dimensional infinitely long axisymmetric nozzle is established.
can be well defined. Moreover, the following conditions hold: (A.1). M (γ) are uniformly bounded byM ; (A.2). |u (γ) | 2 and p (γ) ≥ 0 are uniformly bounded in L 1 loc (Ω); (A.3). e 1 (γ) and curl u (γ) are in a compact set in H −1 loc (Ω); (H). As γ → ∞, Remark 2.2. In the limit γ → ∞, the energy sequence E (γ) may tend to zero. Condition (H) is designed to exclude the case that E (γ) exponentially decays to zero as γ → ∞. In fact, in the two applications in Sects. 3 and 4 below, both of the energy sequences E (γ) go to zero with polynomial rate so that condition (H) is satisfied automatically. It is noted that condition (H) could be replaced equivalently by a pressure condition: Indeed, from (A.1) and (2.2), we have which directly implies the equivalence of the two conditions.
p as bounded measures. (2.5) 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) ) (2.7) 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 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 where ρ (γ) := (p (γ) ) 1 γ . 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 [21] and Tartar [22], the Young measure representation theorem for a uniformly bounded sequence of functions in L p (cf. Tartar [22]; also see Ball [1]), we use (2.1) 1 and (A.3) to obtain the following commutation identity: (2.14) 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 we see that where δ 1 (ρ) is the Delta mass concentrated at ρ = 1.
4. We now show ν(u) is a Dirac measure. 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.
In fact, if this would not be true, 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.
Proof. We follow the same arguments as in the homentropic case. First, the weak convergence of p (γ) is obvious. On the other hand, we observe that (2.7) and (2.9) still hold for the full Euler case. Then, for any γ > q ≥ 1, Thanks to condition (F.1), we obtain Taking q = 1 and q = 2, respectively, and following the same line of argument as in the homentropic case, we conclude that (p (γ) ) 1 γ converges to 1 a.e. in x ∈ Ω as γ → ∞. Then, from condition (F.2), The remaining proof is the same as that for the homentropic case, except the strong convergence of u (γ) only stands on {x :ρ(x) > 0, x ∈ Ω} since the vacuum can not excluded. This completes the proof.
Then, as direct corollaries, we conclude the following propositions. Proof. From Theorem 2.1, we know that (u (γ) , p (γ) ) converges to (ū,p) as γ → ∞. For the approximate continuity equation, we see that, for any test function φ ∈ C ∞ c , Letting γ → ∞, we conclude Then, for any test function φ ∈ C ∞ c , we find 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 Sects. 3 and 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 two-dimensional, infinitely long nozzle.
The infinitely long nozzle is defined as 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. 1. Suppose that the nozzle has impermeable solid walls so that the flow satisfies the slip boundary condition: where ν is the unit outward normal to the nozzle wall. 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 (3.5) and the upstream Bernoulli function is given, i.e., Set 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.
1. From (1.1), we can obtain the following linear transport parts: (3.10) From (3.10) 1 , we can introduce the potential function ψ (γ) : (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.
3. Similar to [7], the vorticity sequence ω (γ) : can be written as (3.15) By direct calculation, we have which implies that ω ε as a measure sequence is uniformly bounded so that it is compact in H −1 loc . Therefore, the flows satisfy condition (A.3).
Remark 3.1. In the two-dimensional homentropic case, the subsonic results in [2,24] 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 The boundary condition is set as follows: Since the nozzle wall is solid, the flow satisfies the slip boundary condition: In Du-Duan [11], axisymmetric flows without swirl are considered for the fluid density ρ = ρ(x 1 , r) and the velocity in the cylindrical coordinates, where U and V are the axial velocity and radial 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: The boundary condition (4.3) becomes whereν is the unit outer normal of the nozzle wall in the cylindrical coordinates. The mass flux condition (4.4) can be rewritten in the cylindrical coordinates as where Σ is any curve transversal to the x 1 -axis direction andl is the unit normal of Σ. Notice that the quantity is constant along each streamline. 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 [11] that (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.

Remark 4.1.
For the full Euler flow case, the subsonic results of [12] can be also extended to the incompressible limit by Proposition 2.4.