The Global Stability of 2-D Subsonic Circulatory Flows for the Steady Isothermal Gas

This paper is a complement of our work in (Cui & Li, 2011) where we have established the global subsonic circulatory solution for the polytropic gas. In this paper, we are concerned with the global stability of the 2-D subsonic circulatory flow around a perturbed circular body for the isothermal gas. The flow is assumed to be isothermal, isentropic, irrotational and described by a steady Euler equations, which can be reduced into a second order quasilinear elliptic equation in a exterior domain with suitable physical conditions. The unique existence and the state of the flow at infinity are obtained under nature physical assumption.


Introduction
We are concerned with the global stability problem of a 2-D perturbed subsonic circulatory flow for the isothermal gas. In (Courant & Friedrichs, 1948) the special subsonic circulatory flows are constructed when the obstacle is regular circular body. If the obstacle is suitably perturbed, the global subsonic circulatory flow is stable? L.Bers (see Bers, 1945) uses the pesudo-complex analysis method to obtain the global existence of the subsonic circulatory flow under the conditions that the adiabatic exponent γ = −1. In (Cui & Li, 2011), we have established the global existence of the subsonic circulatory flows solutions for inviscid gases with adiabatic exponent 1 < γ < 3. In the present paper, our goal is to establish the global existence and stability of subsonic circulatory flows solution for the subsonic isothermal gas around a perturbed circular body. The so-called isothermal gas means that the pressure P and the density ρ of gas are described by the state equation P = Aρ for some constant A > 0 (see Cui & Yin, 2007 and the references therein). In this case, the sound speed is a constant independent of the density ρ.
In addition, we assume the gas is irrotational, that is As in (Courant & Friedrichs, 1948), the last two equations in (1) together (2) yield the Bernoulli's law: here the term A ln ρ is the specific enthalpy for the isothermal gas and B is the Bernoulli' constant.
The solid wall ∂Ω is assumed to be impermeable where ⃗ ν stands for the unit exterior normal to the boundary ∂Ω.
As in (Cui & Li, 2011), through application of Green formula on the first equation in (1), the mass-flux of the circulatory flow should be invariant along each radial ray l which starts from the boundary ∂Ω, so the flow should satisfy the following generalized mass-flux condition lim r→∞ 1 lnr here ⃗ n is the out normal of l.
Based on the first equation in (1) and the condition (4), we can introduced a stream function ψ such that here m is some fixed constant.
Thus the irrotationality can be written as div( 1 ρ ∇ψ) = 0. Notice that from the Bernoulli's law we can solve We can get the circulatory subsonic solution (ψ 0 (r), ρ 0 (r)) of the system (7) in the domain Ω 0 = {x : |x| > 1} by the analogous methods as in (Cui & Li, 2011), with each streamline being a circle and the center being at the origin just as illustrated in (Courant & Friedrichs, 1948). For the specific details, one can see the appendix in this paper.

The Reformulation on Problem (7)
As in (Cui & Li, 2011), it is convenient to use the coordinates transformation: In this case, the exterior domain Ω is changed into Ω 0 and By a tedious computation, the first two equation in (7) can be written as respectively and where Using the notations Φ = ψ − ψ 0 and Ψ = ρ − ρ 0 , then the nonlinear problem (7) can be changed into where and We introduce some weighted Hölder space and corresponding norms which have been used in (Gilbarg & Tudinger, 1998;Chen, 2009;Cui & Li, 2011 and so on). For x, y ∈ Ω 0 , let us write r x = |x|, r y = |y| and r xy = min(r x , r y ). For k ∈ Z + ∪ {0}, 0 < α < 1, l ∈ R and u ∈ C k,α (Ω 0 ), we define and the corresponding function space is defined as Based on above, Theorem 1.1 follows the following conclusion: Theorem 2.1 There exists some positive constants ε 0 , C depending on A, B, such that for any ε < ε 0 , the problem (15) has a unique global solution (Φ, Ψ) ∈ C 2,α (Ω 0 ) × C 1,α (Ω 0 ) with the following estimate where 0 < α < 1.

The Proof of the Theorem 2.1
In order to solve the nonlinear problem (15), a quasilinear elliptic boundary value problem on the unbounded domain Ω 0 coupled with a algebraic equation, the key is to establish the uniform weighted Hölder estimate. As in (Cui & Li, 2011;Gilbarg & Tudinger, 1998), based on this estimate we can use the continuity method to solve the linearized problem of (15). Furthermore, by this estimate together with the standard fixed-point argument in (Gilbarg & Tudinger, 1998), we can arrive at the existence and uniqueness of the solution of the nonlinear problem (15). So we need study the following problem for any σ ∈ [0, 1] : with From the subsonic property (9) together with (16), we have With respect to the source terms F 1 (z, ∇Φ, Ψ), F 2 (z, ∇Φ, Ψ) in (19), we give the detailed estimates as following: Lemma 3.1 For (∇ψ, ρ) ∈ H (1) 1,α × H (0) 1,α , then we have Proof. The Lemma is only a direct computation which we omit, using (13), (14), (17) and (10).
Now we try to obtain the weighted Hölder norm of the solution to (19), which is motivated by Lemma 6.20 in (Gilbarg & Tudinger, 1998). In the following, by the separation variable method as in (Cui & Li, 2011;Li, Xu & Yin, 2015), the estimate of the infinity state of the solution to (19) is established. This together with the weighted Hölder estimate in (Gilbarg & Tudinger, 1998), we can get the uniform weighted Hölder estimate. (19) for any σ ∈ [0, 1], then there exists a generic constant C > 0 independent of σ and δ such that

Lemma 3.2 (Weighted Hölder estimate) For any given
where the generic constant C > 0 is independent of σ, δ.
Proof. We firstly establish the L ∞ bound by the separation variable method as in (Cui & Li, 2011). Set then by (19) we can solve where Then R 0 (r) and R i n (r)(n ≥ 1; i = 1, 2) have the expressions as By the expressions of c i (r)(i = 1, 2) and the subsonic property in (9), one has c 3 (r) = (1 + σc 1 (r)) 1 here c 4 and c 5 are positive constants independent of σ and δ.
Then it follows the maximum principle in (Gilbarg & Tudinger, 1998) that From Lemma 6.20 in (Gilbarg & Tudinger, 1998) together with (30) and the estimate (28), we have and Connecting (31) has a unique solution φ ∈ C 2,α (Ω 0 ) satisfying Proof of Theorem 2.1. Since the uniform weighted Hölder estimate (20) for any σ ∈ [0, 1] is derived, the existence of the solution to the linearized problem of (15) can be solved by the continuity method. Furthermore, using the standard fixed point argument, one can derive the existence and uniqueness of the solution of nonlinear problem (15).