The global stability of 2-D viscous axisymmetric circulatory flows

In this paper, we study the global existence and stability problem of a perturbed viscous circulatory flow around a disc. This flow is described by two-dimensional Navier-Stokes equations. By introducing some suitable weighted energy space and establishing a priori estimates, we show that the 2-D circulatory flow is globally stable in time when the corresponding initial-boundary values are perturbed sufficiently small.

It is not difficult to know that (5)- (6) have the unique solution in Ω (ρ(r),ū r (r),ū θ (r)) = ρ γ−1 If M 0 < c(ρ 0 ) = P (ρ 0 ), which means that the flow is subsonic on the boundary Σ, then the circulatory flow is always subsonic in Ω (i.e., u θ (r) < c(ρ(r)) holds for any r > 1, one can see Figure 1); if M 0 > c(ρ 0 ), which means that the flow is supersonic on the boundary Σ, then the flow is supersonic for 1 < r < r 0 , subsonic for r > r 0 , and sonic on r = r 0 , respectively (see Figure 2), where r 0 = γ + 1 2c 2 (ρ 0 ) + (γ + 1)M 2 0 > 1. Here we point out that the subsonic or supersonicsonic-subsonic properties of viscous circulatory flows do not play the essential roles in the present paper. However, for the inviscid flow, the subsonic or supersonicsonic-subsonic properties are crucial in order to analyze the blowup mechanism and determine the blowup time of the small perturbed solutions to the 2-D compressible Euler system (see [5]). In addition, one can see more mathematical descriptions on the steady circulatory flows of compressible Euler equations in [2,3] and references therein.
Next we focus on the global stability problem of the background solution (8). Namely, the global solution of (2)-(4) is required to be studied under the following perturbed initial-boundary value conditions: . Moreover, the initial-boundary values in (9) are compatible on the circle S 0 = {(t, x) : t = 0, r = 1}.

HUICHENG YIN AND LIN ZHANG
. Then equations (2)-(4) together with (9) are respectively written as and where To state our main results conveniently, we now introduce the following notations: For k ∈ N, we define The main conclusion in our paper is: then problem (10)-(12) together with (13) has a unique global solution where v = (v r , v θ ), and the constant C > 0 depends only on µ, λ, M 0 ,ρ 0 , A and γ.

Remark 1.
For the Navier-Stokes equations (1), one knows from Theorem 1.1 that the global stability of the perturbed axisymmetric circulatory flows holds both for the subsonic background solution and the subsonic-sonic-supersonic background solution. This is somewhat different from the case of compressible Euler system, where the blowup result is shown and the shock wave is formed from the blowup point (see [5]). On the other hand, even if for the small initial data problem of 2-D compressible isentropic Euler equations, the classical axisymmetric solution will develop singularity in finite time (see the details in [1] and [13]).
Remark 2. So far there have been extensive results on the global spherically symmetric weak/strong /classical solutions to the compressible Navier-Stokes equations (in this case, the solution admits a form (ρ(t, x), u(t, x)) = (ρ(t, r), U (t, r) x r ), one can see [7], [15,4], [12,17] and the references therein. Here, we point out that even if u θ ≡ 0, the initial data in (9) does not satisfy the assumptions of the symmetric initial data in [15], where ∞ 1 r|u 0 (x)| 2 dr < ∞ is demanded. However, in our case, Remark 3. For the general Cauchy problem or initial-boundary value problem in exterior domain of 2-D compressible Navier-Stokes equations, when the initial data are in some suitably weighted energy spaces or are of the small perturbations of constant states, many authors have established the local/global existence of weak/strong/classical solutions in appropriate function spaces, one can find the details in [20], [8], [11], [6], [18,14], [16] and so on. If we want to study the general (not axisymmetric) global perturbation problem of the 2-D circulatory flows for (1), the methods applied in the above references cannot be applied directly since our perturbed initial data are rather different from those (for examples, our initial data have not finite energies or are not the small perturbation of the constant states).
On the other hand, motivated by the results and methods in [8] and [9,10], where the global stabilities and large time behaviors of the perturbed plane Couette flow, of the perturbed constant equilibrium on the half space or of the parallel flow in an infinite layer of R 2 are studied respectively when the Reynolds and Mach numbers are sufficiently small, we hope that it will be probably useful for our future research on the global stability of generally perturbed viscous circulatory flows.
Let's recall some previous works which are closely related to our results. For the initial-boundary value problem of (1), the local classical solution of compressible Navier-Stokes equations is obtained in [16] with ρ 0 being positive and bounded. Applying the energy methods in Sobolev spaces, the authors in [14] established the global existence of classical solutions to (1) when the initial data are of small perturbations for a non-vacuum constant state and no slip boundary conditions are given. There are also some interesting results about the existence of global strong solutions to 2-D Navier-Stokes equations (1), for instance, one can see [18] and references therein. For the arbitrary initial data with the finite total energy, the global existence of weak solutions to compressible Navier-Stokes equations has been established by P. L. Lions in [6] for suitably large adiabatic exponent γ, and this result was improved to the case of any γ > 1 for the spherically symmetric solution in [17]. In addition, D. Hoff in [4] showed the existence of spherically symmetric weak solutions for γ = 1 and the discontinuous initial data. It is worth mentioning that the global radially symmetric strong solutions of the Navier-Stokes equations was established in [7]. Here we point out that our problem (10)-(12) has a nontrivial rotation and the related background solution is not a constant state, which are different from those cases mentioned in the previous references.
To prove Theorem 1.1, we require to establish some global weighted energy estimates of the solution (φ, v) to (10)- (12). Thanks to the delicate analysis on system (10)- (12), the uniform weighted estimates of (φ, v) are obtained by making full use of the properties of the background solution and choosing suitable multipliers. Based on this and the local existence result of classical solution to (10)- (12) with (13), Theorem 1.1 is shown by the continuity induction method.
The paper is organized as follows: In §2, we derive some uniform energy estimates from the linearized parts of (10)- (12). Based on this, some uniform weighted energy inequalities of (φ, v) are obtained in §3 and subsequently the proof of Theorem 1.1 is completed.
2. Some elementary estimates. In this section, we will derive some useful inequalities on the solution (φ, v) to (10) -(12) together with (13). For convenience, through the whole section. (10) -(12) with (13), we have that for any t ≥ 0, Proof. Computing Note that Then from (14) we arrive at Also, by (15) we see that It follows from In addition, by ∞ 1 (10) × Aγρ γ−2 rφdr, one immediately obtains Adding (18) to Thus, the proof of Lemma 2.1 is completed.

2-D VISCOUS AXISYMMETRIC CIRCULATORY FLOWS 5073
On the other hand, direct computation yields It is noted that and Combining (30)-(32), we arrive at d dt This, together with Lemma 2.1, yields v r 1/2 Then Lemma 2.2 is shown.
Proof. At first, we see that Note that ). This, together with (41), yields ). (45) Similarly, we see that In addition, by applying Holder inequality, one has Then combining (46) with (43) yields On the other hand, one can rewrite (10) as which implies From (45) and Lemma 2.1-2.2, we see that This, together with (46)-(47), (49) and Lemma 2.3, yields 2µ+λ 2 By equation (48), one arrives at This, together with (52) and direct integration with respect to variable t over (0, τ ), yields In addition, we rewrite (12) as which implies that for k ≥ 1 Then by (56) and (54) Define the energy Next, we show that N (t) is uniformly bounded for any t ≥ 0. In the proof procedure, we will employ the following Gagoliado-Nirenberg's inequality repeatedly: Lemma 3.1. Let 2 ≤ p ≤ +∞ and let j and k be integers satisfying Then there exists a constant C > 0 such that

HUICHENG YIN AND LIN ZHANG
We now state the main result in this section.
we then have At first, we deal with the terms A i (i = 1, 2, 3, 4) in (57). For this purpose, we require to treat the terms f and g = (g 1 , g 2 ) in (10)- (12).
In addition, one has t 0 (g 11 ,ρrv r )dτ We start to treat each integrand in (58). It follows from Lemma 3.1 with p = +∞, By direct computation, we have With respect to the term t 0 (g 12 , rρv r )dτ , we have Next, we analyze each integrand in (63). For the term θ rρv r drdτ , we have that for any η ∈ (0, 1), where C(η) > 0 is a constant depending only on η.

HUICHENG YIN AND LIN ZHANG
Also, we can obtain for η ∈ (0, 1), Similarly to (61), then The other left terms in t 0 (g 12 , rρv r )dτ can be treated analogously. Combining all analysis above, we eventually arrive at Simultaneously, A i (i = 2, 3.4) can be estimated similarly as A 1 . For example, we show the estimate of the term (∂ t g 1 ,ρr∂ tt v r ) in A 4 . We see that t 0 (∂ t g 1 ,ρr∂ tt v r )dτ = t 0 (∂ t g 11 ,ρr∂ tt v r )dτ + t 0 (∂ t g 12 ,ρr∂ tt v r )dτ.
For the t 0 (∂ t g 11 ,ρr∂ tt v r )dτ , one has t 0 (∂ t g 11 ,ρr∂ tt v r )dτ We start to treat each integrand in (68). It follows from Lemma 3.1 and direct computation, we have By Lemma 3.1 with p = +∞, j = 1, k = 2 and direct computation, we can obtain that In addition, With respect to the term t 0 (∂ t g 12 , rρ∂ tt v r )dτ , we have t 0 (∂ t g 12 , rρ∂ tt v r )dτ Next, we analyze each integrand in (73).