Nonlinear asymptotic stability and transition threshold for 2D Taylor-Couette flows in Sobolev spaces

In this paper, we investigate the stability of the 2-dimensional (2D) Taylor-Couette (TC) flow for the incompressible Navier-Stokes equations. The explicit form of velocity for 2D TC flow is given by $u=(Ar+\frac{B}{r})(-\sin \theta, \cos \theta)^T$ with $(r, \theta)\in [1, R]\times \mathbb{S}^1$ being an annulus and $A, B$ being constants. Here, $A, B$ encode the rotational effect and $R$ is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor-Couette flow is asymptotically stable, even at high Reynolds number ($Re\sim \nu^{-1}$), with a sharp exponential decay rate of $\exp(-\nu^{\frac13}|B|^{\frac23}R^{-2}t)$ as long as the initial perturbation is less than or equal to $\nu^\frac12 |B|^{\frac12}R^{-2}$ in Sobolev space. The powers of $\nu$ and $B$ in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor-Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient $B$. The larger the $|B|$, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor-Couette flows.

R −2 t) as long as the initial perturbation is less than or equal to ν 1 2 |B| 1 2 R −2 in Sobolev space. The powers of ν and B in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor-Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient B. The larger the |B|, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor-Couette flows. Reynolds's famous experiment [30] inspired the study of hydrodynamic stability at high Reynolds number. In this regime, the laminer flows could become unstable and transition to turbulence [10,32,35,39].

Contents
With the low Reynolds number, Serrin [33] demonstrated that all equilibria of the forced Navier-Stokes equation on bounded domains are linearly stable. At high Reynolds number, even in the absence of boundaries, the viscosity can significantly complicate the linear problem.
In this paper, we study the 2D (two-dimensional) incompressible Navier-Stokes (NS) equations: where v = (v 1 , v 2 ) ∈ R 2 is the fluid velocity, x = (x 1 , x 2 ) ∈ Ω represents the space variables, Ω ∈ R 2 is an annular region and t ≥ 0 represents the time variable. The unknowns in the equation are the velocity field v(t, x) = (v 1 (t, x), v 2 (t, x)) and the pressure p(t, x) ∈ R. The constant ν > 0 is known as the kinematic viscosity, which is very small in our paper. And the Reynolds number Re is proportional to the inverse of ν.
In the 2D case, the vorticity field is a scalar. By taking the 2D curl of the Navier-Stokes equations, equation (1.2) can be transformed into its vorticity formulation: In this paper, we employ (1.3) to study the asymptotics of solutions to (1.1) around a Taylor-Couette (TC) flow.
1.1. Taylor-Couette flow. TC flow describes a steady circular solution of the viscous fluid bounded between two rotating infinitely long coaxial cylinders. The understanding of solutions' asymptotics around a TC flow has board applications, including desalination, magnetohydrodynamics and viscometric analysis. Despite being a simple type of rotating solutions, perturbation of the TC flow has proven to be a challenging subject and has been extensively studied experimentally, theoretically, and numerically for a long time. Many questions related to TC flow remain unanswered, making it active research field in fluid mechanics [8,11,17,28]. However, rigorous mathematical proofs in this field are still insufficient, even for the 2D case.
In the following, we adopt the convention r = |x| and denote the radial vorticity ω(x) = ω(r) and stream function φ(x) = φ(r). In 2D, the vorticity and the velocity field then take the form (1.4) We first derive the explicit form of the steady TC flow. First noting that if ω = const, then ω is a solution to the 2D NS equation (1.3). For this case, the stream function φ(r) can be determined from (1.4), which yields φ ′′ (r) + 1 r φ ′ (r) = const. (1.5) By employing the polar coordinates, one can relabel v(x 1 , x 2 ) and ω(r) as U (r, θ) and Ω(r), respectively. This allows us to solve equation (1.5) and obtain the expressions for U and Ω as follows: , Ω(r) = 2A. (1.6) Here A, B are constants and spatial variables (r, θ) belong to a domain D = [1, R] × S 1 . The function U (r, θ) in (1.6) is hence a steady state for the 2D incompressible NS equation (1.1), which is commonly called the Taylor-Couette flow.

1.2.
Hydrodynamic stability at high Reynolds number. Reynold's well-known experiment [30] revealed that small perturbations can cause significant change of the flow due to the nonlinear nature of the Navier-Stokes equations. In experiments [9,34] for the TC flow, a small perturbation could lead to instability and transition to turbulence at high Reynolds numbers. This phenomenon is called the subcritical transition, which is a central topic in fluid mechanics.
Numerous efforts have been made to comprehend the subcritical transition mechanism [6]. Kelvin initially suggested that as the Reynolds number Re → ∞, the basin of attraction for the laminar flow diminishes, allowing the flow to become nonlinearly unstable for small but finite perturbations [19]. In [35], Trefethen, Trefethen, Reddy and Driscoll proposed a way to determine the threshold amplitude by considering a perturbation of size Re −β with β ≥ 0 and Re → +∞. In other words, they aimed to identify the lowest possible value of β, which could result in a transition to turbulence with a perturbation of size O(Re −β ). Bedrossian, Germain, and Masmoudi [2] presented a rigorous mathematical formulation of this approach using the fact that Re −1 ∼ ν.
Given a norm · X , we now hope to determine a nonnegative number β = β(X) so that u 0 X ≤ ν β =⇒ stability, The exponent β is referred to as the transition threshold in the applied literature. Significant advancements and findings related to the subcritical transition regime for the 2D incompressible NS equations have emerged recently. These developments include researches on Couette flow without boundaries [4,5,2,25,24] and with boundaries [7,3], on Lamb-Oseen vortices [12,15,20,13,14] and on periodic Kolmogorov flow [38,37]. 1.3. Main theorems. The aim of this paper is to explore both linear and nonlinear stability mechanism of the 2D Taylor-Couette flow using rigorous analytical tools. Specifically, we prove nonlinear asymptotic stability of the TC flows for the 2D NS equations. Furthermore, we identify and trace a critical parameter reflecting the rotating properties of this system.
Our main results are summarized below. The more precise mathematical statements are given in the following two theorems. The first main result can be directly derived from Proposition 4.3 and Lemma 5.4, and describes the asymptotic stability of the linearized TC flow. Theorem 1.2. There exist constants C, c > 0 being independent of ν, A, B, R, such that the solution w for the linear system exists globally in time and for any t ≥ 0. Furthermore, the following stability estimates hold The second main result describes the asymptotic stability of the nonlinear Navier-Stokes equations around the TC flow. Reynolds experiments tell us that, with small viscosity coefficients, even tiny initial perturbations could cause the flow to be chaotic. Therefore, in order to establish the nonlinear asymptotic stability of Navier-Stokes equations, the initial perturbations must be restricted to a certain range with respect to the viscosity coefficient ν. The specific formulation of the result is as follows.
There exist constants ν 0 and c 0 , C, c ′ > 0 independent of ν, A, B, R such that for any 0 < ν ≤ ν 0 , if the initial data satisfies then the solution w(t, r, θ) to the system (1.7) is global in time. Moreover, the following stability estimates hold By setting the radius R to a constant value, such as R = 2, we can immediately derive the following corollary from Theorem 1.3. Theorem 1.4. There exist constants ν 0 and c 0 , C, c ′ > 0 independent of ν, A, B so that for any 0 < ν ≤ ν 0 and |B| ≥ ν 0 , if then the system (1.7) admits a global-in-time solution w(t, r, θ) safisfying the following stability estimates The impact of the rotation coefficient B on stability is mainly manifested in the so-called enhanced dissipation effect. The decay rate of the heat equation is e −νR −2 t , but in the presence of B, the dissipation rate becomes faster and can be described by e −cν (Notice that we are interested in the regime when 0 < ν ≪ 1). This implies that while the heat equation exhibits decay only after the time scale of ν −1 , the antisymmetric part ik B r 2 in the linearized equation results in the energy decay after a shorter time scale of ν − 1 3 |B| − 2 3 R 2 . This phenomenon is known as the enhanced dissipation effect. A shear or a diffusion averaging mechanism to trigger it were investigated in [22,31]. Here we prove this effect in a different setting with rotations. Theorem 1.2 and Theorem 1.3 both indicate that the enhanced dissipation effect becomes stronger as |B| increases, which is quantified by a faster asymptotic decay rate e −cν 1 3 |B| 2 3 R −2 t . This enhanced dissipation has a timescale of t ∼ ν − 1 3 |B| − 2 3 R 2 around the Taylor-Couette flow, with t ∼ ν − 1 3 being consistent with Couette flow [2,3,4,5,7,24,25]. The enhanced dissipation decay rate e −cν 1 3 |B| 2 3 R −2 t for the TC flow in this paper is optimal. In Section 3, we derived the sharp resolvent estimates, which are also known as the optimal pseudospectral bound. It is important to note that although the enhanced dissipation decaying rates for the TC flow (ν  3 ) have the same power of ν, it does not suggest that the corresponding enhanced dissipation can be directly derived through a scaling argument from results of Couette flow. This is because the (normalized) antisymmetric parts of linearized operators for these two flows are still different. Furthermore, with the aid of dimensional analysis or scaling transformations, it can be shown that the sum of the power of ν and B is always 1. Thus, the factor |B| 2 3 is also optimal. Additionally, inspired by the approach used to handle Oseen vortices [15,20,13], in [1] the authors employed self-similar variables and derived an enhanced dissipation decay rate that is independent of the outer radius R.

1.4.2.
Comparison with Couette flow from an operator perspective. Previous studies have investigated plane Couette flow with (x, y) ∈ T × R ( see [4,5,2,25,24]) and Couette flow in a finite channel with (x, y) ∈ T × [−1, 1] (see [7,3]). In these works, the spatial variable x corresponding to the non-shear direction was defined on a torus. However, for the TC flow in this paper, the spatial domain is an annulus region (r, θ) ∈ [1, R] × S 1 .
As both x in Couette flow and θ in TC flow are defined on a torus or S 1 , it is natural to apply Fourier transform on the x or θ direction. The corresponding linearized equations around the Couette flow and TC flow are given as below.
Couette flow: The linearized operators for the Couette flow and TC flow take the different forms: These two operators exhibit different enhanced dissipation rates of ν It can be observed that both the symmetric and antisymmetric parts of the linearized operator for the TC flow have different structures from those of the Couette flow. In the following discussion, we will demonstrate how these differences affect our results and the corresponding proofs for the TC flow.
In previous studies of the planar Couette flow [4,5,2,25,24], where the shear variable y is defined over the entire space R, mathematicians employed the Fourier transform in y to obtain ordinary differential equations with respect to time t. However, in the case of TC flow, the radial variable r is defined in the bounded domain [1, R] with boundaries, which makes it inconvenient to conduct the Fourier transform in r directly. Instead, we adopt the method of resolvent estimates to derive enhanced dissipation, with detailed results and proofs provided in Section 3. Note that this method has also been employed by Chen, Li, Wei and Zhang to study the Couette flow within a finite channel in [7]. 1.4.3. Resolvent estimates. In Section 3, our key results are the resolvent estimates presented in Proposition 3.2. Define Our initial goal is the following inequality: which is a resolvent estimate from L 2 space to L 2 space, as given in Lemma 3.3. This estimate corresponds to the estimate obtained for the Couette flow in [7]: However, due to the specific structure of the TC flow, the corresponding resolvent estimate is weighted in r. If we take c ′ as 1 2C , the inequality (1.9) directly yields the following estimate: which takes the form of a resolvent estimate for rF − c ′ ν Here, rL k r − c ′ ν , which leads to the enhanced dissipation rate for the following linear evolution equation This allows us to obtain the dissipative factor e −cν 1 3 |kB| 2 3 R −2 t in space-time estimates for the fully nonlinear system, as shown in detail in Proposition 4.5.
Next, we derive the resolvent estimate for the stream function ϕ: The proof of this estimate relies on the resolvent estimate for w L 1 , and its details are given in Lemma 3.4. Finally, in Lemma 3.5 and Lemma 3.6 we establish the below estimates for the H 1 norms of w and ϕ, which are controlled by the H −1 norms of the resolvent equations, respectively: The resolvent estimates presented above are crucial for our analysis, and are summarized in Proposition 3.2. Notably, all of the estimates in Proposition 3.2 have been carefully derived in preparation for the subsequent discussion in Section 4.
Based on the above conclusions, we summarize the following table which illustrates the relation between coefficients of the terms on the left of resolvent estimates and the norm of F − c ′ ν Pseudospectrum and enhanced dissipation. In the section 4, the pseudospectral bound of L is defined as Recall that in Proposition 3.2 we establish the following resolvent estimates This provides a lower bound for the pseudospectrum of L k : Applying the Gearhart-Prüss type lemma established by Wei in [36] and by Helffer-Sjöstrand in [16], we can obtain the pointwise estimate of the semigroup bound in Proposition 4.3 where w l k (t) satisfies the homogeneous linear equation It is worth noting that here, the exponential decay factor e −c(νk 2 ) The precise estimate is given in Lemma 4.6, which states that, for any k ∈ Z and |k| ≥ 1, it holds Note that ν is much smaller than 1, and B is a fixed constant, the leading terms on the right of above inequality are Thus, both the coefficients in front of ϕ l k terms on the left and the main initial-vorticity terms on the right are independent of the viscosity coefficient ν. This phenomenon is commonly refer to as the effect of inviscid damping. Therefore, Lemma 4.6 presents a space-time version of linear inviscid damping around 2D TC flow.
To prove this result, in Lemma 4.8 we find and formulate a new set of explicit orthonormal basis of (weighted) eigenfunctions corresponding to the Laplace operator for circular flows, namely −∂ 2 r + (k 2 − 1 4 )/r 2 . This basis can be represented in the closed form and satisfies the equation Note that there is the 1/r 2 on the right and −( lπ log R ) 2 − k 2 is not the canonical eigenvalue. As far as we know, our explicit constructions here are new. With this basis, we move to prove a crucial estimate where k ∈ Z, B and R are constants with R > 1, w k (0) ∈ L 2 ([1, R]), andφ l 0 k is defined as the solution to the elliptic equations below: Notice that the Laplace operator for the 2D Couette flow is given by −∂ 2 y + k 2 within the domain y ∈ [−1, 1]. To find corresponding orthonormal eigenfunctions, it is natural to introduce the set {sin (πj(y + 1)/2)} ∞ j=1 , as described in [7]. However, for the circular Laplace operator ∂ 2 r − k 2 − 1 4 r 2 , we need to construct a set of more complicated eigenfunctions. The main issue is that if we directly consider the following eigenvalue problem: it can be shown that there is no closed-form solution. In this paper, we instead solve the weighted eigenvalue problem: Utilizing change of variables, we obtain a family of closed-form solutions to this boundary value problem, which can be written in the following explicit expressions These functions form an orthonormal basis of the weighted space L 2 w ([1, R]). Therefore, with the aid of {ψ l } ∞ l=1 , we can evaluate wherew l 0 k := e −ikB t r 2 w k (0). Notice that Thus, employing Fourier transform and Plancherel's formula, the desired estimate (1.11) follows. It is worth noting that Lemma 4.6 also plays an important role in improving the nonlinear transition threshold of Theorem 1.3.
1.4.6. Space-time estimates for the linearized Navier-Stokes equations. In Proposition 5.1 of Section 5, we establish the space-time estimates for the linearized 2D Navier-Stokes equation written in the vorticity formulation (2.2): with the stream function ϕ k satisfying To achieve so, we employ the estimates in Section 3 and Section 4.
(1) The space-time estimate for the zero frequency part (i.e. k = 0) can directly obtained in view of the equation structure and integration by parts, as detailed in Lemma 5.4.
(2) For the nonzero frequency part with k ∈ Z\{0}, we decompose the solution w k = w l k +w n k , where w l k obeys the homogeneous linear equation ∂ t w l k + L k w l k = 0, w l k (0) = w k (0), while w n k is the solution to the inhomogeneous linear equation with zero initial condition Here we denote the nonlinear forms by f 1 and f 2 , which are expressed as As a result, we also decompose ϕ k as ϕ k = ϕ l k + ϕ n k , where ϕ l k and ϕ n k fulfill r 2 )ϕ n k = w n k , ϕ n k | r=1,R = 0, with r ∈ [1, R] and t ≥ 0. For our space-time estimate, the first step is to provide a control of w l k (t) based on the initial data w l k (0). With the aid of the pseudospectrum bounds and enhanced dissipation effect proved in Section 3, we establish the following inequality in Lemma 4.4, which holds for any k ∈ Z and |k| ≥ 1: To proceed, we derive an integrated linear invisid damping estimate, in which ϕ l k can be controlled with the initial data of the vorticity as detailed in Lemma 4.6. Note that ν is much smaller than 1, and B is a fixed constant, so the initial vorticity has an upper bound independent of the viscosity coefficient ν.
By combining this observation with the estimates for w l k , we obtain the following inequality It is worth mentioning that the coefficients in front of the terms on both sides of the inequality are independent of viscosity. Therefore, similar estimates can also hold for the inviscid flow (i.e. ν = 0). By applying the resolvent estimates from Proposition 3.2, we then arrive at the conclusion stated in Proposition 4.5 for the nonlinear part, which involves the inhomogeneous linear equation with zero initial data. This result allows us to bound w n k and ϕ n k by the nonlinear terms f 1 and f 2 as follows Finally, by combining the aforementioned three estimates, we derive the crucial proposition of Section 5, namely Proposition 5.1. This proposition states that the following inequality holds We refer to it as the space-time estimates for the vorticity w k and the stream function ϕ k .
1.4.7. Nonlinear stability. Finally, we close the energy estimate with a bootstrap argument. By utilizing the spacetime estimates established in Proposition 5.1 from Section 5, we derive the result of Proposition 6.1: Here, we define µ k := max (νk 2 ) as it is necessary to use distinct energy estimates for the high-frequency case of νk 2 ≥ |B| and the low-frequency case of νk 2 ≤ |B| separately. In the case of 2D Couette flow [7], the discussions on different frequencies k correspond to νk 2 ≥ 1 and νk 2 ≤ 1. It is important to mention that the frequency classification of k for the 2D TC flow also relies on the rotation parameter B.
We then define the energy functional E k as follows: According to Proposition 6.1, it can be infered that for k ∈ Z\{0}, it holds Utilizing Lemma 6.3, we can control nonlinear forms f 1 , f 2 by This allows us to obtain Lemma 6.4, which states that Simultaneously, we can derive analogous estimates for E 0 by introducing M 0 (0) = w 0 (0) L 2 . Incorporating all estimates above through a bootstrap argument, we finally arrive at the main theorem of this paper: Theorem 6.5.
1.4.8. Subcritical transition and transition threshold. Theorem 1.2 and Theorem 1.3 follow from Proposition 4.3 and Theorem 6.5, respectively. Here are some further remarks regarding these two main theorems. On one hand, Theorem 1.2 shows that the linearized Navier-Stokes equations around the 2D Taylor-Couette flow is dynamically stable at any Reynolds number (including the inviscid case), and the vorticity at any time can be controlled by the given initial value in L 2 norm. On the other hand, for the full Navier-Stokes equation, we only expect nonlinear asymptotic stability within the range of small perturbation of size ν β(s) (β(s) > 0) in H s space.
This proved in our Theorem 1.3, provided the initial perturbation does not exceed ν The exponent 1 2 of ν agrees with that of the 2D Couette flow [5,7], and provides an upper bound 1 2 for the corresponding transition threshold β. The dependency of the subcritical transition on the rotational speed B is captured by |B| 1 2 . This indicates that the system can achieve global stability under a wider range of initial data via taking account of the rotating effect.

Derivation of the perturbative equation
We take the Fourier transform in the θ direction and denote the Fourier coefficients of w and ϕ byŵ k andφ k , respectively. Using this notation, the equation (1.7) can be rewritten as follows: To eliminate the first derivative 1 r ∂ r , we introduce the weight r 1 2 and define w k and ϕ k as w k := r 1 2 e ikAtŵ k and ϕ k := r 1 2 e ikAtφ k . As a consequence, the equation (2.1) is transferred into Note that the Navier boundary condition w k | r=1,R = 0, ϕ k | r=1,R = 0 for any k ∈ Z implies that ∂ t w k and the nonlinear terms in (2.2) vanish on the boundary of interval [1, R]. This forces the second derivative of w k to be zero on the boundary, i.e. ∂ 2 r w k | r=1,R = 0 for any k ∈ Z. Up to this point, we convert our problem to exploring the dynamics and long-time behaviors of (2.2). Remark 2.1. In below, for two quantities A and B, we frequently use A B in short to stand for the inequality A ≤ CB with some universal constant C > 0, that is independent of ν, k, B, λ and R. Additionally, we also write A ≈ B to indicate that both A B and B A are true.

Resolvent estimates
To establish decays of the linearized equation of (2.2) as below the key step is to derive the resolvent estimates that will be presented in Proposition 3.2.
More precisely, we want to study the resolvent equation subject to the following (Navier) boundary condition for any λ ∈ R. After taking the Fourier transform with respect to time t, the resolvent equation becomes The domain of the operator is defined as Note that for any |k| ≥ 1, it holds We also introduce Here , represents the canonical inner product in L 2 (R + , dr).
In this section, we will present the resolvent estimates for w ′ , w r , ϕ ′ and ϕ r with respect to F in both L 2 norm and H −1 r norm.
3.1. Coercive estimates of the real part. We start with the coercive estimates for the real part of F, w , which will be used to obtain the desired resolvent estimates for w.
Proof. Via integration by parts, one can directly check 3.2. Resolvent estimates. For equation (3.1), this subsection aims to establish upper bounds for w and ϕ in terms of F . We prove the following inequalities.
Proposition 3.2. For any |k| ≥ 1, λ ∈ R and w ∈ D k , there exist constants C, c > 0 independent of ν, k, B, λ, R, such that for any 0 ≤ c ′ ≤ c, we have Moreover, it also holds The proof of this proposition can be separated into four parts. We first derive the resolvent estimates for w from L 2 to L 2 . Lemma 3.3. For any |k| ≥ 1, λ ∈ R and w ∈ D k , there exist constants C > 0 independent of ν, k, B, λ, R, such that the following estimate holds Proof. We first prove (3.3). The second inequality readily follows if one selects w r L 2 . By utilizing Lemma 3.1 and Cauchy-Schwarz inequality, we obtain To proceed, we then first prove the statement The mathematical discussion can be divided into three cases: Here 0 < δ ≪ 1 is a small constant, which will be determined later. ( Taking δ = ν , it yields .
The rest part of w r L 2 can be controlled as below w r Combining these two bounds, we deduce w r It is then inferred from Lemma A.1 and Lemma 3.1 that w r The desired estimate (3.5) follows by picking δ = ν .
When r ∈ (1, 1+δ λ ), we bound w r by the L ∞ norm of w r 1 2 and it holds w r Summing two inequalities above renders w r In view of Lemma A.1 and Lemma 3.1, we further deduce w r By setting δ = ν , we can optimize the inequality above and thus obtain for small δ > 0, it can be seen that ) such that the following inequalities hold In order to control w r L 2 ((1,r − )∪(r + ,R)) , we examine the inner product of F and w(χ (1,r − ) − χ (r + ,R) ). Via integration by parts, we write Taking the imaginary part of above equality, it indicates that Thanks to the choice of r − and r + , we can derive the estimate for w (3.8), the second term on the right-hand side of (3.9) can be bounded by Thus, we obtain the following estimate for w r L 2 : w r where in the last line we employ Lemma A.6 from the Appendix. Utilizing Lemma 3.1 and Lemma A.1, it can be further deduced that w r This leads to the conclusion One can obtain an optimized form of the above inequality if δ is chosen as δ = ν .
It then follows that w r L 2 . Therefore, we arrive at Notice that Lemma 3.1 also implies ℜ F, w ≥ ν w ′ 2 L 2 . Along with (3.10) this yields the desired estimate on w ′ : The remaining task is to control r( 1 r 2 − λ)w L 2 , and we appeal to exploring the imaginary part of F, r 2 ( 1 r 2 − λ)w . Integration by parts gives Then we take the imaginary part of both sides and rewrite −λrw = r( 1 In view of (3.10) and (3.11), the L 2 norm of r( 1 r 2 − λ)w obeys the following bounds: provided that ν |kB| ≪ 1. This completes the proof.
Relying on the above L 2 estimates for w and w ′ , we can derive the below bounds for ϕ and ϕ ′ .
Lemma 3.4. For any |k| ≥ 1, λ ∈ R and w ∈ D k , there exist constants C, c > 0 independent of ν, k, B, λ, R, such that for any 0 ≤ c ′ ≤ c, it holds Proof. DenoteF := rF − c ′ ν On the other hand, utilizing Lemma 3.3 again as well as Lemma A.7, it can be inferred that w r 3 2 Thus we deduce w r 3 2 The controls for ϕ ′ and ϕ r also follow from Lemma A.3. And we have This completes the proof.
Now we turn to estimate the H −1 r norm of w through the H 1 r norm of F . Lemma 3.5. For any |k| ≥ 1, λ ∈ R and w ∈ D k , there exist a constants C, c > 0 independent of ν, k, B, λ, such that for 0 ≤ c ′ ≤ c, the following estimate holds Proof. By Lemma 3.1, one first obtains In below, we utilize a similar method as in the proof of Proposition 3.3 to demonstrate To achieve this, we will examine the following three cases: Taking δ = ν , together with (3.12), we obtain we then deduce .
The remaining part of w r L 2 can be bounded in terms of w Combining these two estimates, together with (3.12), we get w r Applying Lemma A.1 and (3.12), we provide the estimate for w r L 2 by w r Picking δ = ν , we then obtain The additional w r L 2 terms on the right can be absorbed to the left by choosing a sufficiently small c ′ > 0, and noting that 0 < ν, δ ≪ 1. Hence, we also get the desired estimate (3.13).
ii) λ ∈ [1, 1 + δ]. Using that we can bound the imaginary part of G, w as below .
Given 0 < δ ≪ 1, it then holds We further choose For the next step, we will construct an appropriate multiplier. To do so, we first define a piecewise C 1 cutoff function ρ with domain (1, R) as follows Via integration by parts, we obtain the following expression Taking the imaginary part of above equality leads to Based on our choice of ρ and the definition of the H 1 r norm, it follows that .
Hence, plugging in (3.12) and (3.14), we then derive the estimate w r .
Meanwhile, it can be deduced from Lemma A.6, Lemma A.1 and (3.12) that Taking δ = ν , we can then bound the L 2 norm of w r as below w r 2 Picking c ′ > 0 sufficiently small and noting that 0 < δ ≪ 1, we can proceed to the conclusion We have therefore proved (3.13) in all three scenarios. This together with (3.12) yields the desired bounds for w H 1 r : For future use, we also establish the resolvent estimate for ϕ ′ .
Lemma 3.6. For any |k| ≥ 1, λ ∈ R and w ∈ D k , there exists a constant C > 0 independent of ν, k, B, λ, R, such that there holds Proof. If ν|k| 3 > |B|, by Lemma 3.5 and Lemma A.3 we obtain wϕdr| ≤ r 2 w r To bound the second term on the right of (3.16), we use a piecewise C 1 cut-off function χ : (0, ∞) → R from Lemma A.8, which is defined as with r ± = 1±δ λ . Conducting integration by parts allows us to write Due to the fact that χ( 1 (3.17) Applying Lemma A.1 and Lemma A.8, we further have Plugging the above inequalities into (3.17), together with Lemma 3.5, we then deduce Therefore, the term a = ϕ ′ L 2 + |k| ϕ r L 2 now obeys a 2 (δν − 5 Substituting δ = ( ν |kB| ) 1 3 and noticing ν|k| 3 ≤ |B|, we thus arrive at This completes the proof of this lemma.

Enhanced dissipation and invisid damping
4.1. Pseudospectral bound. The direct implication of the resolvent estimates is to provide controls for the semigroup by using pseudospectral bounds.
As in [27], we call an operator L in a Hilbert space H is accretive if ℜ Lf, f ≥ 0, for any f ∈ D(L).
The operator L is said to be m-accretive if, in addition, all ℜλ < 0 belong to the resolvent set of L (see [18] for more details). The pseudospectral bound of L is defined as r is an operator with the compact resolvent. Since L k is a relatively compact perturbation of −ν∂ 2 r in the domain D k , it is hence clear that the operators L k also has the compact resolvent, this indicates that the operator L k has only point spectrum.
In Lemma 3.1 we have obtained The above inequality indicates L k being accretive, and furthermore m-accretive. Recall that in Proposition 3.2 we establish the following resolvent estimates In view of the fact r ∈ [1, R], this provides a lower bound for the pseudospectrum of L k : We summarize it into Lemma 4.1. Let Ψ be defined as in (4.1). There exists some C > 0 independent of ν, k, B, R such that

Semigroup bound and enhanced dissipation.
It is convenient to use the below space-time norm .
To obtain decaying semigroup bounds from pseudospectral bounds, we appeal to the following Gearhart-Prüss type lemma established by Wei in [36]. (See also [16] by Helffer and Sjöstrand for relevant reference.) for any t ≥ 0.
We proceed to studying the homogeneous linear equation Utilizing semigroup theory, we can express w l k as w l k (t) = e −tL k w k (0). We start to derive the space-time estimate for w l k . Proposition 4.3. Let w l k be the solution to (4.2) with w k (0) ∈ L 2 . Then for any k ∈ Z and |k| ≥ 1, there exist constants C, c > 0 being independent of ν, k, B, R, such that the following inequality holds Moreover, for any c ′ ∈ (0, c), we have Then we integrate above inequality with respect to t to deduce This implies the estimate in (4.4).

Homogeneous linear equation with nonzero initial data.
Based on the results of Proposition 4.3 and the structure of the linear equation itself, we can obtain the following L ∞ L 2 energy estimate for the homogeneous linear equation with respect to w l k . Lemma 4.4. Let w l k be the solution to (4.2) with initial data w k (0) ∈ L 2 . Considering c to be the same as in Proposition 4.3 and c ′ ∈ (0, c) . For any k ∈ Z and |k| ≥ 1, it holds We first conduct the integration by parts and get By multiplying e 2c ′ (νk 2 ) This implies a space-time estimate for w l k : Combining with Proposition 4.3, we arrive at This completes the proof of Lemma 4.4.

4.4.
Inhomogeneous linear equation with zero initial data. We then derive the spacetime estimates for inhomogeneous equations with zero initial conditions, which will be frequently used in later sections. The derivation of this lemma heavily relies on established resolvent estimates in Proposition 3.2 from Section 3.
We then take the Fourier transform in t and define 1 The inhomogeneous equation (4.7) can thus be transferred into the form We further decomposeŵ asŵ =ŵ 1 +ŵ 2 , whereŵ 1 andŵ 2 solve We also define the corresponding stream functionsφ 1 andφ 2 via the below linear elliptic equations This enables us to writeφ =φ 1 +φ 2 . In view of Proposition 3.2, we then obtain the estimate for w 1 and ϕ 1 : as well as the control for w 2 and ϕ 2 : Here we utilize the definition of H −1 r norm to bound Combining the above inequalities, we arrive at and ν 1 6 |kB| According to the Plancherel's theorem, we have the following equivalence relations based on L 2 norms Plugging in all estimates above, we thus deduce and ν 1 6 |kB| Applying the integration by parts, we also get By employing the Cauchy-Schwarz inequality, we further obtain Together with (4.8), the above inequality yields This finishes the proof of Proposition 4.5.

4.5.
Integrated invisid damping. This subsection is devoted to providing estimates for the stream function ϕ l k in terms of initial vorticity w k (0), with ϕ l k admitting the linear elliptic equation Lemma 4.6. Let w l k be the solution to (4.2) with initial data w k (0) ∈ L 2 and ϕ l k solves (4.9). Considering c to be the same as in Proposition 4.3 and c ′ ∈ (0, c). For any k ∈ Z and |k| ≥ 1, the following inequality holds (4.10) Remark 4.7. Note that ν is much smaller than 1, and B is a fixed constant, so (4.10) can also be written in the following expression We can see that both sides of the above inequality are independent of viscosity coefficient ν, which indicates that the result is also valid for the Euler equation. This type of estimate is called inviscid damping, and here we establish its integrated form. Thus we refer to it as the integrated invisid damping estimates.
In order to prove (4.10), we perform a suitable decomposition of ϕ l k . Definẽ and denotew l 2 k to be the solution to the inhomogeneous linear equation equation with zero initial conditions as below: As a result, the corresponding stream functions can be defined by Then it is easily seen that w l k =w l 1 k +w l 2 k and ϕ l k =φ l 1 k +φ l 2 k . We start with the estimates forφ l 1 k .
Lemma 4.8. Let w l k be the solution to (4.2) with initial data w k (0) ∈ L 2 . Considering c to be the same as in Proposition 4.3, c ′ ∈ (0, c) andφ l 1 k defined in (4.13). For any k ∈ Z and |k| ≥ 1, the following inequality holds (4.14) k and c ′ ∈ (0, c), to obtain the above conclusion it suffices to prove Here we develop some new ideas. With w = 1/r 2 , we introduce the weighted Hilbert space We find a set of explicit orthonormal basis for L 2 w ([1, R]) as below: ψ l (r) = ( 2 log R ) 1 2 r 1/2 sin( lπ log R log r), for all l ∈ N + .
Here ψ l satisfies the following property As far as we know, our explicit constructions above are new. For the sake of convenience, we denote α := ( 2 log R ) 1 2 , β := π log R and λ k,l := (βl) 2 + k 2 . Since (4.17) In view of the definitionw l 0 k = e −ikB t r 2 w k (0), we have the expression where F represents the canonical Fourier transform over R, and W 1 k,l (r) = 1 This together with (4.17) yields On the other hand, observing This completes the proof of this lemma.
To control the L 2 L 2 norm ofφ l 2 k , we utilize resolvent estimates in Section 3 together with Lemma 4.4. Lemma 4.9. Let w l k be the solution to (4.2) with initial data w k (0) ∈ L 2 . Considering c to be the same as in Proposition 4.3 and c ′ ∈ (0, c). For any k ∈ Z and |k| ≥ 1, there holds Proof. First recall thatw l 2 k satisfies the inhomogeneous linear equation with the stream functionφ l 2 k solving the boundary value problem k and g(r) = 1 yields Notice that Thus we arrive at Therefore, the desired bound (4.10) can be readily deduced from Lemma 4.8 and Lemma 4.9. This completes the proof of Lemma 4.6.

Space-time estimates for the linearized Navier-Stokes equations
In this section, we establish the space-time estimates for the linearized 2D Navier-Stokes equation in the vorticity formulation (1.7): Recall in Section 2 we convert this equation to the following system  r 2 ) + i kB r 2 and f 1 , f 2 stand for nonlinear forms given by 5.1. Space-time estimates for non-zero frequency. We decompose the solution w k to (5.2) into two parts. Let w k = w l k + w n k , with w l k fulfilling the homogeneous linear equation ∂ t w l k + L k w l k = 0, w l k (0) = w k (0), and w n k solving the inhomogeneous linear equation with zero initial data Correspondingly, we also decompose ϕ k as ϕ k = ϕ l k + ϕ n k , where ϕ l k and ϕ n k satisfy r 2 )ϕ l k = w l k , ϕ l k | r=1,R = 0, with r ∈ [1, R] and t ≥ 0, and (∂ 2 r − k 2 − 1 4 r 2 )ϕ n k = w n k , ϕ n k | r=1,R = 0, with r ∈ [1, R] and t ≥ 0. Now we state the main conclusion of this section, which is also named as the space-time estimate for equation (5.2).
Proposition 5.1. Assume w k is a solution to (5.2) with w k (0) ∈ L 2 , then there exists a constant c ′ > 0 independent of ν, B, k, R so that it holds Proof. Employing Lemma 4.4 and Lemma 4.6, we can bound the linear part w l k and ϕ l k : To obtain the desired bounds for the remainder terms w n k and ϕ n k , we utilize Proposition 4.5 with the choices of Combining these two estimates, we obtain the desired inequality.
When log R ν − Notice that when i.e., νk 2 ≥ |B|, the heat dissipation effect becomes more significant compared with the enhanced dissipation. Therefore, in the regime where νk 2 ≥ |B|, the following space-time estimates offer a more precise control of w k .
Proof. Conducting the integration by parts, we obtain

It infers
By applying Cauchy-Schwarz inequality, we then deduce Noticing that w k r L 2 ≥ R −1 w k L 2 , it follows Therefore, we can multiply e 2c ′ νk 2 R −2 t on both sides of above inequality. With c ′ being a small constant independent of ν, B, k, R, we obtain . In view of Lemma A.3, it also holds Combining these two estimates above yields which completes the proof.

5.2.
Space-time estimates for zero frequency. To establish the space-time estimate for zero mode of solutions, we directly utilize the heat dissipative structure of equation and then perform integration by parts.
Recall that the nonlinear perturbation equation reads where w| r=1,R = 0 and (∂ 2 r + 1 r ∂ r + 1 r 2 ∂ 2 θ )ϕ = w. Denote the zero mode of function f (r, θ) to be We then have that the zero frequency part of equation (5.6) takes the form Subsequently, we derive the spacetime estimate for w = .
Proof. Employing integration by parts for equation (5.7), we deduce Conducting integration by parts again, we then arrive at . This leads to the following inequality We proceed to integrate the above inequality in t variable and get In view of the definition for w k , ϕ k from (5. This completes the proof of (5.8).

Nonlinear stability
This section is devoted to the proof of our main Theorem 1.3 using a bootstrap argument. Thanks to the transformation conducted in Section 2, we can focus on analyzing the system (5.2), which is presented in the following form   Here L k = −ν(∂ 2 r − k 2 − 1 4 r 2 ) + i kB r 2 and f 1 , f 2 have the nonlinear structure of Note that ϕ k satisfies r 2 )ϕ k = w k , ϕ k | r=1,R = 0, with r ∈ [1, R] and t ≥ 0.
For notational simplicity, we set µ k := max (νk 2 ) We then define the initial energy M 0 (0) = w 0 (0) L 2 and for k ∈ Z\{0} we let As a result of Proposition 5.2 and Proposition 5.3, we readily deduce Proposition 6.1. For k ∈ Z\{0}, let w k be the solution to (5.2) with w k (0) ∈ L 2 . Given log R ν − 1 3 |B| 1 3 , then there exists a constant c ′ > 0 independent of ν, B, k, R, such that it holds We proceed to construct and control below the energy functionals, with The derivation of a priori estimates for E k crucially relies on the space-time estimates obtained in Section 5. We start from presenting the bound for the energy at zero frequency: . We have the following energy inequality Proof. Directly applying Lemma 5.4, we obtain which proves Lemma 6.2.
Next, we turn to control E k for k = 0. We first show that the nonlinear terms f 1 and f 2 obey the below L 2 estimates. Lemma 6.3. For k ∈ Z\{0}, we have Proof. With the expressions of nonlinear terms we get where in the third line we employ Lemma A.3 and Lemma A.5. And the L 2 norm of f 2 can be directly controlled by The above proposition leads to a control of E k with k ∈ Z\{0}.
Thanks to the definition this yields (6.1) µ k ≤ µ l + µ k−l for any k, l ∈ Z by considering the scenarios νk 2 < |B| and νk 2 ≥ |B| separately. In view of Proposition 6.1, Lemma A.1 and Lemma A.3, we now have Utilizing Lemma 6.3, it holds where in the last line we use the fact |k| ≤ 2|l||k − l| for any k ∈ Z, l ∈ Z\{0, k}.
Note that To estimate the term involving f 2 on the right of (6.2), we apply Lemma 6.3 again to deduce ν − 1 2 e c ′ µ k t f 2 L 2 L 2 ≤ν − 1 2 l∈Z\{0} |l| e c ′ µ l t ϕ l r 1 2 E l E k−l .

(6.4)
Together with (6.2), the above two bounds give the desired control for E k when |k| ≥ 1.
Gathering all above estimates, we now conclude our main theorem: Theorem 6.5. Given log R ν − 1 3 |B| 1 3 . There exist constants ν 0 and c 0 , C, c ′ > 0 independent of ν, B, R, such that if then the solution w to the system (2.2) exists globally in time and satisfies the decaying and stability estimates k∈Z\{0} w k (t) L 2 ≤ Ce −c ′ ν Proof. Employing Lemma 6.2 and Lemma 6.4, we have We perform a bootstrap argument to prove our main theorem. Assume that for a small constant c 0 > 0 there holds we then obtain Choosing c 0 ≤ 1 3 C −1 , we arrive at which improves the bootstrap assumption (6.5). By a standard continuity argument, we therefore prove that the system (2.2) admits a global-in-time solution, and its k-th mode w k obeys the desired bounds k∈Z\{0} w k (t) L 2 ≤ Ce −c ′ ν This completes the proof of our main theorem.

Appendix A. Basic Estimates
In the appendix, we provide the proof of several elementary but fundamental calculus lemmas which are continually utilized throughout this paper. The following two Sobolevtype inequalities are crucial in the derivation of resolvent estimates in Section 3.  When w| r=1,R = 0, it can be inferred that |w(r)| 2 = r 1 ∂ s (|w(s)| 2 )ds = r 1 (w ′ (s)w(s) + w(s)w ′ (s))ds ≤ 2 w r 1 2 For the weighted quantity r  To control the stream function ϕ k with the vorticity w k , we develop a series of (weighted) elliptic estimates as below. r 2 )ϕ with ϕ| r=1,R = 0. Then the following inequality holds Proof. Noting that − w, ϕ = ϕ ′ 2 L 2 + (k 2 − 1 4 ) ϕ r 2 L 2 , we immediately obtain the first inequality. It then follows from Cauchy-Schwarz inequality that |k| 2 ϕ r 2 L 2 rw L 2 ϕ r L 2 , which implies |k| 2 ϕ r L 2 rw L 2 , and | w, ϕ | ≤ rw L 2 ϕ r L 2 |k| −2 rw L 2 . On the other hand, in view of Cauchy-Schwarz inequality again and using Lemma A.1, it can be estimated , which yields the second inequality. For the last inequality, we observe that By applying Lemma A.1 and inequalities above, we then prove To establish the elliptic estimate for the zero mode of stream function, we require the below Poincaré-type inequalities.
Lemma A.4. For any w ∈ L 2 (1, R), w| r=1,R = 0 with R > 1, the following Poincaré type inequalities hold w L 2 ≤ 2 rw ′  Consequently, we obtain the following elliptic estimate which involves the outer radius R.
For the second term, a direct computation yields which gives the desired inequality.