Stability threshold of the 2D Couette flow in Sobolev spaces

We study the stability threshold of the 2D Couette flow in Sobolev spaces at high Reynolds number $Re$. We prove that if the initial vorticity $\Omega_{in}$ satisfies $\|\Omega_{in}-(-1)\|_{H^{\sigma}}\leq \epsilon Re^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to Couette flow for time $t\gg Re^{1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$.


Introduction
In this paper, we consider the 2D incompressible Navier-Stokes equations in T × R: where ν denote the viscosity which is the multiplicative inverse of the Reynolds number Re. V = (U 1 , U 2 ) and P denote the velocity and the pressure of the fluid respectively. Let Ω = ∂ x U 2 − ∂ y U 1 be the vorticity, which satisfies (1.2) Ω t + V · ∇Ω − ν∆Ω = 0.
In [30], Orr observed an important phenomenon that the velocity will tend to 0 as t → ∞, even for a time reversible system such as the Euler equations(ν = 0). This phenomenon is socalled inviscid damping, which is the analogue in hydrodynamics of Landau damping found by Landau [24], which predicted the rapid decay of the electric field of the linearized Vlasov equation around homogeneous equilibrium. Mouhot and Villani [29] made a breakthrough and proved nonlinear Landau damping for the perturbation in Gevrey class(see also [3]).In this case, the mechanism leading to the damping is the vorticity mixing driven by shear flow or Orr mechanism [30]. See [32] for similar phenomena in various system. We point out that the inviscid damping for general shear flow is a challenge problem even in linear level due to the presence of the nonlocal operator for general shear flow. For the linear inviscid damping we refer to [39,34,20,17] for the results of general monotone flows. For non-monotone flows such as the Poiseuille flow and the Kolmogorov flow, another dynamic phenomena should be taken into consideration, which is so called the vorticity depletion phenomena, predicted by Bouchet and Morita [11] and later proved by Wei, Zhang and Zhao [35,36]. Due to possible nonlinear transient growth, it is a challenging task extending linear damping to nonlinear damping. Even for the Couette flow there are only few results. Moreover, nonlinear damping is sensitive to the topology of the perturbation. Indeed, Lin and Zeng [26] proved that nonlinear inviscid damping is not true for the perturbation of the Couette flow in H s for s < 3 2 . Bedrossian and Masmoudi [7] proved nonlinear inviscid damping around the Couette flow in Gevrey class 2 − . Recently Deng and Masmoudi [14] proved that the instability for initial perturbations in Gevrey class 2 + . We refer to [19,21] and references therein for other related interesting results. Moreover it is also observed by Orr that, if we rewrite the linearized system by the change of coordinates f (t, z, y) = ω(t, z + ty, y), then the Fourier transform of the stream function φ(t, z, y) = ψ(t, z + yt, y) iŝ φ(t, k, η) =f (t, k, η) (η − kt) 2 + k 2 , (1.7) The denominator of (1.7) is minimized at t = η k which is known as the Orr critical times. The second phenomenon -enhanced dissipation is sometimes referred to modern authors as the 'shear-diffusion mechanism'. This decay rate is much faster than the diffusive decay of e −νt . The mechanism leading to the enhanced dissipation is also due to vorticity mixing.
However, for the nonlinear system, the Orr mechanism is known to interact poorly with the nonlinear term, creating a weakly nonlinear effect referred to as an echo. The basic mechanism is straight-forward: a mode which is near its critical time is creating most of the velocity field and at this point can interact with the enstrophy which as already mixed to transfer enstrophy to a mode which is un-mixing. When this third mode reaches its critical time, the result of the nonlinear interaction becomes very strong (the time delay explains the terminology 'echo'). There are two necessary ways to control(compete against) the echo cascades. One is to assume enough smallness of the initial perturbations such that the rapid growth of the enstrophy may not happen before enhanced dissipative time-scale ν − 1 3 . The other is to assume enough regularity (Gevery class) of the initial perturbations such that one can pay enough regularity to control the growth caused by the echo cascade.
In this work, we are interested in the first method to stabilize the system and studying the long time behavior of (1.3) for small initial perturbations ω in . We are aimed at finding the largest perturbation (threshold) in Sobolev spaces below which the Couette flow is stable. More precisely, we are studying the following classical question: Given a norm · X , find a β = β(X) so that Another interesting question which is related to this problem is the nonlinear enhanced dissipation and inviscid damping which can be proposed in the following two ways: 1. Given a norm · X (X ⊂ L 2 ), determine a β = β(X) so that for the initial vorticity ω in X ≪ ν β and for t > 0 ω = L 2 x,y ≤ C ω in X e −cν 1 3 t and V = L 2 t,x,y ≤ C ω in X , (1.8) or the weak enhanced dissipation type estimate ω = L 2 t L 2 x,y ≤ Cν − 1 6 ω in X (1.9) holds for the Navier-Stokes equation (1.3).
2. Given β, is there an optimal function space X ⊂ L 2 so that if the initial vorticity satisfies ω in X ≪ ν β , then (1.8) or (1.9) hold for the Navier-Stokes equation (1.3)?
These two problems(find the smallest β or find the largest function space X) are related to each other, since one can gain regularity in a short time by a standard time-weight argument if the initial perturbation is small enough.
We summarize the results as follows: • For β = 0, Bedrossian, Masmoudi and Vicol [8] showed that if X is taken as Gevery-m with m < 2, then the Couette flow is stable and (1.9) holds. • For β = 1 2 , Bedrossian, Vicol and Wang [9] proved the Couette flow is stable as well as the nonlinear enhanced dissipation and inviscid damping for the perturbation of initial vorticity in H s , s > 1.
• For β = 1 2 , recently in [28], we proved the nonlinear enhanced dissipation and inviscid damping for the perturbation of initial vorticity in the almost critical space H log x L 2 y ⊂ L 2 x,y . Let us also mention some other recent progress [4,5,6,13,15,17,25,27,18,33,36,38] on the stability problem of different types of shear flows in different domains.
In this paper, we find a smaller β(= 1 3 ) such that the Couette flow is stable and the nonlinear enhanced dissipation and inviscid damping hold, when X takes a Sobolev spaces. Our main result is stated as follows. Theorem 1.1. For σ ≥ 40, ν > 0, there exist 0 < ǫ 0 , ν 0 < 1, such that for all 0 < ν ≤ ν 0 and 0 < ǫ ≤ ǫ 0 , if ω in satisfies ω in H σ ≤ ǫν 1 3 , then the solution ω(t) of (1.3) with initial data ω in satisfies the following properties: 1. Global stability in H σ , where Φ(t, y) is given explicitly by 2. Inviscid damping, 3. Weak enhanced dissipation, The constant C is independent of ν and ǫ.
Remark 1.2. By replacing D(t, η) by D(t, η) α with α ≥ 1 in the proof and assuming σ large enough(depending on α), one can obtain the stronger enhanced dissipation of the following from: However, the weak enhanced dissipation of the same decay rate as in the Theorem 1.1 is enough for the proof of the Sobolev stability. Both (1.12) and (1.13) are far from the exponential decay of the linear case.
Let us now outline the main ideas in the proof of Theorem 1.1. First, we provide a (well chosen) change of variable that adapts to the solution as it evolves and yields a new 'relative' velocity which is time-integrable. Second, we will construct a new multiplier which can be regarded as a ghost weight in phase space and helps us control the growth caused by echo cascades.

Proof of Theorem 1.1
In this section, we will present several key propositions and complete the proof of Theorem 1.1 by admitting those propositions.
2.1. Notation and conventions. See Section 11.1 for the Fourier analysis conventions we are taking. A convention we generally use is to denote the discrete x (or z) frequencies as subscripts. By convention we always use Greek letters such as η and ξ to denote frequencies in the y or v direction and lowercase Latin characters commonly used as indices such as k and l to denote frequencies in the x or z direction (which are discrete). Another convention we use is to denote M, N, K as dyadic integers M, N, K ∈ D where When a sum is written with indices K, M, M ′ , N or N ′ it will always be over a subset of D. We will mix use same A for Af = (A(η)f (η)) ∨ or Af = A(η)f (η), where A is a Fourier multiplier.
We use the notation f g when there exists a constant C > 0 independent of the parameters of interest such that f ≤ Cg(we analogously g f define). Similarly, we use the notation f ≈ g when there exists C > 0 such that C −1 g ≤ f ≤ Cg.

2.2.
Coordinate transform. We will use the same change of coordinates as in [8] which allows us to simultaneously 'mod out' by the evolution of the time-dependent background shear flow and treat the mixing of this background shear as a perturbation of the Couette flow (in particular, to understand the nonlinear effect of the Orr mechanism).
The change of coordinates used is (t, x, y) → (t, z, v), where z(t, x, y) = x − tv(t, y) and v(t, y) satisfies Define the following quantities Thus we get We also obtain that which implies that and that g satisfies It gives that By the changing of the coordinates we deduce our problem to studying the following system: (2.26) The following proposition follows from the bootstrap hypotheses, elliptic estimates and the properties of the multipliers: A σ and A s E . Proposition 2.2. Under the bootstrap hypotheses, the following inequalities hold: For the enhanced dissipation and the inviscid damping in Sobolev norm, we also have the following remark.
By Lemma 2.1, for the rest of the proof we may focus on times t ≥ 1. Let I * be the connected set of times t ≥ 1 such that the bootstrap hypotheses (2.22)-(2.26) are all satisfied. We will work on regularized solutions for which we know E σ (t) takes values continuously in time, and hence I * is a closed interval [1, T * ] with T * ≥ 1. The bootstrap is complete if we show that I * is also open, which is the purpose of the following proposition, the proof of which constitutes the majority of this work. Proposition 2.4. For σ ≥ 40, ν > 0 and 8 ≤ s ≤ σ − 10, there exist 0 < ǫ 0 , ν 0 < 1, such that for all 0 < ν ≤ ν 0 and 0 < ǫ ≤ ǫ 0 , such that if on [1, T * ] the bootstrap hypotheses (2.22)-(2.26) hold, then for any t ∈ [1, T * ], 1. Vorticity boundedness, 2. Control of coordinates system, The rest part of this section will give an outline of the proof of (2.30). The proof of (2.31) can be found in Section 9.1.1; The proof of (2.32) and (2.33) can be found in Section 9.2; The proof of (2.34) can be found in Section 10.1; The proof of (2.35) can be found in Section 9.3.1; The proof of (2.36) can be found in Section 9.3.2; The proof of (2.37) can be found in Section 10.2; The proof of (2.38) and (2.39) can be found in Section 9.1.2. Form the time evolution of E H,f we get where the CK stands for 'Cauchy-Kovalevskaya' To treat the second term in (2.40), we have Notice that the relative velocity is not divergence free: The first term is controlled by the bootstrap hypothesis (2.25). For the second term we use the elliptic estimates, Lemma 4.1, which shows that under the bootstrap hypotheses we have Therefore, by the Sobolev embedding, σ > 40 and the bootstrap hypotheses, (2.43) To handle the commutator, we use a paraproduct decomposition. Precisely, we define three main contributors: transport, reaction and remainder: Here N ∈ D = { 1 2 , 1, 2, 4, ..., 2 j , ...} and g N denote the N -th Littlewood-Paley projection and g <N means the Littlewood-Paley projection onto frequencies less than N .
For the last term, we get The next four propositions together with (2.43) imply (2.30). At first, we deal with the dissipation term. In Section 5, we will prove the following proposition.
Proposition 2.5. Under the bootstrap hypotheses, Next we control the transport part. In Section 6, we will prove the following proposition.
Proposition 2.6. Under the bootstrap hypotheses, Next we control the remainder part. In Section 7, we will prove the following proposition.
Proposition 2.7. Under the bootstrap hypotheses, At last, we control the reaction part. In Section 8, we will prove the following proposition.
Proposition 2.8. Under the bootstrap hypotheses, Let us admit the above propositions and finish the proof of (2.30).
Proof. We then get by (2.40) that Thus by (2.43) and the above propositions, we have Thus by taking ǫ small enough, we proved (2.30).

Toy model and the nonlinear growth
3.1. The toy model. According to the change of coordinate, the relative velocity now is time integrable. The growth may come from the reaction term. In each time interval I m,η which contain only one Orr critical time t = η m , it is necessary to study the following toy model At last we use the following model to control the entropy growth in each critical time region.
We need to point out that in the toy model e −cν 1 3 t is replaced by ν 1 3 t m,η −(1+β) with 0 < β ≤ 1 2 due to some technical reasons when we deal with zero mode (see (8.6)). The condition β > 0 ensures the total growth is bounded (see Lemma 3.3).
With w k (t, η), now we can define our key multiplier A σ k (t, η),

3.3.
Basic estimate for the multiplier. The following lemma expresses the well-separation of critical times.

Lemma 3.2 ([7]
). Let ξ, η be such that there exists some α ≥ 1 with α −1 |ξ| ≤ |η| ≤ α|ξ| and let k, n be such that t ∈ I k,η ∩ I n,ξ , then k ≈ n and moreover at least one of the following holds: Now we will present a lemma about the upper and lower bounds estimates of w(t, η).
Thus we proved the lemma.
The above lemma gives that for all t, (3.10) A σ k (t, η) ≈ k, η σ . Next we introduce several lemma related to the properties of D. The first lemma can be found in [8] which will be useful in the proof of the commutator estimate in Section 10.

Lemma 3.4 ([8])
. Uniformly in ν, η, ξ and t ≥ 1 we have: Next lemma we will introduce the product lemma related to D which is a Sobolev type estimates comparing to the Lemma 3.7 in [8].
Lemma 3.5. The following holds for all q 1 and q 2 and γ > 1, Proof. We use the dual method. By Lemma 3.4, we get Thus we proved the lemma.

Elliptic estimate
The purpose of this section is to provide a thorough analysis of ∆ t . Lemma 4.1. Under the bootstrap hypotheses, for ν sufficiently small and s ′ ∈ [0, 2], it holds that for 2 ≤ γ ≤ σ − 1 Proof. We get that for s ′ ∈ [0, 2] and s ≥ 0 that We write ∆ t as a perturbation of ∆ L via Thus we get then by using the fact that v ′′ = (h + 1)∂ v h, (11.3) and the bootstrap hypotheses, we get and v ′′ are zero mode, by the same argument as the proof, we can easily get that for γ ≤ σ − 1 Under the bootstrap hypotheses, it holds that Proof. By the definition of u we get Here we use the same argument as (4.1) and get that Then by Lemma 4.1 and the bootstrap hypotheses, we have The first inequality follows from (4.3) with s = σ − 2.
We also have Therefore by Lemma 4.1 and the bootstrap hypotheses, we get Thus we proved the lemma.
Proof. We have We also have Therefore by Lemma 4.1 and the bootstrap hypotheses, we get By taking ǫ small enough, we get the first inequality.
In what follows we use the shorthand and then Thus we get Then we get similarly we have At last we deal with T 2 HL , we have ∂tw k (t,ξ) w k (t,ξ) kt η and then get Thus we proved the lemma.
By the fact that u = (0, g) Lemma 4.4. Under the bootstrap hypotheses for ǫ sufficiently small, for s ≤ σ − 7 it holds that Proof. By Lemma 3.4, we have By Lemma 4.1, we get By Lemma 4.1, we then obtain that Thus we proved the lemma.

Dissipation error term
In this section, we will deal with the dissipation error term in (2.45).

5.2.
Treatment of the non-zero mode. We use a paraproduct decomposition in v. Then we have
Proof. We get by (2.45) that Then by (5.1) and (5.2), we obtain that Thus by taking ǫ small enough and using Proposition 2.2, we get Thus we proved the proposition.

Transport
To treat the transport term, we need consider the commutator. The following lemma gives the key commutator estimate. Lemma 6.1. Assume that |ξ − η| ≤ 1 10 |η|, then it holds that Let us admit the lemma and finish the estimate of transport term first. Then proof of the lemma will be present at the end of this section.
We then set more restrictions on the support of the integrand to make k, η and l, ξ be closer. We get |l, ξ| .
Then we get by | w(t,ξ) w(t,η) | 1 that We rewrite T 2 N,D as follows.
We get by Lemma 6.1 and Lemma 3.1 that . By the fact that We then get by Lemma 4.1(by taking s ′ = 2 in the lemma) that Therefore we get Now we are able to prove Proposition 2.6.
6.3. Proof of Lemma 6.1. We end this section by proving Lemma 6.1.
Proof. Without loss of the generality, we assume 0 < η < ξ. Then according to the relation between t and ξ, η, we need to consider following 5 cases.
Then the lemma follows from the following inequalities which follow from (6.2) and the fact that where we use the fact that η k 2 . Then the lemma follows from (6.3), (6.4) and the following two inequalities where we use the fact that |k − l| ξ−η t , l ≈ η t . (3c.) |ξ − η| ξ l ≈ η k . In this case, we have Case 4. For 2η ≤ t ≤ 2ξ, then t ∈ I 1,ξ and Thus the lemma follows from (6.3), (6.4) and the following inequalities Case 5. For t ≥ 2ξ. We get by (6.3) and (6.4) that Thus we proved the lemma.

Remainder
In this section we deal with the remainder and prove Proposition 2.7. Now the commutator can not gain us anything so we may as well treat each term separately. We rewrite both terms on the Fourier side: On the support of the integrand, |l, ξ| ≈ |k − l, η − ξ| thus Therefore we get by (4.4) which gives Proposition 2.7.

Treatment of the zero mode
We have According to the relation between t and ξ, we have the following 5 cases.
Since we restrict the integrand in D 1 , it holds that |l, ξ|.

8.4.
Corrections. In this section we treat R ǫ,1 N which is higher order in ν N,LH + R ǫ,1 N,HL + R ǫ,1 N,HH . We recall that χ N denotes the Littlewood-Paley cut-off to the N -th dyadic shell in Z × R; see Section 11.1.
Begin first with R ǫ,1 N,LH . On the support of the integrand Thus A σ k (η) ≈ A σ l (ξ ′ ) and From here we may proceed analogous to treatment of R 1 N with (l, ξ ′ ) playing the role of (l, ξ). We omit the details and simply conclude the result is Turn now to R ǫ,1 N,HL . On the support of the integrand, it holds that k, η σ ≈ l, ξ σ ≈ l, ξ ′ − ξ σ , Thus we get that Thus we divide R ǫ,1,z N,HL into two parts and then by the bootstrap hypotheses and the Young's inequality, it holds that Thus we have proved Proposition 2.8. 9. Coordinate system 9.1. Higher regular controls. In this subsection we will study the energy estimate for g in H σ and h,h in H σ−1 and H σ .
9.1.1. Energy estimate of g. In this section, we will prove (2.31). We need to mention that the result of (2.31) is not optimal, however it is enough. It is natural to computer the time evolution of g 2 H σ . We get To treat V H,g 1 , we get by using integration by parts, By Lemma 11.1 and the Sobolev embedding theory, we get . We now use the fact that Then by the bootstrap hypotheses, we get At last we treat the dissipation term V H,g

3
. We have We use the same argument as in the treatment of V H,g 1 and get that Thus by the fact that we get Next we turn to V H,h and Putting together and using the bootstrap assumption, we get We also get We the get We have Next we treat Vh ,LH For the first term, by Lemma 6.1, we have |ξ| ≈ |η| and then M ≥8 |ξ−η|≤ 1 10 |η| Then the estimate is similar to Rh M,N R and we get The remainder term is easy to dealt with. We use the fact that |η| |η − ξ| + |ξ| and |ξ| ≈ |η − ξ| and get that To treat the dissipation error term Vh 3,σ , we have     The energy for h in H σ is similar to the estimate of H σ−1 . We have Treatment of T 0 N . We get that  We make use of the notation for some constant C which is independent of M . Generally the exact value of C which is being used is not important; what is important is that it is finite and independent of M . With this notation, we also have During much of the proof we are also working with Littlewood-Paley decompositions defined in the (z, v) variables, with the notation conventions being analogous. Our convention is to use N to denote Littlewood-Paley projections in (z, v) and M to denote projections only in the v direction.
Another key Fourier analysis tool employed in this work is the paraproduct decomposition, introduced by Bony [10](see also [1]). Given suitable functions f, g we may define the paraproduct decomposition (in either (z, v) or just v), f g = T f g + T g f + R(f, g) where all the sums are understood to run over D. In our work we do not employ the notation in the first line since at most steps in the proof we are forced to explicitly write the sums and treat them term-by-term anyway. This is due to the fact that we are working in non-standard regularity spaces and, more crucially, are usually applying multipliers which do not satisfy any version of AT f g ≈ T f Ag. Hence, we have to prove almost everything 'from scratch' and can only rely on standard para-differential calculus as a guide.
We also show some product estimates(or Young's inequality) based on Sobolev embedding. It holds for s > 1 that f g H s (T×R) f H s (T×R) g H s (T×R) , f * g 2 f 2 g H s (T×R) , f * g * h 2 f 2 g 2 h H s (T×R) .

Composition lemma.
According to the coordinate transform, we need the following composition lemma.