Enhanced dissipation and blow-up suppression for an aggregation equation with fractional diffusion and shear flow
Niu Binqian Binbin Shi Weike Wang




In this paper, we consider an aggregation equation with fractional diffusion and large shear flow, which arise from modelling chemotaxis in bacteria. Without the advection, the solution of aggregation equation may blow up in finite time. First, we study the enhanced dissipation of shear flow by resolvent estimate method, where the fractional Laplacian (-Δ)^α/2 is considered and α∈ (0,2). Next, we show that the enhanced dissipation of shear flow can suppress blow-up of solution to aggregation equation with fractional diffusion and establish global classical solution in the case of α≥ 3/2. Here we develop  some new technical to overcome the difficult of low regularity for fractional Laplacian.




[
    Brent Mittelstadt
    
=====================








§ INTRODUCTION


Aggregation-diffusion-type equations arise in a wide variety of biological applications, such as Keller-Segel models of chemotaxis and migration patterns in ecological systems. In this paper, we consider the following aggregation equation on torus 𝕋^2 with fractional diffusion and large shear flow

    ∂_tn+Au(y)∂_xn+(-Δ)^α/2 n+∇·(n 𝐁(n))=0,
    
    n(t,x,y)|_t=0=n_0(x,y),   (t,x,y)∈ℝ^+×𝕋^2.

Here the n(t,x,y) is nonnegative unknown functions which represent the density, the smooth function u(y) represents the underlying fluid velocity and A is a positive constant. The domain

    𝕋^2={(x,y)|x,y∈𝕋_2π},

where 𝕋_2π=[-π,π) is a periodic interval. The fractional Laplacian  (-Δ)^α/2 is defined via the Fourier transform, it is as follows

    (-Δ)^α/2 n=∑_(k,l)∈ℤ^2(k^2+l^2)^α/2n(k,l)e^ikx+ily,   0<α≤2,

the n denotes the Fourier transform of n. The fractional Laplacian is a nonlocal operator and its kernel representation is used in this paper, see Section <ref>. The linear vector operator 𝐁(n) is called attractive kernel, which could be formally represented as

    𝐁(n)=∇ (-Δ)^-1(n-n),

the n denotes the average of n. In this paper, we study the global well-posedness of equation (<ref>) for some large shear flows.

.05in

Without the advection (A=0), the equation (<ref>) is an aggregation equation with fractional diffusion

    ∂_tn+(-Δ)^α/2 n+∇·(n 𝐁(n))=0,

which is used as a model for various biological and physical phenomena, see <cit.>. When α=2, the equation (<ref>) goes back to the classical parabolic-elliptic Keller-Segel model. It is well-known that the solutions for Keller-Segel model in high dimensional may blow up in finite time if the initial data n_0 is large in L^1 norm.  More precisely, for two dimensional case, if the L^1 norm of initial data n_0 is less than 8π, there exists a unique global solution; and if the L^1 norm of initial data n_0 exceeds 8π, the solution may blow up in finite time, one could refer to <cit.> for more details. In three and higher dimensional cases, the solution may blow up even for initial data n_0 arbitrary small in L^1 norm, see <cit.>. When 0<α<2, the solution of (<ref>) may blow up in finite time for high dimensions and large initial data n_0, see <cit.>. In addition, the blow-up solution has also been studied in <cit.> for more general aggregation-diffusion equations.

.05in

The case A≠0 is corresponding to aggregation progress in the background of a shear flow. A realistic scenario is that chemotactic processes take place in a moving fluid, and the possible effects and related problems resulting from the interactions between the chemotactic process and the fluid transport have been widely investigated, see <cit.>. The study of aggregation equation with an incompressible flow is one of those attempts, the model is as follows

    ∂_tn+A u·∇ n+(-Δ)^α/2 n+∇·(n 𝐁(n))=0,

where u is divergence free vector field. An interesting question arising is whether one can suppress the possible finite time blow-up by the mixing effect coming from the fluid transport. Recently, some progresses have been made for the suppression of blow-up by incompressible flow. When α=2, the equation (<ref>) is classical Keller-Segel model with incompressible flow. Kiselev, Xu <cit.> and Hopf, Rodrigo <cit.> considered that the u is the relaxation enhancing flow <cit.>, they proved that the solution of the advective Keller-Segel equation does not blow-up in finite time provided the amplitude of the relaxation enhancing flow is large enough. Bedrossian and He <cit.> found that shear flows (u=(u(y),0)) have a different suppression effect in the sense that sufficiently large shear flows could prevent the blow-up in two dimensions but could not guarantee the global existence in three dimensions if the initial mass is greater than 8π. When  0<α<2, the equation (<ref>) is an aggregation equation with fractional diffusion (also known as generalized Keller-Segel model) with incompressible flow. Hopf, Rodrigo <cit.> and Shi, Wang <cit.> proved that the solution of (<ref>) does not blow-up in finite time by the large relaxation enhancing flow, where the range of α need to be considered. For the blow-up phenomenon can be suppressed through fluid transport progress, some other problems and models can refer to <cit.>. However, the equation (<ref>) becomes (<ref>) if u is shear flow, and it is currently unclear whether the shear flow can suppress the blow-up in the case of 0<α<2.


.05in


The additional flows studied in those references are found to provide an enhanced dissipation effect from fluid, which could help the dissipation terms dominate even in the nonlinear level. In this paper, we study that the blow-up solution of (<ref>) can  be suppressed by enhanced dissipation of shear flow. First, we need to consider the enhanced dissipation of shear flow in the case of fractional dissipation, the model is as follows

    ∂_tg+u(y)∂_x g+ν(-Δ)^α/2g=0,   g(0,x,y)=g_0(x,y),

where ν>0. The meaning of enhanced dissipation is that the dissipation effect can be enhanced and the L^2 norm of solution to equation (<ref>) has a faster decaying rate if ν is small enough. When α=2, Bedrossian and Zelati <cit.> studied the enhanced dissipation of (<ref>) by hypocercivity, and Wei <cit.> studied the enhanced dissipation of (<ref>) by resolvent estimate. Recently, some special shear flows have been widely studied, such as Couette flow <cit.>, Poiseuille flow <cit.> and Kolmogorov flow <cit.>. When 0<α<2, Zelati, Delgadino and Elgindi <cit.> given a enhanced dissipation rate of (<ref>). He <cit.> obtained an almost sharp enhanced dissipation rate of (<ref>) by resolvent estimate, where α>1 and the fractional Laplacian operator was written as the form of anisotropic. Li and Zhao <cit.> considered the linearized critical surface quasi-geostrophic equation around the Kolmogorov flow. As a toy model, they studied the equation (<ref>) in the case of α=1 and u(y)=cos y, and obtained the sharp enhanced dissipation rate by hypocercivity. In <cit.>, the authors also given some useful comments for the range of α.


.05in

In this paper, we consider the equation (<ref>), the goal is to show that the blow-up solution of an aggregation  equation with fractional diffusion can be suppressed through some shear flows. We will prove the global existence of solution to aggregation equation with fractional diffusion and large shear flow for some α. This question is motivated by the works of <cit.>. Here we study the sharp enhanced dissipation rate of (<ref>) in the case of α>0. As an application, we show that the enhanced dissipation of shear flow can suppressed the blow-up of an aggregation  equation with fractional diffusion. 

.05in

First, we study the enhanced dissipation of shear flow in an aggregation  equation with fractional diffusion, the equation (<ref>) is written as

    ∂_tg+ℒ_ν,α g=0,  g(0,x,y)=g_0(x,y),

where

    ℒ_ν,α=ν(-Δ)^α/2+u(y)∂_x,   0<α<2.

In this paper, we study the enhanced dissipation of shear flow by the equation (<ref>). In fact, we only to study semigroup with the operator -ℒ_ν,α as generator and give the semigroup estimate. Let denote P_0 and P_≠ are  projection operator, which are defined that for any g(x,y)

    g^0=P_0g=1/2π∫_𝕋g(x,y)dx,    g_≠=P_≠g=g-g^0,

where g^0 is zero node and g_≠ is nonzero mode.

.05in

The e^-tℒ_ν,α is semigroup with the operator -ℒ_ν,α as generator, the first main theorem  is the following


Assume that the shear flow u(y)∈ C^∞(𝕋), there exist constants m, N∈ℕ, c_1>0, δ_0>0 with the property that, for
any λ∈ℝ and δ∈(0,δ_0), there exist finitely many points y_1,… y_n∈𝕋 with n≤ N, such that

    |u(y)-λ|≥ c_1 δ^m,     ∀  |y-y_j|≥δ,   ∀ j∈{1,… n}.

Then there exists a ν_0=ν(u) such that if ν<ν_0, for any t≥0, the following enhanced dissipation estimate hold,

    e^-tℒ_ν,αP_≠_L^2→ L^2≤ e^-λ'_ν,α t+π/2,     λ'_ν,α=ϵ_0 ν^m/m+α,

where the ℒ_ν,α and P_≠ are defined in (<ref>) and (<ref>) respectively, the ϵ_0 is small enough and α∈ (0,2).


In Theorem <ref>, we obtain the sharp enhanced dissipation rate of shear flow, this result seems to improve previous works <cit.>, where general shear flow u(y) satisfies the assumption in Theorem <ref> and α∈ (0,2).  In this paper, we study enhanced dissipation of shear flow based on the assumption in Theorem <ref> and resolvent estimate, which is inspired by the works of <cit.>.

.05in


The assumption of shear flow in Theorem <ref>
is reasonable, we can easily check that the Kolmogorov flow u(y)=cos y satisfies the assumption by Taylor expansion and m=2. In the case of α=2, u=cos y, the enhanced dissipate rate has been studied in <cit.>. If we consider the case of 0<α<2 and u(y)=cos y, we also have following corollary by Theorem <ref>,


Assume that the shear flow u(y)=cos y is Kolmogorov flow, then there exists a ν_0=ν(u) such that if ν<ν_0, for any t≥0, the following enhanced dissipation estimate hold,

    e^-tL_ν,αP_≠_L^2→ L^2≤ e^-λ_ν,α t+π/2,     λ_ν,α=ϵ_0 ν^2/2+α,

where

    L_ν,α=ν(-Δ)^α/2+cos y∂_x,

the P_≠ are defined in (<ref>), the ϵ_0 is small enough and α∈ (0,2).



.05in

Next, we study that the enhanced dissipation of shear flow can suppress blow up in an aggregation equation with fractional diffusion, and we establish the global classical solution with large initial data. Here we study the enhanced dissipation of (<ref>) by operator ℒ_ν,α and Theorem <ref>, thus we modify the  (<ref>) by time rescaling. If taking t=Aτ and denoting ν=A^-1, the equation (<ref>) is written as

    ∂_tn+u(y)∂_xn+ν(-Δ)^α/2 n+ν∇·(n 𝐁(n))=0,
    
    n(t,x,y)|_t=0=n_0(x,y),   (t,x,y)∈ℝ^+×𝕋^2.

Since equations (<ref>) and (<ref>) are equivalent in the sense of time rescaling, we mainly consider the (<ref>) and establish the global classical solution by enhanced dissipation of shear flow. Based on the Theorem <ref>, we imply that

    ν^m/m+α→ν,    m→∞,

then the enhanced dissipation of shear  flow is very weak for m is large enough, it is difficult for suppressing the blow-up. In this paper, we consider that the shear flow u(y)=cos y is Kolmogorov flow, and we imply m=2 to the assumption in Theorem <ref>.


.05in

The second main theorem of this paper read as follow.


Let α∈ [3/2,2) and initial data n_0≥0, n_0∈ H^γ(𝕋^2)∩ L^1(𝕋^2), γ>1+α.  the u(y)=cos y is Kolmogorov flow. Then there exists a ν_0=ν(α,n_0), such that if ν<ν_0, the unique nonnegative classical solution n(t,x,y) of equation (<ref>) is global in time.


In Theorem <ref>, we show that the shear flow can suppress the blow-up of aggregation equation with fractional diffusion in the case of α≥ 3/2, it seems to be the first result about shear flow suppress blow up in nonlinear equation with fractional diffusion. I believe that the range of α is not sharp, which is only technical one. We know by (<ref>) that the aggregation equation with fractional diffusion  is scaling invariant, thus the critical space of (<ref>) is L^d/α, where d represents the spatial dimension. It is well know that the solution is global smooth if we have global supercritical estimate. By the definition of P_0, P_≠ and Theorem <ref>, we deduce that the zero mode of (<ref>) is one dimension which is no enhanced dissipation, and the L^1 is supercritical for α>1. Therefore, I speculate that the Theorem <ref> still holds in the case of α>1. In addition, the L^2 estimate of (<ref>) is supercritical in the case of α≥ 3/2, thus the key point of this paper is to establish global L^2 estimate of (<ref>) through the enhanced dissipation of shear flow.

.05in

If consider the equation (<ref>), we have the following corollary by Theorem <ref>.


Let α∈ [3/2,2) and initial data n_0≥0, n_0∈ H^γ(𝕋^2)∩ L^1(𝕋^2), γ>1+α, the u(y)=cos y is Kolmogorov flow. Then there exists a A_0=A(α,n_0), such that if A>A_0, the unique nonnegative classical solution n(t,x,y) of equation (<ref>) is global in time.


In the following, we briefly state our main ideas of the proofs to Theorem <ref> and <ref>. Firstly, we need to proved the enhanced dissipation of equation (<ref>) by resolvent estimate and semigroup estimate, see Theorem <ref>. We assume that the shear flow satisfies the assumption in Theorem <ref> and consider the operator ℒ_ν,α in (<ref>). In the proof, we establish the pseudospectral bound estimate by resolvent estimate and can obtain the semigroup estimate of e^-tℒ_ν,α by the Gearchart-Prüss type theorem (see Lemma <ref>). Next, we prove that the shear flow suppress the blow-up of equation (<ref>) and establish global classical solution, see Theorem <ref>. Since L^2 is supercritical estimate in the case of α>3/2, we only need to establish global L^2 estimate. We decompose equation (<ref>) into one dimensional zero mode equation and two dimensional nonzero mode equation, see (<ref>)-(<ref>). Since L^1 is supercritical in one dimension zero mode equation, the solution is global existence. And two dimensional nonzero mode equation has enhanced dissipation, then we can establish global L^2 estimate by bootstrap argument.


.05in


In this paper, we study the enhanced dissipation of shear flow in (<ref>) and blow-up suppression in an aggregation equations with fractional diffusion, see equation (<ref>). Some proofs technical and ideas are inspired by <cit.>. Here we use semigroup theory to study the enhanced dissipation of (<ref>) and the key point is to obtain  pseudospectral bound by resolvent estimate method. Since the fractional Laplacian is nonlocal operator in the case of α∈(0,2), these technical is not obvious. Our strategy is to transform the operator ℒ_ν,α into the case similar to α=2 through transformation and calculation, the details can refer to Section <ref>. We study that the shear flow suppress the blow-up of (<ref>) by bootstrap argument, where we only consider the nonzero mode equation, see Assumption <ref> and Proposition <ref>. Here the estimates of (<ref>) and (<ref>) are also needed, see Lemma <ref>,  <ref> and  <ref>. Some mathematical methods are used in the proof, such as energy methods, nonlinear maximum principle, semigroup theory and Duhamel's principle, the details can refer to Section <ref>. Compared to the case of α=2, the enhanced dissipation estimate of n_≠ is difficult in the case of α<2, the main reason is that the low regularity of dissipative terms cannot control nonlinear terms in the (<ref>). In this paper, we establish some new techniques to overcome this difficulty, the details can refer to the proof of Proposition <ref> and Appendix <ref>.

.05in

The rest of this paper is arranged as follows. In Section <ref>, we introduce some preparations and give the bootstrap argument. In Section <ref>, we prove the enhanced dissipation effect of shear flow. In Section <ref>, we prove  the Theorem <ref> to establish the global well-posedness of. In the Appendix <ref> and <ref>, we provide necessary supplements and useful tools in this paper.





§ PRELIMINARIES AND  BOOTSTRAP ARGUMENT


In what follows, we provide some  notations and auxiliary results, which is helpful for the proof of this paper. In addition, we set up the bootstrap argument in this section. The details are as follows.



 §.§ Notations and auxiliary results

Throughout the paper, we use the standard notations to denote function spaces and use C to denote a generic constant which may vary from line to line. Given quantities X,Y, if there exists a positive constant C such that X≤ CY, we write X≲ Y. If there exist positive constants C_1,C_2 such that C_1Y≤ X≤ C_2Y, we write X∼ Y.

.05in

In this paper, we study the enhanced dissipation of shear flow by resolvent estimate, where the fractional Laplacian is considered. Therefore, we need to introduce some the operator theory. Let (𝒳,·) be a complex Hilbert space and let H be a closed linear operator in 𝒳 with domain D(H). H is m-accretive if the left open half-plane is contained in the resolvent set with

    (H+λ I)^-1∈ℬ(𝒳) ,  (H+λ I)^-1≤ (λ)^-1,  λ >0,

where ℬ(𝒳) denotes the set of bounded linear operators on 𝒳 with operator norm · and I is the identity operator.

.05in

We denote e^-tH is a  semigroup with -H as generator and define pseudospectral bound

    Ψ(H)=inf{(H-iλ I)f: f∈ D(H), λ∈ℝ, f=1 }.

The following result is the Gearchart-Prüss type theorem for m-accretive operators, see <cit.>.


Let H be an m-accretive operator in a Hilbert space 𝒳. Then for any t≥ 0, we have

    e^-tH_𝒳→𝒳≤ e^-tΨ(H)+π/2,

where Ψ(H) is defined in (<ref>).


The fractional Laplacian in (<ref>) is a nonlocal operator, it also has the following kernel representation on 𝕋^d, see <cit.>

    Λ^α f(x)=C_α,d∑_k∈ℤ^dP.V. ∫_𝕋^df(x)-f(y)/|x-y+k|^d+αdy,

where Λ=(-Δ)^1/2, α∈ (0,2), x,y∈𝕋^d, C_α,d>0. In this paper, we denote Λ_x,Λ_y as one dimension fractional Laplacian operator. Next, we present some lemmas related to the fractional Laplacian,  which is helpful in this paper, the details are as follows. 
 
Let α∈[0,2], for any f, g∈ C^∞(𝕋^d), one has

    ∫_𝕋^dΛ^α f(x)g(x)dx=∫_𝕋^d f(x)Λ^α g(x)dx.





Let α∈(0,2), f∈ C^∞(𝕋^d) and denote by x the point such that

    f(x)=max_x∈𝕋^df(x),

and f(x)>0. Then for any 1≤ p<∞, we have

    Λ^αf(x)≥ C(α,d,p)f(x)^1+pα/d/f_L^p^pα/d,  or  f(x)≤ C(d,p)f_L^p.





Let s≥0, p∈(1,∞), for any f, g∈ C^∞(Ω), one has

    Λ^s(fg)_L^p≲Λ^sf_L^p_1g_L^p_2+Λ^sg_L^p_3f_L^p_4,

where 1<p_i≤∞ (i=1,2,3,4), 1/p=1/p_1+1/p_2=1/p_3+1/p_4.







Let α∈(0,1),  for any f∈ C^∞(𝕋^d) and ε>0, one has

    Λ^α f_L^∞≲ε^1-α∇ f_L^∞+ε^-αf_L^∞.





Let f,g∈ C^∞(𝕋^d) satisfy f(0)g(0)=0, then for any s∈ [0,2], one has

    ∫_𝕋^d f(x)g(x)dx=∫_𝕋^dΛ^-sf(x)Λ^s g(x)dx,

and

    |∫_𝕋^df(x)g(x)dx|≲Λ^sf_L^2Λ^-sg_L^2,

where f and g are the Fourier transform.



In Lemma <ref>, combining f(0)g(0)=0, one has 

    ∫_𝕋^df(x)g(x)dx=∑_k∈ℤ^df(k)g(k)=∑_k≠ 0|k|^-sf(k)|k|^sg(k)
    =∫_𝕋^dΛ^-sf(x)Λ^s g(x)dx.

and the (<ref>) was established in <cit.>.


In this paper, we only need to establish global L^2 estimate of (<ref>), the main reason is that there are the following local existence and regularity criterion.



Let α∈[3/2,2) and initial data n_0≥0, n_0∈ H^γ(𝕋^2)∩ L^1(𝕋^2), γ>1+α, the shear flow u(y)=cos y is Kolmogorov flow, there exists a time T_*=T(n_0,α,ν)>0 such that the nonnegative solution of (<ref>)

    n(t,x,y)∈ C([0,T_*], H^γ(𝕋^2)∩ L^1(𝕋^2)).

Moreover, if for a given T, the solution of (<ref>) verifies the following bound

    lim_t→ Tsup_0≤τ≤ tn(τ,·)_L^2<∞,

then the solution can be extended up to time T+δ for sufficiently small δ>0. If n_0∈ L^1(𝕋^2), the solution of (<ref>) is L^1 conservation, namely,

    M=n_L^1=n_0_L^1.





The Proposition <ref> tell us that we only need to have certain control of spatial L^2 norm of the solution for establishing the classical solution of equation (<ref>). The proofs of local existence and L^1 conservation is standard method, the proof of regularity criterion can refer to <cit.>.





 §.§ Bootstrap argument
 We know that the enhanced dissipation of shear flow occurs in nonzero mode. Similar to <cit.>, denote

    n^0=P_0n,   n_≠=P_≠n,

where the P_0 and P_≠ are defined in (<ref>), the solution of equation (<ref>) is decomposed  into x-independent part and x-dependent part. Since

    (-Δ)^α/2 n=∑_(k,l)∈ℤ^2(k^2+l^2)^α/2n(k,l)e^ikx+ily,

and combining the definition of  P_0 and P_≠ in (<ref>), one gets 

    P_0((-Δ)^α/2 n)=(-∂_yy)^α/2n^0,  P_≠((-Δ)^α/2 n)=(-Δ)^α/2 n_≠.

Then we obtain the one dimensional zero mode equation

    ∂_tn^0+ν(-∂_yy)^α/2n^0+ν∂_y(n^0𝐁_1(n^0))+ν(∇·(n_≠𝐁_2(n_≠)))^0=0,

and the two dimensional nonzero mode equation

    ∂_tn_≠+u(y)∂_xn_≠   +ν(-Δ)^α/2 n_≠+ν∇ n^0·𝐁_2(n_≠)+ν∂_yn_≠𝐁_1(n^0)
        +ν(∇·(n_≠𝐁_2(n_≠)))_≠-ν n^0n_≠-ν n_≠(n^0-n)=0,

where

    𝐁_1(n^0)=∂_y(-∂_yy)^-1(n^0-n),    𝐁_2(n_≠)=∇(-Δ)^-1n_≠.

By local estimate (see Lemma <ref>), we know that there exists

    t_0=t(n_0, α, ν)=O(1/ν),

such that for any 0≤ t≤ t_0, one has

    n^2_L^2=n^0^2_L^2+n_≠^2_L^2≤ 4n_0^2_L^2,

where the definition of t_0 can be seen in (<ref>).

.05in

In this paper, we establish the global well-posedness based on the standard bootstrap argument. We denote

    s_0=t_0/2,

then we list the bootstrap assumptions as below.


Let α∈ [3/2,2) and n_≠ be the solution of (<ref>) with initial data n_0, the positive constant C_≠ be determined by the proof. Assume that the u(y)=cos y is Kolmogorov flow, define T^∗=T(n_0,α,ν)>0 to be the maximum time such that following assumptions hold,


  (A-1) Nonzero mode L^2Ḣ^α/2 estimate: for any s_0≤ s≤ t≤ T^∗

    ν∫_s^tΛ^α/2 n_≠^2_L^2dτ≤ 16C_≠e^-λ_ν,α (s-s_0)n_0^2_L^2,


  (A-2)Nonzero mode enhanced dissipation estimate: for any s_0≤ t≤ T^∗

    n_≠(t)^2_L^2≤ 4C_≠e^-λ_ν,α (t-s_0)n_0^2_L^2,


where λ_ν,α is defined in (<ref>) and s_0 is defined in (<ref>).



We aim to show T^∗=∞, this is achieved through the bootstrap argument. To be specific, we will prove the following refined estimates hold on [s_0, T^∗] by choosing proper ν.


Let α∈ [3/2,2) and n_≠ be the solution of (<ref>) with initial data n_0 and satisfy the Assumption <ref>. Then there exist ν_0=ν(n_0,α), such that ν<ν_0, one has


  (P-1) Nonzero mode L^2Ḣ^α/2 estimate: for any s_0≤ s≤ t≤ T^∗

    ν∫_s^tΛ^α/2 n_≠^2_L^2dτ≤ 8C_≠e^-λ_ν,α (s-s_0)n_0^2_L^2,


  (P-2)Nonzero mode enhanced dissipation estimate: for any s_0≤ t≤ T^∗

    n_≠(t)^2_L^2≤ 2C_≠e^-λ_ν,α (t-s_0)n_0^2_L^2,


where λ_ν,α is defined in (<ref>) and s_0 is defined in (<ref>).




In the proof of Proposition <ref>, the estimate of term ν∂_yn_≠𝐁_1(n^0) to equation (<ref>) is difficult. We need to obtain the enhanced dissipation decay this term by Assumption <ref>,  the ∂_yn_≠ contains a first-order derivative, while the Λ^α/2n_≠ only has α/2-order derivative in Assumption <ref>, which from the fractional Laplacian. This is a technical obstacle in the case of α<2. In this paper, we develop  some new technical to overcome the difficult of the low regularity for fractional Laplacian, the details can see Appendix <ref>.




Combining Assumption <ref> and Proposition <ref>, we know that the n_≠  has enhanced dissipation after time s_0, the main reason is that we need the following estimate in the proof

    [ν∫_s^t+s(∫_τ^t+sΛ^α/2𝒮_τ_0P_≠dτ_0
    )^2dτ]^1/2≲(tλ_ν,α+1)^1/2,

see (<ref>), and it holds in the case of s≥ s_0. I believe that this is technical.




We know that the time T^∗ in Assumption <ref> is large than s_0+8λ^-1_ν by local estimate, the details can see Lemma <ref>. The L^∞ L^2 estimate of n^0, L^∞ L^∞ estimate of n and L^∞Ḣ^α/2 estimate of n^0 can be obtained by the Assumption <ref>, see Section <ref>. Combining the Assumption <ref> and Proposition <ref>, we imply that the T^∗ is infinity by bootstrap argument and the global L^2 estimate of n is established. Based on the local solution and uniform L^2 estimate of n, the global classical solution of equation (<ref>) can be established by Proposition <ref>, we can finish the proof of Theorem <ref>. In this paper, the Proposition <ref> is the most important and we can prove it by enhanced dissipation of shear flow, the details of proof can be seen in Section <ref>.




§ ENHANCED DISSIPATION OF SHEAR FLOW IN FRACTIONAL DIFFUSION



In this section, we study the enhanced dissipation of shear flow in (<ref>) and finish the proof of Theorem <ref>. Here we consider the equation (<ref>) and operator ℒ_ν,α in (<ref>). In addition, the shear flow u(y) satisfies the assumption in Theorem <ref>, the specific details are as follows



Let u(y)∈ C^∞(𝕋), there exist constants m, N∈ℕ, c_1>0, δ_0>0 with the property that, for
any λ∈ℝ and δ∈(0,δ_0), there exist finitely many points y_1,… y_n∈𝕋 with n≤ N, such that

    |u(y)-λ|≥ c_1 δ^m,     ∀  |y-y_j|≥δ,   ∀ j∈{1,… n}.



Next,we prove the Theorem <ref>. If g(t,x,y) is the solution of equation (<ref>), taking the Fourier transform in x, it is as follows

    g_k(t,y)=1/2π∫_𝕋g(t,x,y)e^-ik xdx,   k≠0,

then g_k(t,y) satisfy

    ∂_tg_k+ℒ_ν,α,kg_k=0,   g_k(0,y)=g_0,k(y),

and

    ℒ_ν,α,k=ν(k^2-∂_yy)^α/2+iku(y),

where α∈ (0,2), k0. We study the semigroup with the operator -ℒ_ν,α,k as generator and establish a lower bound of Ψ(ℒ_ν,α,k), which is defined by (<ref>) and (<ref>), it is as follows

    Ψ(ℒ_ν,α,k)=inf{(ℒ_ν,α,k-iλ I)f: f∈ D(ℒ_ν,α,k), λ∈ℝ, f=1 }.

We definite the operator (k^2-∂_yy)^α/2  by Fourier series that

    (k^2-∂_yy)^α/2f(y)=∑_l∈ℤ(k^2+l^2)^α/2f̂(l)e^ily,   f̂(l)=1/2π∫_𝕋f(y)e^-il ydy.

Since

    1/2((l^2)^α/2+(k^2)^α/2)
    ≤(k^2+l^2)^α/2≤2((l^2)^α/2+(k^2)^α/2),

one get by (<ref>) and Fourier series that

    (k^2-∂_yy)^α/2f(y)   =∑_l∈ℤ(k^2+l^2)^α/2f̂(l)e^ily
       ∼∑_l∈ℤ(k^2)^α/2f̂(l)e^ily+∑_l∈ℤ(l^2)^α/2f̂(l)e^ily
       =|k|^α f(y)+(-∂_yy)^α/2f(y).

Similar to <cit.>, we know that for any f∈ D(ℒ_ν,α,k)=H^α(𝕋), we deduce by (<ref>) and (<ref>) that

    ⟨ℒ_ν,α,k f, f⟩   =⟨ν(k^2-∂_yy)^α/2f+iku(y)f ,f⟩
       ∼⟨ν|k|^α f+ν(-∂_yy)^α/2f,f⟩
       =ν|k|^αf^2_L^2+νΛ^α/2_yf^2_L^2≥ 0,

and

    (λ)f_L^2^2   ≤1/2(ν|k|^αf^2_L^2+νΛ^α/2_yf^2_L^2)+(λ)f_L^2^2
       ≤⟨ℒ_ν,α,k f, f⟩+⟨λ I f,f⟩
       =Re⟨ (ℒ_ν,α,k+λ I)f, f⟩
       ≤(ℒ_ν,α,k+λ I)f_L^2f_L^2,

therefore, the ℒ_ν,α,k is m-accretive operator. Based on the Assumption <ref>, we have the following Gearchart-Prüss type theorem and the lower bound of Ψ(ℒ_ν,α,k) in (<ref>).



Let ℒ_ν,k be an m-accretive operator in (<ref>), then one has

    e^-tℒ_ν,α,k_L^2→ L^2≤ e^-tΨ(ℒ_ν,α,k)+π/2.

If the shear flow u(y) satisfy the Assumption <ref>, k≠  0 and ν|k|^-1<1. Then there exists a positive constant ϵ_0 independent of ν and k, such that

    Ψ(ℒ_ν,α,k)≥ϵ_0 ν^m/m+α|k|^α/m+α,

where Ψ(ℒ_ν,α,k) is defined in (<ref>) and α∈ (0,2)



Since ℒ_ν,α,k is an m-accretive operator, the estimate of (<ref>) is trivial by Lemma <ref>. Here we only need to prove (<ref>). For any fixed λ∈ℝ, define

    ℒ_ν,α,k=ℒ_ν,α,k-iλ I =ν(k^2-∂_yy)^α/2+ik(u(y)-λ),

where λ=λ/ k. Taking the set as follows

    E={y∈𝕋: |y-y_j|≥δ,   ∀ j∈{1,…, n}},

and the y_j satisfy the Assumption <ref>. Since u(y) is a continuous function, we define the function

    χ: 𝕋→ [-1,1]

as a smooth approximation of sign(u(y)-λ), and there exists a constant c_2>0, such that for any y∈𝕋, one has

    |χ'(y)|≤ c_2δ^-1,     |χ”(y)|≤ c_2δ^-2,

and

    χ(y) (u(y)-λ)≥ 0.

In addition, for any y∈ E, one has

    χ(y)(u(y)-λ)=|u(y)-λ| .

For f∈ D(ℒ_ν,α,k) and f_L^2=1, we obtain by the definition of ℒ_ν,α,k in (<ref>) that

    ⟨ℒ_ν,α,kf, χ f ⟩ =ν⟨(k^2-∂_yy)^α/2f,  χ f⟩+ik⟨(u(y)-λ)f,χ f⟩,

and consider the imaginary part of (<ref>), one has

    ⟨ℒ_ν,α,kf, χ f ⟩ =ν⟨(k^2-∂_yy)^α/2f,  χ f⟩+k⟨(u(y)-λ)f,χ f⟩.

Since u(y) satisfy the Assumption <ref>, one has

    |u(y)-λ|≥ c_1 δ^m,    y∈ E,

then we deduce by (<ref>) and (<ref>) that

    ⟨(u(y)-λ)f,χ f⟩≥∫_E|u(y)-λ||f(y)|^2dy≥ c_1 δ^m∫_E|f(y)|^2dy.

Thus we know ⟨ (u(y)-λ)f,χ f⟩ is nonnegative, and we can easily get

    |k|⟨(u(y)-λ)f,χ f⟩   =|k⟨(u(y)-λ)f,χ f⟩|
       =|⟨ℒ_ν,α,kf, χ f ⟩-ν⟨(k^2-∂_yy)^α/2f,  χ f⟩|.

Combining ⟨|k|^αf,  χ f⟩ is real and (<ref>), we have

    ⟨(k^2-∂_yy)^α/2f,  χ f⟩∼⟨(-∂_yy)^α/2f,  χ f⟩.

By Cauchy-Schwartz inequality and Lemma <ref>, one has

    |⟨(-∂_yy)^α/2f, χ f⟩|   =|⟨Λ_y^α/2f,  Λ_y^α/2(χ f)⟩|
       ≲Λ_y^α/2f_L^2Λ_y^α/2(χ f)_L^2,

and by Lemma <ref> and <ref>, we obtain

    Λ_y^α/2(χ f)_L^2   ≲Λ_y^α/2χ_L^∞f_L^2+χ_L^∞Λ_y^α/2 f_L^2
       ≲ c_2δ^-α/2f_L^2+Λ_y^α/2f_L^2.

Then we imply from (<ref>) that

    |k|⟨(u(y)-λ)f,χ f⟩   =|⟨ℒ_ν,α,kf, χ f ⟩-ν⟨(k^2-∂_yy)^α/2f,  χ f⟩|
       ≲ℒ_ν,α,kf_L^2f_L^2+νΛ_y^α/2f^2_L^2
    + c_2νδ^-α/2f_L^2Λ_y^α/2 f_L^2.

Combining (<ref>) and (<ref>), we obtain

    ∫_E|f(y)|^2dy   ≤ c^-1_1δ^-m⟨(u(y)-λ)f,χ f⟩
       ≲ c^-1_1δ^-m|k|^-1(ℒ_ν,α,kf_L^2f_L^2
    +νΛ_y^α/2f^2_L^2
    + c_2νδ^-α/2f_L^2Λ_y^α/2 f_L^2).

Since

    ⟨ℒ_ν,α,kf,  f ⟩ =ν⟨(k^2-∂_yy)^α/2f,  f⟩+ik⟨ (u(y)-λ)f,f⟩,

and combining (<ref>) and Lemma <ref>, one gets

    ⟨ℒ_ν,α,kf, f ⟩ =ν⟨(k^2-∂_yy)^α/2f,  f⟩∼ν|k|^αf^2_L^2+νΛ_y^α/2f^2_L^2.

Then we deduce by Cauchy-Schwartz inequality that

    Λ_y^α/2f_L^2≲ν^-1/2( Re⟨ℒ_ν,α,kf,  f ⟩)^1/2≲ν^-1/2ℒ_ν,α,kf^1/2_L^2f^1/2_L^2,

and one gets

    ∫_E|f(y)|^2dy
    ≤ Cc^-1_1δ^-m|k|^-1(ℒ_ν,α,kf_L^2f_L^2
    +c_2ν^1/2δ^-α/2ℒ_ν,α,kf^1/2_L^2f^3/2_L^2).

Since

    Cc^-1_1δ^-m|k|^-1c_2ν^1/2δ^-α/2ℒ_ν,α,kf^1/2_L^2f^3/2_L^2
    ≤   1/4f^2_L^2+C^2c^-2_1δ^-2m-α|k|^-2c^2_2νℒ_ν,α,kf_L^2f_L^2,

there exists a constant C=C(c_1,c_2), such that the (<ref>) can be written as

    ∫_E|f(y)|^2dy
    ≤C(δ^-m|k|^-1+|k|^-2νδ^-2m-α)ℒ_ν,α,kf_L^2f_L^2+1/4f^2_L^2.

Denoted E^c as the complement of E, then |E^c|≤ 2Nδ by the definition of E in (<ref>). For any α∈ (0,2), let p=4/(2-α),q=4/α, one has 

    ∫_E^c|f(y)|^2dy≲(∫_E^c1^qdy)^2/q(∫_E^c|f(y)|^pdy)^2/p≲ (2Nδ)^α/2f^2_L^p,

and 

    f_L^p≲f^1/2_L^2Λ_y^α/2f^1/2_L^2.

Then we deduce by (<ref>) that 

    ∫_E^c|f(y)|^2dy
       ≲(2Nδ)^α/2f_L^2Λ_y^α/2f_L^2
       ≲(2Nδ)^α/2ν^-1/2ℒ_ν,α,kf^1/2_L^2f^3/2_L^2
       ≤1/4f^2_L^2+ C(Nδ)^αν^-1ℒ_ν,α,kf_L^2f_L^2.

Combining (<ref>) and (<ref>), we have

    f^2_L^2≤ 2(Cδ^-m|k|^-1+C|k|^-2νδ^-2m-α+C(Nδ)^αν^-1)ℒ_ν,α,kf_L^2f_L^2.

Taking δ small enough and

    δ=c_3(ν/|k|)^1/m+α,

where c_3>0 is a small constant. Then there exists a constant C_0=C(c_1,c_2,c_3,α,m,N), such that

    Cδ^-m|k|^-1+C|k|^-2νδ^-2m-α+C(Nδ)^αν^-1≤ C_0ν^-m/m+α|k|^-α/m+α.

Therefore, we can imply by (<ref>) that

    ℒ_ν,α,kf_L^2≥ϵ_0ν^m/m+α|k|^α/m+αf_L^2,

where ϵ_0=1/2C_0. Since f is arbitrary, we deduce by the definition of Ψ(ℒ_ν,α,k) in  (<ref>) that

    Ψ(ℒ_ν,α,k)≥ϵ_0ν^m/m+α|k|^α/m+α.

This completes the proof of Lemma <ref>.



Next, we give the proof of Theorem <ref> based on the Lemma <ref>.



For any h(x,y)∈ L^2(𝕋^2), we consider the equation (<ref>) with initial data g_0(x,y)=P_≠h(x,y), then the solution can be written as

    g(t,x,y)=e^-tℒ_ν,αg_0=e^-tℒ_ν,αP_≠h,

and we can easily get P_0g=0. Here the operator ℒ_ν,α is defined in (<ref>), the operator P_0 and P_≠ are defined in (<ref>). Through the Fourier series,  one has 

    g(t,x,y)=e^-tℒ_ν,αP_≠h=∑_k≠0g_k(t,y)e^ikx,

where g_k(t,y) is defined in (<ref>) and is the solution of equation (<ref>). Then we have

    g_k(t,y)=e^-tℒ_ν,α,kg_0,k,

and the operator ℒ_ν,α,k is defined in (<ref>). By Lemma <ref>, one gets

    e^-tℒ_ν,α,k_L^2→ L^2≤ e^-λ_ν,α,kt+π/2,

where λ_ν,α,k=ϵ_0ν^m/m+α|k|^α/m+α. Then we deduce by Plancherel equality and (<ref>) that

    g^2_L^2   =e^-tℒ_ν,αP_≠h^2_L^2=∑_k≠ 0g_k^2_L^2=∑_k≠ 0e^-tℒ_ν,α,kg_0,k^2_L^2
       ≤∑_k≠ 0e^-2λ_ν,α,k t+πg_0,k^2_L^2≤max_k≠ 0e^-2λ_ν,α,kt+π∑_k≠ 0g_0,k^2_L^2
       ≤ e^-2λ'_ν,α t+πg_0^2_L^2≤ e^-2λ'_ν,α t+πh^2_L^2.

Since h is arbitrary, we have

    e^-tℒ_ν,αP_≠_L^2→ L^2≤ e^-λ'_ν,αt+π/2.

This completes the proof of Theorem <ref>.







§ SUPPRESSION OF BLOW-UP AND GLOBAL L^2 ESTIMATE



In this section, we will prove the  Theorem <ref>. We know from the Remark <ref> that we only need to show the Proposition <ref>. Some estimates are used in the proof. First, we give the following lemma.



Let α∈ [3/2,2),  the n^0 and n_≠ are the solutions of (<ref>) and (<ref>) with initial data n_0. If n_≠ satisfy the Assumption <ref>, then there exist ν_0=ν(n_0,α) and a positive constant B_1=B(n_0_L^2, M, C_≠,α), if ν<ν_0, we have

    n^0_L^∞(0, T^∗;L^2)≤ B_1.





Let us multiply both sides of (<ref>) by n^0 and integrate it over 𝕋, one has

    1/2d/dtn^0^2_L^2+νΛ_y^α/2 n^0^2_L^2+ν∫_𝕋∂_y(n^0 𝐁_1(n^0))n^0dy+ν∫_𝕋(∇·(n_≠𝐁_2(n_≠)))^0n^0dy=0.

Using integral by part, Lemma <ref> and energy estimate, one has

    | ∫_𝕋∂_y(n^0 𝐁_1(n^0))n^0dy|
    =|1/2∫_𝕋∂_y 𝐁_1(n^0)(n^0)^2dy|≲∂_y 𝐁_1(n^0)_L^2n^0^2_L^4,

and

    ∂_y 𝐁_1(n^0)_L^2n^0^2_L^4≲n^0^3-1/α_L^2Λ^α/2_y n^0^1/α_L^2≤1/4Λ^α/2_y n^0^2_L^2+Cn^0^3+1/2α-1_L^2.

Then the third term of (<ref>) is estimated as

    |ν∫_𝕋∂_y(n^0 𝐁_1(n^0))n^0dy|
    ≤ν/4Λ^α/2_y n^0^2_L^2+Cνn^0^3+1/2α-1_L^2.

Since

    ∫_𝕋(∇·(n_≠𝐁_2(n_≠)))^0n^0dy
       =1/2π∫_𝕋^2∇·(n_≠𝐁_2(n_≠))n^0dxdy
       =1/2π∫_𝕋^2∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))n^0dxdy,

where ∇^-1=(∂_x^-1,∂_y^-1), we deduce by Lemma <ref> and <ref> that

    |1/2π∫_𝕋^2∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))n^0dxdy|
    ≲Λ^-α/2∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2Λ^α/2_y n^0_L^2,

and 

    Λ^-α/2∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2≲Λ^1-α/2(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2
    ≲   Λ^1-α/2n_≠_L^2∂_y∇^-1·𝐁_2(n_≠)_L^∞
    +n_≠_L^2Λ^1-α/2∂_y∇^-1·𝐁_2(n_≠)_L^∞.

Combining α≥ 3/2 and Lemma <ref>, one has

    Λ^1-α/2n_≠_L^2≲Λ^α/2n_≠^2/α-1_L^2n_≠^2-2/α_L^2,
       Λ^1-α/2∂_y∇^-1·𝐁_2(n_≠)_L^∞≲∂_y∇^-1·𝐁_2(n_≠)^α-1_L^∞∂_y𝐁_2(n_≠)^2-α_L^4≲n_≠_L^4,
       n_≠_L^4≲Λ^α/2 n_≠^1/α_L^2n_≠^1-1/α_L^2,

and

    ∂_y∇^-1·𝐁_2(n_≠)_L^∞≲∂_y𝐁_2(n_≠)_L^4≲n_≠_L^4.

Therefore, we can easily get

    Λ^-α/2∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2Λ^α/2_y n^0_L^2
    ≲   Λ^α/2 n_≠^3/α-1_L^2n_≠^3-3/α_L^2Λ^α/2_y n^0_L^2
    +Λ^α/2 n_≠^1/α_L^2n_≠^2-1/α_L^2Λ^α/2_y n^0_L^2.

If α>3/2, we imply

    Λ^α/2 n_≠^3/α-1_L^2n_≠^3-3/α_L^2Λ^α/2_y n^0_L^2
    ≤   1/8Λ^α/2_y n^0^2_L^2+CΛ^α/2 n_≠^6/α-2_L^2n_≠^6-6/α_L^2
    ≤   1/8Λ^α/2_y n^0^2_L^2+1/128C_≠Λ^α/2 n_≠^2_L^2+C n_≠^6α-6/2α-3_L^2,

and

    Λ^α/2 n_≠^1/α_L^2n_≠^2-1/α_L^2Λ^α/2_y n^0_L^2
    ≤   1/8Λ^α/2_y n^0^2_L^2+CΛ^α/2 n_≠^2/α_L^2n_≠^4-2/α_L^2
    ≤   1/8Λ^α/2_y n^0^2_L^2+1/128C_≠Λ^α/2 n_≠^2_L^2+C n_≠^4α-2/α-1_L^2,

Then the fourth term of (<ref>) is estimated as

    |ν∫_𝕋(∇·(n_≠𝐁_2(n_≠)))^0n^0dy|
    ≤   ν/8Λ^α/2_y n^0^2_L^2+ν/64C_≠Λ^α/2 n_≠^2_L^2
       +Cν n_≠^6α-6/2α-3_L^2
    +Cνn_≠^4α-2/α-1_L^2.

Combining (<ref>), (<ref>) and (<ref>), one has

    d/dtn^0^2_L^2+νΛ_y^α/2 n^0^2_L^2≤    Cνn^0^3+1/2α-1_L^2+ν/32C_≠Λ^α/2 n_≠^2_L^2
       +Cν n_≠^6α-6/2α-3_L^2
    +Cν n_≠^4α-2/α-1_L^2.

By Nash inequality and  n^0_L^1≤ CM, one has

    -Λ_y^α/2 n^0^2_L^2≤ -n^0^2α+2_L^2/CM^2α.

Then we have

    d/dtn^0^2_L^2   ≤-νn^0^3+1/2α-1_L^2/CM^2α(n^0^2α-1-1/2α-1_L^2-CM^2α)
       +ν/32C_≠Λ^α/2 n_≠^2_L^2
    +Cν n_≠^6α-6/2α-3_L^2
    +Cν n_≠^4α-2/α-1_L^2.

Define

    G(t)=∫_0^tν/32C_≠Λ^α/2 n_≠^2_L^2
    +Cν n_≠^4α-2/α-1_L^2
    +Cν n_≠^6α-6/2α-3_L^2dτ,

and for any t∈ [0,T^∗], we deduce by Assumption <ref>, (<ref>), (<ref>) and ν is small enough that

    G(t)   ≤∫_0^s_0ν/32C_≠Λ^α/2 n_≠^2_L^2
    +Cνn_≠^4α-2/α-1_L^2
    +Cνn_≠^6α-6/2α-3_L^2dτ
          +∫_s_0^tν/32C_≠Λ^α/2 n_≠^2_L^2
    +Cν n_≠^4α-2/α-1_L^2
    +Cν n_≠^6α-6/2α-3_L^2dτ
       ≤ K_0n_0^2_L^2,

where K_0=K(n_0_L^2,α, C_≠,M). Then we obtain from (<ref>) that

    d/dt(n^0^2_L^2-G(t))
    ≤-νn^0^3+1/2α-1_L^2/CM^2α(n^0^2α-1-1/2α-1_L^2-G(t)-CM^2α).

Combining (<ref>) and (<ref>), we imply that

    n^0^2_L^2≤( C M^2α+n_0^2_L^2)^2α-1/4α^2-4α+(K_0+1)n_0^2_L^2≜ B^2_1.

This completes the proof of Lemma <ref>.




In the (<ref>), we used the Young's inequality, then

    6/α-2<2⇒α>3/2.

And for α=3/2, the (<ref>) can be written as

    Λ^α/2 n_≠^3/α-1_L^2n_≠^3-3/α_L^2Λ^α/2_y n^0_L^2≤1/8Λ^α/2_y n^0^2_L^2+CΛ^α/2 n_≠^2_L^2n_≠^2_L^2,

and by Assumption <ref> and (<ref>), one has

    Cν∫_0^tΛ^α/2 n_≠^2_L^2n_≠^2_L^2dτ
       ≤ Cν∫_0^s_0Λ^α/2 n_≠^2_L^2n_≠^2_L^2dτ
    +Cν∫_s_0^tΛ^α/2 n_≠^2_L^2n_≠^2_L^2dτ
       ≤ C( n_0^α/α-1_L^2+n)n_0^2_L^2
    +4CC_≠n_0^2_L^2ν∫_0^tΛ^α/2 n_≠^2_L^2dτ
       ≤ C( n_0^α/α-1_L^2+n)n_0^2_L^2+64 CC^2_≠n_0^4_L^2.

Thus, we can also obtain the estimate of n^0_L^2. In this paper, we use energy inequality in the case of α≥3/2, which will not be pointed out in later sections.





In the (<ref>), for any t∈ [0,s_0], we also have

    G(t)≤∫_0^s_0ν/32C_≠Λ^α/2 n_≠^2_L^2
    +Cνn_≠^4α-2/α-1_L^2
    +Cνn_≠^6α-6/2α-3_L^2dτ≤ K_0n_0^2_L^2.




.05in

To improve the Assumptions <ref>, we are left to complete the L^∞ L^∞ estimate of n. This will be achieved by the nonlinear maximum principle, see Lemma <ref>.




Let α∈ [3/2,2),  the n, n^0 and n_≠ are the solutions of(<ref>), (<ref>) and (<ref>) with initial data n_0. If n_≠ satisfy the Assumption <ref>, then there exist ν_0=ν(n_0,α) and a positive constant B_2=B(n_0_L^2,n_0_L^∞, B_1, M, C_≠,α), if ν<ν_0, we have

    n_L^∞(0, T^∗;L^∞)≤ B_2.





Define

    ℰ(t)=n(t,x_t,y_t)=sup_(x,y)∈𝕋^2n(t,x,y).

For any fixed t≥0, using the vanishing of a derivation at the point of maximum, we can observe that

    ∂_t n(t,x_t,y_t)=d/dtℰ(t), u(y)∂_x n(t,x_t,y_t)=0,

we obtain from (<ref>) that

    ∇·(n 𝐁(n))(t,x_t,y_t)=-ℰ^2(t)+nℰ(t),

and we denote

    (-Δ)^α/2n(t,x)|_x=x_t,y=y_t
    =(-Δ)^α/2ℰ(t),

Then combining (<ref>), we deduce that the ℰ=ℰ(t) follows

    d/dtℰ+ν(-Δ)^α/2ℰ
    -νℰ^2+νnℰ=0.

Combining Lemma <ref>, Assumption <ref> and Lemma <ref>, we know that ℰ(t) satisfies

    ℰ(t)≲n_L^2≲ (4C_≠)^1/2n_0_L^2+2n_0_L^2+B_1≜ K_1,

or ℰ(t) satisfies

    (-Δ)^α/2ℰ(t)≥ C(α)ℰ^1+α/n_L^2^α≥ C(α)ℰ^1+α/K_1^α.

Combining (<ref>) and (<ref>), one has

    d/dtℰ≤ -ν C(α)ℰ^1+α/K_1^α+νℰ^2-νnℰ≤-ν C(α)ℰ^1+α/K_1^α+νℰ^2.

Then we deduce by α≥3/2 and (<ref>) that

    ℰ(t)≤n_0_L^∞+K_1^α/α-1C(α)^-1/α-1.

Combining the definition of ℰ(t), (<ref>) and (<ref>), we have

    n_L^∞≤n_0_L^∞+K_1^α/α-1C(α)^-1/α-1+K_1≜ B_2.

This completes the proof of Lemma <ref>.



Next, we consider the L^∞Ḣ^α-1 estimate of n^0 in the following lemma.



Let α∈ [3/2,2),  the n^0 and n_≠ are the solutions of (<ref>) and (<ref>) with initial data n_0. If n_≠ satisfy the Assumption <ref>, then there exist ν_0=ν(n_0,α) and a positive constant B_3=B(Λ_y^α-1 n_0_L^2, n_0_L^2, B_1, B_2, C_≠, α), if ν<ν_0, we have

    Λ_y^α-1n^0_L^∞(0, T^∗;L^2)≤ B_3.





Let β=α-1 and applying Λ^β_y to (<ref>), one gets

    ∂_tΛ^β_yn^0+ν(-∂_yy)^α/2Λ^β_yn^0
    +ν∂_yΛ^β_y(n^0𝐁_1(n^0))+νΛ^β_y(∇·(n_≠𝐁_2(n_≠)))^0=0.

Multiplying both sides of (<ref>) by Λ^β_yn^0 and integrate over 𝕋, to obtain

    1/2d/dtΛ^β_y n^0^2_L^2+νΛ^β+α/2_y n^0^2_L^2   +ν∫_𝕋∂_yΛ^β_y(n^0𝐁_1(n^0))Λ^β_y n^0dy
       +ν∫_𝕋Λ^β_y(∇·(n_≠𝐁_2(n_≠)))^0Λ^β_y n^0dy=0.

We deduce by Lemma <ref> and <ref> that

    |∫_𝕋∂_yΛ^β_y(n^0𝐁_1(n^0))Λ^β_y n^0dy|
       ≲∂_yΛ^β-α/2_y(n^0𝐁_1(n^0))_L^2Λ^β+α/2_y n^0_L^2
       ≲Λ^α/2_y(n^0𝐁_1(n^0))_L^2Λ^β+α/2_y n^0_L^2,

and 

    Λ^α/2_y(n^0𝐁_1(n^0))_L^2≲Λ^α/2_yn^0_L^2𝐁_1(n^0)_L^∞
    +n^0_L^∞Λ^α/2_y𝐁_1(n^0)_L^2.

Combining energy estimate and the definition of 𝐁_1(n^0) in (<ref>), one has

    Λ^α/2_y𝐁_1(n^0)_L^2≲𝐁_1(n^0)^1/2_L^2Λ^α_y𝐁_1(n^0)^1/2_L^2≲𝐁_1(n^0)^1/2_L^∞Λ^β_yn^0^1/2_L^2,

and

    Λ^α/2_yn^0_L^2≲Λ^β_yn^0^2β/α_L^2Λ^β+α/2_yn^0^1-2β/α_L^2.

Thus, we deduce by Lemma <ref>, <ref> and <ref> that

    Λ^α/2_yn^0_L^2𝐁_1(n^0)_L^∞Λ^β+α/2_y n^0_L^2
    ≲   Λ^β_yn^0^2β/α_L^2Λ^β+α/2_yn^0^2-2β/α_L^2n^0_L^2
    ≤   1/8Λ^β+α/2_yn^0^2_L^2
    +K_2Λ^β_yn^0^2_L^2,

and

    n^0_L^∞Λ^α/2_y𝐁_1(n^0)_L^2Λ^β+α/2_y n^0_L^2
    ≲   n^0_L^∞n^0^1/2_L^2Λ^β_yn^0^1/2_L^2Λ^β+α/2_y n^0_L^2
    ≤   1/8Λ^β+α/2_yn^0^2_L^2
    +K_3Λ^β_yn^0_L^2,

where K_2=K(B_1,α), K_3=K(B_1,B_2). Then the third term of (<ref>) is estimated as

    |ν∫_𝕋∂_yΛ^β_y(n^0𝐁_1(n^0))Λ^β_y n^0dy|
    ≤ν/4Λ^β+α/2_yn^0^2_L^2+ν K_2Λ^β_yn^0^2_L^2
    +ν K_3Λ^β_yn^0_L^2.

By Lemma <ref>, <ref> and <ref>, one gets

    |∫_𝕋Λ^β_y(∇·(n_≠𝐁_2(n_≠)))^0Λ^β_y n^0dy|
    
    =   |1/2π∫_𝕋^2∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))Λ^2β_y n^0dxdy|
    ≲   Λ^α/2-1∂_y(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2Λ^2β+1-α/2_y n^0_L^2
    ≲   Λ^α/2(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2Λ^β+α/2_y n^0_L^2,

and 

    Λ^α/2(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2≲   Λ^α/2n_≠_L^2∂_y∇^-1·𝐁_2(n_≠)_L^∞
       +n_≠_L^∞Λ^α/2∂_y∇^-1·𝐁_2(n_≠)_L^2.

Combining

    ∂_y∇^-1·𝐁_2(n_≠)_L^∞≲n_≠_L^4,  n_≠_L^4≲ n_≠_L^∞,

and Lemma <ref>, one has

    Λ^α/2∂_y∇^-1·𝐁_2(n_≠)_L^2   ≲∂_y∇^-1·𝐁_2(n_≠)^1-α/2_L^2∂_y 𝐁_2(n_≠)^α/2_L^2
       ≲∂_y∇^-1·𝐁_2(n_≠)^1-α/2_L^∞∂_y 𝐁_2(n_≠)^α/2_L^2
       ≲n_≠^1-α/2_L^4n_≠^α/2_L^2.

Then we can easily get 

    Λ^α/2(n_≠∂_y∇^-1·𝐁_2(n_≠))_L^2Λ^β+α/2_y n^0_L^2
    ≲   Λ^α/2n_≠_L^2n_≠_L^4Λ^β+α/2_y n^0_L^2
    +n_≠_L^∞n_≠^1-α/2_L^4n_≠^α/2_L^2Λ^β+α/2_y n^0_L^2
    ≤   1/4Λ^β+α/2_y n^0_L^2+ K_4(Λ^α/2n_≠^2_L^2+n_≠^α_L^2),

where K_4=K(B_2). Therefore, the fourth term of (<ref>) is estimated as

    |ν∫_𝕋Λ^β_y(∇·(n_≠𝐁_2(n_≠)))^0Λ^β_y n^0dy|≤ν/4Λ^β+α/2_y n^0_L^2+ ν K_4(Λ^α/2n_≠^2_L^2+n_≠^α_L^2).

Combining (<ref>), (<ref>) and (<ref>), one has

    d/dtΛ^β_y n^0^2_L^2+νΛ^β+α/2_y n^0^2_L^2   ≤ν K_5Λ^β_yn^0^2_L^2
    +ν K_5Λ^β_yn^0_L^2
          +ν K_5(Λ^α/2n_≠^2_L^2+n_≠^α_L^2),

where K_5=K_2+K_3+K_4. Since n^0_L^2≤ B_1, we imply 

    -Λ^β+α/2_yn^0^2_L^2≤ -Λ^β_yn^0^3α-2/α-1_L^2/Cn^0_L^2^α/α-1≤ -Λ^β_yn^0^3α-2/α-1_L^2/CB^α/α-1_1.

Then the (<ref>) can be written as

    d/dtΛ^β_y n^0^2_L^2   ≤-Λ^β_yn^0^2_L^2/CB^α/α-1_1(Λ^β_y n^0^1+1/α-1_L^2-CK_5B_1^α/α-1-CK_5B_1^α/α-1Λ^β_y n^0^-1_L^2)
            +ν K_5(Λ^α/2n_≠^2_L^2+n_≠^α_L^2).

Define

    G(t)=ν K_5∫_0^tΛ^α/2n_≠^2_L^2+n_≠^α_L^2dτ,

and by Assumption <ref>, (<ref>), (<ref>) and ν is small, one has

    G(t)   ≤ν K_5 ∫_0^s_0Λ^α/2n_≠^2_L^2+n_≠^α_L^2dτ+ν
    K_5∫_s_0^tΛ^α/2n_≠^2_L^2+n_≠^α_L^2dτ
       ≲ K_5(n_0^α/α-1_L^2+M)n_0^2_L^2
    +K_5n_0^α_L^2
           +16C_≠K_4n_0^2_L^2+2νλ^-1_ν,αα^-1K_5(4C_≠)^α/2n_0^α_L^2
       ≤ K_5(n_0^α/α-1_L^2+M+32C_≠)n_0^2_L^2
    +K_5n_0^α_L^2
       ≤ K_6n_0^2_L^2+K_5n_0^α_L^2,

where K_6=K(n_0_L^2, α,C_≠, M, K_5). Combining (<ref>) and (<ref>), we obtain

    d/dt(Λ^β_y n^0^2_L^2-G(t))
    ≤   -Λ^β_yn^0^2_L^2/CB^α/α-1_1(Λ^β_y n^0^1+1/α-1_L^2-CK_5B_1^α/α-1-CK_5B_1^α/α-1Λ^β_y n^0^-1_L^2)
    ≤   -Λ^β_yn^0^2_L^2/CB^α/α-1_1(Λ^β_y n^0^1+1/α-1_L^2-G(t)-CK_5B_1^α/α-1-CK_5B_1^α/α-1Λ^β_y n^0^-1_L^2).

If

    Λ^β_y n^0^1+1/α-1_L^2-G(t)-CK_5B_1^α/α-1-CK_5B_1^α/α-1Λ^β_y n^0^-1_L^2≥0,

then we deduce by (<ref>) that

    Λ^β_y n^0^2_L^2≤Λ^β_y n_0^2_L^2+K_6n_0^2_L^2+K_5n_0^α_L^2.

If

    Λ^β_y n^0^1+1/α-1_L^2-G(t)-CK_5B_1^α/α-1-CK_5B_1^α/α-1Λ^β_y n^0^-1_L^2<0,

we imply that there exist a positive constant

    K_7=K(α,B_1, K_5, C_≠, n_0_L^2,M)<∞,

such that

    Λ^β_y n^0^2_L^2≤ K_7.

Combining (<ref>) and (<ref>), we have

    Λ^β_y n^0^2_L^2≤Λ^β_y n_0^2_L^2+CK_6n_0^2_L^2+CK_5n_0^α_L^2
    +K_7≜ B_3^2.

This completes the proof of Lemma <ref>.



Finally, we prove the Proposition <ref> by the enhanced dissipation of shear flow.



We will prove (P-1) and (P-2) in the Proposition <ref> respectively.

.05in

(1) Nonzero mode L^2Ḣ^α/2 estimate of n_≠.  Let us multiply both sides of (<ref>) by n_≠ and integrate over 𝕋^2, we obtain

    1/2d/dtn_≠^2_L^2+νΛ^α/2n_≠^2_L^2
    +ν∫_𝕋^2F(n^0,n_≠)n_≠dxdy=0,

where

    F(n^0,n_≠)   =∇ n^0·𝐁_2(n_≠)+∂_yn_≠𝐁_1(n^0)
          +(∇·(n_≠𝐁_2(n_≠)))_≠-n^0n_≠-n_≠(n^0-n),

and 𝐁_1(n^0) and 𝐁_2(n_≠) are defined in (<ref>). By Lemma <ref>, we obtain

    |∫_𝕋^2∇ n^0·𝐁_2(n_≠)n_≠dxdy|
       =|∫_𝕋^2∂_y n^0∂_y∇^-1·𝐁_2(n_≠)n_≠dxdy|
       ≲Λ_y^βn^0_L^2Λ^1-β(∂_y∇^-1·𝐁_2(n_≠)n_≠)  _L^2.

Combining energy estimate Lemma <ref> and <ref>, one has

    Λ^1-β(∂_y∇^-1·𝐁_2(n_≠)n_≠)_L^2   ≲n_≠_L^4Λ^1-β∂_y∇^-1·𝐁_2(n_≠)_L^4
          +∂_y∇^-1·𝐁_2(n_≠)_L^∞Λ^1-βn_≠_L^2
       ≲n_≠^2_L^4+n_≠_L^4Λ^1-βn_≠_L^2,

and

    Λ_y^βn^0_L^2Λ^1-β(∂_y∇^-1·𝐁_2(n_≠)n_≠)_L^2
    ≲   Λ_y^βn^0_L^2n_≠^2_L^4
    +Λ_y^βn^0_L^2n_≠_L^4Λ^1-βn_≠_L^2
    ≲   Λ_y^βn^0_L^2n_≠^2-2/α_L^2Λ^α/2n_≠^2/α_L^2
    +Λ_y^βn^0_L^2n_≠^4-5/α_L^2Λ^α/2n_≠^5/α-2_L^2
    ≤   1/2Λ^α/2n_≠^2_L^2+C(Λ^β n^0^α/α-1_L^2+Λ^β n^0^2α/4α-5_L^2)n_≠^2_L^2

Thus, we have

    |ν∫_𝕋^2∇ n^0·𝐁_2(n_≠)n_≠dxdy|
    ≤ν/2Λ^α/2n_≠^2_L^2+Cν(Λ^β n^0^α/α-1_L^2+Λ^β n^0^2α/4α-5_L^2)n_≠^2_L^2.

By Lemma <ref>, one gets

    |ν∫_𝕋^2∂_yn_≠𝐁_1(n^0)n_≠dxdy|
    =|ν/2∫_𝕋^2n^2_≠∂_y𝐁_1(n^0)dxdy|
    ≲νn_≠^2_L^2n^0_L^∞.

Combining the definition of P_≠ in (<ref>) and Lemma <ref>, one has

    |ν∫_𝕋^2(∇·(n_≠𝐁_2(n_≠)))_≠n_≠dxdy|
       =|ν∫_𝕋^2∇·(n_≠𝐁_2(n_≠))n_≠dxdy|
       =|ν/2∫_𝕋^2∇·𝐁_2(n_≠)n^2_≠dxdy|
       ≲νn_≠^2_L^2n^0_L^∞.

And we can easily get

    |ν∫_𝕋^2-n^0(n_≠)^2-(n_≠)^2(n^0-n)dxdy|
    ≲νn^0_∞ n__L^2^2.

Thus, we obtain from (<ref>)-(<ref>) that

    |ν∫_𝕋^2F(n^0,n_≠)n_≠dxdy|
    ≤   ν/2Λ^α/2n_≠^2_L^2+Cν(Λ^β n^0^α/α-1_L^2+Λ^β n^0^2α/4α-5_L^2+ n^0_∞)n_≠^2_L^2.

Combining (<ref>), (<ref>), Lemma <ref> and <ref>, one has

    d/dtn_≠^2_L^2+νΛ^α/2n_≠^2_L^2≤ν K_8n_≠^2_L^2,

where

    K_8=K(α, B_2,B_3)<∞.

Then by time integral in (<ref>) from s to t, the ν is small and Assumption <ref>, one gets

    n_≠(t)^2_L^2+ν∫_s^tΛ^α/2 n_≠(τ)^2_L^2dτ
    ≤    4C_≠ν K_8  n_0^2_L^2∫_s^te^-λ_ν,α (τ-s_0)dτ+n_≠(s)^2_L^2
    ≤    4C_≠C ν K_8 λ^-1_ν,α e^-λ_ν,α (s-s_0)n_0^2_L^2 +4C_≠e^-λ_ν,α (s-s_0)n_0^2_L^2
    ≤    8C_≠e^-λ_ν,α (s-s_0)n_0^2_L^2.

Thus, we have

    ν∫_s^tΛ^α/2 n_≠(τ)^2_L^2dτ≤ 8C_≠e^-λ_ν,α (s-s_0)n_0^2_L^2.



.05in

(2) Nonzero mode L^∞ L^2 estimate of n_≠. Denote

    𝒮_t=e^-tL_ν,α,

where the L_ν,α is defined in (<ref>). Combining Duhamel's principle, (<ref>) and (<ref>), one has

    n_≠(t+s)=𝒮_tn_≠(s)
    -ν∫_s^t+s𝒮_t+s-τF(n^0,n_≠)dτ.

Then we have

    n(t+s)_L^2≤   𝒮_tn_≠(s)_L^2+Cν∫_s^s+t∇ n^0·𝐁_2(n_≠)_L^2
    +(∇·(n_≠𝐁_2(n_≠)))_≠_L^2
       +n^0n_≠_L^2+n_≠(n^0-n)_L^2dτ
    +Cν∫_s^t+s𝒮_t+s-τ(∂_yn_≠𝐁_1(n^0))_L^2dτ.

By Corollary <ref>, one gets

    𝒮_tn_≠(s)_L^2≤ e^-λ_ν,α t+π/2n_≠(s)_L^2.

Combining Lemma <ref> and energy estimate, one has

    ∇ n^0·𝐁_2(n_≠)_L^2≲∂_y n^0_L^2n_≠_L^4≲∂_y n^0_L^2n_≠_L^2+∂_y n^0_L^2Λ^α/2n_≠_L^2,

then we deduce by Assumption <ref> and Lemma <ref> that

    ν∫_s^s+t∇ n^0·𝐁_2(n_≠)_L^2dτ
    ≲   ν∫_s^s+t∂_y n^0_L^2n_≠_L^2dτ+ν∫_s^s+t∂_y n^0_L^2Λ^α/2n_≠_L^2dτ
    ≲   (ν∫_s^t+s∂_y n^0^2_L^2dτ)^1/2(ν∫_s^t+sn_≠^2_L^2dτ)^1/2
       +(ν∫_s^t+s∂_y n^0^2_L^2dτ)^1/2(ν∫_s^t+sΛ^α/2 n_≠^2_L^2dτ)^1/2
    ≲   (ν^6/7t)^1/2((λ^-1_ν,αν)^1/2+4)C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

By Lemma <ref> and energy estimate, one has

    (∇·(n_≠𝐁_2(n_≠)))_≠_L^2   ≲∇ n_≠_L^2n_≠_L^4+n_≠_L^∞n_≠_L^2
       ≲∇ n_≠_L^2n_≠_L^2+∇ n_≠_L^2Λ^α/2n_≠_L^2+n_≠_L^∞n_≠_L^2,

and we deduce by Assumption <ref>, Lemma <ref> and <ref> that

    ν∫_s^t+s∇ n_≠_L^2n_≠_L^2dτ   ≲(ν∫_s^t+s∇ n_≠^2_L^2dτ)^1/2(ν∫_s^t+sn_≠^2_L^2dτ)^1/2
       ≲(ν^6/7t)^1/2(λ^-1_ν,αν)^1/2C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2,


    ν∫_s^t+s∇ n_≠_L^2Λ^α/2n_≠_L^2dτ   ≲(ν∫_s^t+s∇ n_≠^2_L^2dτ)^1/2(ν∫_s^t+sΛ^α/2n_≠^2_L^2dτ)^1/2
       ≲(ν^6/7t)^1/2C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2,

and

    ν∫_s^t+sn_≠_L^∞n_≠_L^2dτ   ≲(ν∫_s^t+s n_≠^2_L^∞dτ)^1/2(ν∫_s^t+sn_≠^2_L^2dτ)^1/2
       ≲ B_2ν (λ^-1_ν,α t)^1/2C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

Then we have

    ν∫_s^s+t(∇·(n_≠𝐁_2(n_≠)))_≠_L^2dτ
    ≲   ((ν^6/7t)^1/2(λ^-1_ν,αν)^1/2
    +(ν^6/7t)^1/2+B_2ν (λ^-1_ν,α t)^1/2)C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

Combining Assumption <ref> and Lemma <ref>, one has

    ν∫_s^s+tn^0n_≠_L^2+n_≠(n^0-n)_L^2dτ   ≲ν∫_s^s+tn^0_L^∞n_≠_L^2dτ
       ≲ B_2λ^-1_ν,αν C_≠^1/2e^-λ_ν,α(s-s_0)/2n_0_L^2.

Next, we consider the last term of the right-hand side of (<ref>). Since

    𝒮_t+s-τ(∂_yn_≠𝐁_1(n^0))
    =𝒮_t+s-τ∂_y(n_≠𝐁_1(n^0))
    -𝒮_t+s-τ(n_≠∂_y𝐁_1(n^0)),

one has

    ν∫_s^t+s𝒮_t+s-τ(∂_yn_≠𝐁_1(n^0))_L^2dτ
    ≲   ν∫_s^t+s𝒮_t+s-τ∂_y(n_≠𝐁_1(n^0))_L^2dτ
    +ν∫_s^t+s𝒮_t+s-τ(n_≠∂_y𝐁_1(n^0))_L^2dτ.

By Lemma <ref>, we have

    𝒮_t+s-τ∂_y(n_≠𝐁_1(n^0))(τ)
    =   -Λ^α/2_y𝒮_t+s-τΛ^-α/2_y∂_y(n_≠𝐁_1(n^0))(τ)
       -∫_τ^t+se^-(t+s-τ_0)ℒ_ν,αℛ_n_≠𝐁_1(n^0)(τ_0)dτ_0,

where

    ℛ_n_≠𝐁_1(n^0)=C_α∑_k∈ℤP.V.∫_𝕋𝒦(y,y)
    Λ_x^α/2-1∂_x𝒮_τ_0(Λ^1-α/2_xΛ^-α/2_y∂_y(n_≠𝐁_1(n^0)))dy.

Then we have

    𝒮_t+s-τ∂_y(n_≠𝐁_1(n^0))(τ)_L^2≲   Λ^α/2_y𝒮_t+s-τΛ^-α/2_y∂_y(n_≠𝐁_1(n^0))(τ)_L^2
       +∫_τ^t+sℛ_n_≠𝐁_1(n^0)(τ_0)_L^2dτ_0.

By Lemma <ref>, one has

    Λ^α/2_y𝒮_t+s-τΛ^-α/2_y∂_y(n_≠𝐁_1(n^0))(τ)_L^2≲Λ^α/2_y𝒮_t+s-τΛ^1-α/2_y(n_≠𝐁_1(n^0))(τ)_L^2,

we deduce by Lemma <ref> that

    Λ^1-α/2_y(n_≠𝐁_1(n^0))(τ)_L^2   ≲Λ^1-α/2n_≠_L^2𝐁_1(n^0)_L^∞
    +n_≠_L^2Λ^1-α/2_y𝐁_1(n^0)_L^∞
       ≲n_≠^2-2/α_L^2Λ^α/2n_≠^2/α-1_L^2n^0_L^2+n_≠_L^2n^0_L^2.

Since

    ν∫_s^t+sΛ^α/2_y𝒮_t+s-τn_≠^2-2/α_L^2Λ^α/2n_≠^2/α-1_L^2n^0_L^2dτ
    ≲    B_1(ν∫_s^t+sΛ^α/2𝒮_t+s-τ^2 dτ)^1/2(ν∫_s^t+sn_≠^4-4/α_L^2Λ^α/2n_≠^4/α-2_L^2dτ)^1/2
    ≲    B_1(ν∫_s^t+sn_≠^2_L^2dτ)^1-1/α(ν∫_s^t+sΛ^α/2n_≠^2_L^2dτ)^1/α-1/2
    ≲    (νλ^-1_ν,α)^1-1/αB_1C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2,

and

    ν∫_s^t+sΛ^α/2𝒮_t+s-τn_≠_L^2n^0_L^2dτ
    ≲    B_1(ν∫_s^t+sΛ^α/2𝒮_t+s-τ^2 dτ)^1/2(ν∫_s^t+sn_≠^2_L^2dτ)^1/2
    ≲   (νλ^-1_ν,α)^1/2 B_1 C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2,

we have

    ν∫_s^s+tΛ^α/2_y𝒮_t+s-τΛ^-α/2_y∂_y(n_≠𝐁_1(n^0))(τ)_L^2dτ
    ≲   ((νλ^-1_ν,α)^1-1/α+(νλ^-1_ν,α)^1/2)B_1 C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

By Hölder's inequality and (<ref>), we have

    ℛ_n_≠𝐁_1(n^0)(τ_0) _L^2   ≲(C_α∑_k∈ℤP.V.∫_𝕋^2𝒦^2(y,y)dydy)^1/2
         ·(∫_𝕋^2[Λ_x^α/2-1∂_x𝒮_τ_0(Λ^1-α/2_xΛ^-α/2_y∂_y(n_≠𝐁_1(n^0)))]^2dydx)^1/2
       ≲Λ_x^α/2-1∂_x𝒮_τ_0(Λ^1-α/2_xΛ^-α/2_y∂_y(n_≠𝐁_1(n^0))) _L^2
       ≲Λ^α/2𝒮_τ_0P_≠Λ^2-α(n_≠𝐁_1(n^0)) _L^2,

and combining lemma <ref> and <ref>, one has

    Λ^2-α(n_≠𝐁_1(n^0)) _L^2   ≲Λ^2-α n_≠_L^2𝐁_1(n^0)_L^∞
    +n_≠_L^2Λ^2-α𝐁_1(n^0)_L^∞
       ≲n_≠^3-4/α_L^2Λ^α/2n_≠^4/α-2_L^2n^0_L^2
    +n_≠_L^2n^0_L^2.

Since s≥ s_0, we have

    ν∫_s^t+s∫_τ^t+sΛ^α/2𝒮_τ_0P_≠n_≠^3-4/α_L^2Λ^α/2n_≠^4/α-2_L^2n^0_L^2dτ_0dτ
    ≲   B_1[ν∫_s^t+s(∫_τ^t+sΛ^α/2𝒮_τ_0P_≠dτ_0
    )^2dτ]^1/2[ ν∫_s^t+sn_≠^6-8/α_L^2Λ^α/2n_≠^8/α-4_L^2dτ]^1/2
    ≲   B_1(tλ_ν,α+1)^1/2(  ν∫_s^t+sn_≠^2_L^2
    dτ)^3/2-2/α(ν∫_s^t+sΛ^α/2n_≠^2_L^2dτ)^2/α-1
    ≲   B_1(tλ_ν,α+1)^1/2(νλ^-1_ν,α)^3/2-2/α C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2

and

    ν∫_s^t+s∫_τ^t+sΛ^α/2𝒮_τ_0P_≠n_≠_L^2n^0_L^2dτ_0dτ
    ≲    B_1[ν∫_s^t+s(∫_τ^t+sΛ^α/2𝒮_τ_0P_≠dτ_0
    )^2dτ]^1/2[ ν∫_s^t+sn_≠^2_L^2dτ]^1/2
    ≲    B_1(tλ_ν,α+1)^1/2(νλ^-1_ν,α)^1/2 C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2,

where we use that for  s≥ s_0, one has

    [ν∫_s^t+s(∫_τ^t+sΛ^α/2𝒮_τ_0P_≠dτ_0
    )^2dτ]^1/2
    ≲   (t/λ_ν,α+1/λ^2_ν,α)^1/2e^-λ_ν,αs/2≲(tλ_ν,α+1)^1/2.

Therefore, we have

    ν∫_s^t+s∫_τ^t+sℛ_n_≠𝐁_1(n^0)(τ_0)_L^2dτ_0dτ
    ≲   B_1(tλ_ν,α+1)^1/2((νλ^-1_ν,α)^3/2-2/α+(νλ^-1_ν,α)^1/2) C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

By Lemma <ref>, one has

    ν∫_s^t+s𝒮_t+s-τ(n_≠∂_y𝐁_1(n^0))_L^2dτ≲   ν∫_s^t+sn_≠_L^2n^0_L^∞dτ
    ≲   (ν∫_s^t+sn_≠^2_L^2dτ)^1/2(ν∫_s^t+sn^0^2_L^∞dτ)^1/2
    ≲   ν B_2(λ^-1_ν,α t)^1/2C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

Combining (<ref>), (<ref>) and (<ref>), and denote

    K_7=   ((νλ^-1_ν,α)^1-1/α+(νλ^-1_ν,α)^1/2)B_1
       +B_1(tλ_ν,α+1)^1/2((νλ^-1_ν,α)^3/2-2/α
    +(νλ^-1_ν,α)^1/2)+ν B_2(λ^-1_ν,α t)^1/2,

then we have

    ν∫_s^t+s𝒮_t+s-τ(∂_yn_≠𝐁_1(n^0))_L^2dτ
    ≲ K_7C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

Combining (<ref>)-(<ref>) and (<ref>), we have

    n(t+s)_L^2≤   e^-λ_ν,α t+π/2n_≠(s)_L^2+(ν^6/7t)^1/2((λ^-1_ν,αν)^1/2+4)C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2
       +((ν^6/7t)^1/2(λ^-1_ν,αν)^1/2
    +(ν^6/7t)^1/2+B_2ν (λ^-1_ν,α t)^1/2)C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2
       +B_2λ^-1_ν,αν C_≠^1/2e^-λ_ν,α(s-s_0)/2n_0_L^2+ K_7C^1/2_≠e^-λ_ν,α(s-s_0)/2n_0_L^2.

Taking τ^∗=8λ^-1_ν,α, combining (<ref>), α≥3/2 and ν is small, one has

    n_≠(τ^∗+s)_L^2≤1/4e^-4n_≠(s)_L^2+1/2e^-4e^-λ_ν,α (s-s_0)/2n_0_L^2.

Then we can easily get

    n_≠(τ^∗+s_0)_L^2≤ e^-4n_0_L^2.

Assume that for k=m∈ℤ^+,  one has

    n_≠(mτ^∗+s_0)_L^2≤ e^-4mn_0_L^2,

then for k=m+1, we have

    n_≠((m+1)τ^∗+s)_L^2   ≤1/4e^-4n_≠(mτ^∗+s_0)_L^2+1/2e^-4e^-4mn_0_L^2
       ≤ e^-4(m+1)n_0_L^2.

By same argument, we know that for any s_0≤ t≤ T^∗, there exist τ_0 and m∈ℤ^+, such that

    t=mτ_∗+τ_0+s_0,   0≤τ_0≤τ^∗,

then by the local estimate of (<ref>) and (<ref>), one has

    n_≠(mτ^∗+s_0+τ_0)_L^2≤ 2n_≠(mτ^∗+s_0)_L^2≤2e^-4mn_0_L^2.

Then we obtain

    n_≠(t)^2_L^2≤ 4e^-8mn_0^2_L^2≤ 4e^8e^-λ_ν,α (t-s_0)n_0^2_L^2≤ 2C_≠e^-λ_ν,α (t-s_0)n_0^2_L^2,

where C_≠≥ 2e^8. This completes the proof of Proposition <ref>.



In this paper, we choose C_≠≥ 2e^8, and combining local estimate, Assumption <ref> and Proposition <ref>, we obtain the global L^2 estimate of solution to equation (<ref>).







§ ENERGY ESTIMATES


In this section, we establish some useful energy estimate for this paper.




 §.§ Attractive kernel estimate

The attractive kernel operator 𝐁_1(n^0) and 𝐁_2(n_≠) are defined in (<ref>).
We have following estimates.


The operator 𝐁_1(n^0) and 𝐁_2(n_≠) are defined in (<ref>), one has


  (1) The estimates of 𝐁_1(n^0):

    ∂_y𝐁_1(n^0)_L^p   ≲n^0_L^P,       1≤ p≤∞,
    Λ^β_y𝐁_1(n^0)_L^2   ≲Λ^β-1n^0_L^2,      β>1,
    𝐁_1(n^0)_L^∞+Λ^β_y𝐁_1(n^0)_L^∞   ≲n^0_L^2,         0<β≤1/2.


  (2)The estimates of 𝐁_2(n_≠)

    ∇·𝐁_2(n_≠)_L^p+∂_x𝐁_2(n_≠)_L^p
    +∂_y𝐁_2(n_≠)_L^p   ≲n_≠_L^p,     1<p<∞,
    𝐁_2(n_≠)_L^∞   ≲n_≠_L^4.






We finish the proof by following two steps.

(1) The estimates of 𝐁_1(n^0).
According to the definition of 𝐁_1(n^0) in (<ref>), we have

    ∂_y𝐁_1(n^0)_L^p=n^0-n_L^p≲n^0_L^p+n.

and by  Hölder's inequality, one has

    n=1/4π^2∫_𝕋^2n(t,x,y)dxdy=1/2π∫_𝕋n^0(t,y)dy≲n^0_L^p,

thus we have

    ∂_y𝐁_1(n^0)_L^p≲n^0_L^p.

Combining the definition of 𝐁_1(n^0) and β>1, we can easily get

    Λ^β_y𝐁_1(n^0)_L^2=Λ^β-1_y∂_y𝐁_1(n^0)_L^2
    =Λ^β-1_y(n^0-n)_L^2≲Λ^β-1_yn^0_L^2.

By Gagliardo-Nirenberg inequality, one has

    𝐁_1(n^0)_L^∞≲𝐁_1(n^0)^1/2_L^2∂_y𝐁_1(n^0)^1/2_L^2≲𝐁_1(n^0)^1/2_L^∞∂_y𝐁_1(n^0)^1/2_L^2,

then we deduce by (<ref>) that

    𝐁_1(n^0)_L^∞≲∂_y𝐁_1(n^0)_L^2≲n^0_L^2.

By Gagliardo-Nirenberg inequality, (<ref>), (<ref>) and 0<β≤ 1/2, one has

    Λ^β_y𝐁_1(n^0)_L^∞   ≲𝐁_1(n^0)^1/3-2β/3_L^1∂_y 𝐁_1(n^0)^2/3+2β/3_L^2
       ≲𝐁_1(n^0)^1/3-2β/3_L^∞∂_y 𝐁_1(n^0)^2/3+2β/3_L^2≲n^0_L^2.


.05in

(2) The estimates of 𝐁_2(n_≠).
Obviously, for 1<p<∞, one has

    ∇·𝐁_2(n_≠)_L^p=n_≠_L^p.

Since for 0<p<∞, one has

    ∂_x𝐁_2(n_≠)_L^p+∂_y𝐁_2(n_≠)_L^p≲∑_i,j=1^2∂_i∂_j(-Δ)^-1n_≠_L^p,

and

    ∂_i∂_j(-Δ)^-1n_≠_L^p=ℛ_iℛ_jn_≠_L^p≲n_≠_L^p,

where ∂_1=∂_x, ∂_2=∂_y, ℛ_i=∂_iΛ^-1 is Riesz transforms, we have

    ∂_x𝐁_2(n_≠)_L^p+∂_y𝐁_2(n_≠)_L^p≲n_≠_L^p.

By Gagliardo-Nirenberg inequality, one has

    𝐁_2(n_≠)_L^∞≲𝐁_2(n_≠)^1/2_L^2∇·𝐁_2(n_≠)^1/2_L^4≲𝐁_2(n_≠)^1/2_L^∞∇·𝐁_2(n_≠)^1/2_L^4,

then we deduce by (<ref>) that

    𝐁_2(n_≠)_L^∞≲∇·𝐁_2(n_≠)_L^4≲n_≠_L^4.


Combining (<ref>)-(<ref>), we finish the proof of Lemma <ref>.




In the proof of Lemma <ref>, we used the properties of Riesz transforms. Namely, the Riesz transforms is bounded in L^p for 1<p<∞, the details can see <cit.>.




 §.§ Local estimate

The local estimate makes the Assumption <ref> reasonable, which is the premise of bootstrap argument. Here we give the local estimate of solution of equation (<ref>).


Let α∈[3/2,2) and n_≠ be a solution of equation (<ref>) with initial data n_0≥0, n_0∈ H^γ(𝕋^2)∩ L^1(𝕋^2), γ>1+α. Assume that the u(y)=cos y is Kolmogorov flow. Then there exist time s_0>0, t^∗=s_0+8λ^-1_ν,α, ν is small enough and a  positive constant C_≠, such that for any s_0≤ s≤ t≤ t^∗, one has

    ν∫_s^tΛ^α/2 n_≠^2_L^2dτ≤ 16C_e^-λ_ν,α(s-s_0)n_0_L^2^2,

and for any s_0≤ t≤ t^∗, one has

    n_≠(t)^2_L^2≤ 4C_e^-λ_ν,α(t-s_0)n_0^2_L^2,

where λ_ν,α is defined in (<ref>).



Let us multiply both sides of (<ref>) by n and integrate over 𝕋^2, to obtain

    1/2d/dtn^2_L^2+νΛ^α/2n^2_L^2
    +ν∫_𝕋^2∇·(n𝐁(n))ndxdy.

Using integral by part, Lemma <ref> and energy estimate, one has

    |∫_𝕋^2∇·(n𝐁(n))ndxdy|
    =|1/2∫_𝕋^2∇·𝐁(n)n^2dxdy|
    ≤n^3_L^3+nn^2_L^2,

and 

    n^3_L^3≲Λ^α/2n^2/α_L^2n^3-2/α_L^2≤1/2Λ^α/2n_L^2^2+Cn_L^2^3α-2/α-1.

The (<ref>) can be written as

    d/dtn_L^2^2+νΛ^α/2n_L^2^2≤ Cνn_L^2^3α-2/α-1+2νnn_L^2^2.

Then we deduce by (<ref>) that there exists

    t_0=α-1/nανln(4^α/2α-2(2n
    +Cn_0_L^2^α/2α-2)/2n+4^α/2α-2
    Cn_0_L^2^α/2α-2)=O(1/ν),

such that for any 0≤ t≤ t_0, one has

    n^2_L^2≤ 4n_0^2_L^2.

Taking s_0=t_0/2 and t^∗=s_0+8λ^-1_ν,α, we know that for ν is small enough, one has

    s_0≤ t^∗≤ t_0.

Then for any s_0≤ t≤ t^∗, one has

    n^0^2_L^2+n_≠^2_L^2≤ 4n_0^2_L^2≤ 4C_e^-2N_0n_0_L^2^2≤ 4C_e^-λ_ν,α(t-s_0)n_0^2_L^2,

where C_≥ e^8. Combining (<ref>) and (<ref>), we imply that for any s_0≤ s≤ t≤ t^∗ and ν is small, one has

    ν∫_s^tΛ^α/2n_L^2^2dτ   ≤ Cν∫_s^tn_L^2^3α-2/α-1dτ+2νn∫_s^tn_L^2^2dτ
    +n(s)_L^2^2
       ≤  Cν(4C_≠n_0^2_L^2)^3α-2/2α-2∫_s^te^-λ_ν,ατdτ e^λ_ν,α s_0
          +8nν C_≠n_0^2_L^2∫_s^te^-λ_ν,ατdτ e^λ_ν,α s_0+4C_e^-λ_ν,αsn_0_L^2^2e^λ_ν,α s_0
       ≤ Cν(4C_≠n_0^2_L^2)^3α-2/2α-2λ^-1_ν,α e^-λ_ν,α se^λ_ν,α s_0
          +8nν C_≠n_0^2_L^2λ_ν,α^-1e^-λ_ν,α se^λ_ν,α s_0+4C_e^-λ_ν,αsn_0_L^2^2e^λ_ν,α s_0
       ≤ 16C_e^-λ_ν,αsn_0_L^2^2e^λ_ν,α s_0.

Since

    Λ^α/2n_L^2^2=Λ^α/2n_≠_L^2^2+Λ_y^α/2 n^0^2_L^2_y,

we deduce by (<ref>) that

    ν∫_s^tΛ^α/2n_≠_L^2^2dτ≤ν∫_s^tΛ^α/2n_L^2^2dτ≤ 16C_e^-λ_ν,α(s-s_0)n_0_L^2^2.

Combining (<ref>) and (<ref>), we finish the proof of Lemma <ref>.



Based on the (<ref>), (<ref>) and (<ref>), we deduce that for any 0≤ t≤ s_0, one has

    n_≠_L^2≤n_L^2≤2n_0_L^2,

and for any 0≤ s≤ t≤ s_0, one has

    ν∫_s^tΛ^α/2n_≠_L^2^2dτ≤ν∫_s^tΛ^α/2n_L^2^2dτ≲( n_0^α/α-1_L^2+n)n_0^2_L^2.






 §.§ High order estimate

Here we establish the H^1 estimate of n to equation (<ref>) in the case of n∈ L^1∩ L^∞, it is as follows


Let α∈ [3/2,2) and n be the solution of (<ref>) with initial data n_0≥0, n_0∈ H^γ(𝕋^2)∩ L^1(𝕋^2), γ>1+α. Assume that the u(y)=cos y is Kolmogorov flow. If n∈ L^1∩ L^∞, then

    n_H^1≲ν^-1/14.





Applying ∂_x to (<ref>) and multiplying both sides by ∂_x n and integrating over 𝕋^2, we obtain

    1/2d/dt∂_x n^2_L^2+νΛ^α/2∂_x n^2_L^2
    +ν∫_𝕋^2∂_x(∇·(n 𝐁(n)))∂_x ndxdy=0.

Since

    ∫_𝕋^2∂_x(∇·(n 𝐁(n)))∂_x ndxdy=∫_𝕋^2∂_x(∇ n ·𝐁(n))∂_x ndxdy
    +∫_𝕋^2∂_x(n ∇·𝐁(n))∂_x ndxdy,

and one has from Lemma <ref> and <ref> that

    |∫_𝕋^2∂_x(∇ n ·𝐁(n))∂_x ndxdy|
    ≲Λ^1-α/2(∇ n ·𝐁(n))_L^2Λ^1+α/2n_L^2,

and

    Λ^1-α/2(∇ n ·𝐁(n))_L^2≲Λ^1-α/2∇ n_L^2𝐁(n)_L^∞
    +∇ n_L^2Λ^1-α/2𝐁(n)_L^∞.

Combining

    Λ^1-α/2∇ n_L^2≲Λ^2-α/2 n_L^2≲Λ^1+α/2n^2/α-1_L^2Λ n^2-2/α_L^2,


    𝐁(n)_L^∞≲n_L^4≲1,


    Λ^1-α/2𝐁(n)_L^∞≲𝐁(n)^α/2-1/2_L^4Λ𝐁(n)^3/2-α/2_L^4≲n_L^4≲1,

we obtain

    Λ^1-α/2(∇ n ·𝐁(n))_L^2Λ^1+α/2n_L^2
    ≲   Λ^1+α/2n^2/α_L^2Λ n^2-2/α_L^2
    +Λ^1+α/2n_L^2Λ n_L^2
    ≤   1/4Λ^1+α/2n^2_L^2+CΛ n^2_L^2.

and

    |∫_𝕋^2∂_x(n ∇·𝐁(n))∂_x ndxdy |
    ≲n_L^∞∂_x n^2_L^2≲Λ n^2_L^2.

Then one gets

    |ν∫_𝕋^2∂_x(∇·(n 𝐁(n)))∂_x ndxdy|
    ≤ν/4Λ^1+α/2n^2_L^2+CνΛ n^2_L^2.

Combining (<ref>) and (<ref>), we have

    1/2d/dt∂_x n^2_L^2+νΛ^α/2∂_x n^2_L^2≤ν/4Λ^1+α/2n^2_L^2+CνΛ n^2_L^2.

Applying ∂_y to (<ref>), multiplying both sides by ∂_y n and integrating over 𝕋^2, we obtain

    1/2d/dt∂_y n^2_L^2+νΛ^α/2∂_y n^2_L^2   +∫_𝕋^2∂_y(u(y) ∂_xn)∂_y ndxdy
       +ν∫_𝕋^2∂_y(∇·(n 𝐁(n)))∂_y ndxdy=0.

Obviously, one has

    |∫_𝕋^2∂_y(u(y) ∂_xn)∂_y ndxdy|
    =|∫_𝕋^2u'(y)∂_xn∂_y ndxdy|
    ≲Λ n^2_L^2.

Similar to (<ref>), we obtain

    |ν∫_𝕋^2∂_y(∇·(n 𝐁(n)))∂_y ndxdy|
    ≤ν/4Λ^1+α/2n^2_L^2+CνΛ n^2_L^2.

Then we deduce by (<ref>) that

    1/2d/dt∂_y n^2_L^2+νΛ^α/2∂_y n^2_L^2≤ν/4Λ^1+α/2n^2_L^2+CνΛ n^2_L^2+CΛ n^2_L^2.

Combining (<ref>) and (<ref>), we have

    d/dtΛ n^2_L^2+νΛ^α/2+1 n^2_L^2≲νΛ n^2_L^2+Λ n^2_L^2.

Since n_L^1+n_L^∞≲1, one has

    -Λ^α/2+1 n^2_L^2≲ -Λ n^16_L^2.

Therefore, the (<ref>) is written as

    d/dtΛ n^2_L^2≲ -νΛ n^16_L^2
    +νΛ n^2_L^2+2Λ n^2_L^2,

Then we imply

    Λ n_L^2≲ν^-1/14.

This completes the proof of Lemma <ref>





§ SEMIGROUP THEORY


In this section, we establish some useful semigroup theory, define semigroup operator

    𝒮_t=e^-tℒ_ν,α,

where the ℒ_ν,α is defined in (<ref>) and shear flow u(y) satisfies the assumption in Theorem <ref>. We give the following two lemmas.



Let 𝒮_t be defined in (<ref>), then for 0≤ s≤ t<∞, one has

    ν∫_s^tΛ^α/2𝒮_τP_≠^2dτ≲ e^-2λ'_ν,αs,    ν∫_s^tΛ^α/2𝒮_τ^2dτ≲ 1,

where λ'_ν,α=ϵ_0 ν^m/m+α, α∈ (0,2) and

    Λ^α/2𝒮_tP_≠=sup_g_0_L^20Λ^α/2𝒮_tP_≠ g_0_L^2/g_0_L^2,  Λ^α/2𝒮_t=sup_g_0_L^20Λ^α/2𝒮_t g_0_L^2/g_0_L^2.




Consider the equation (<ref>), (<ref>) and (<ref>), one has

    ∂_tg_≠+u(y)∂_xg_≠+ν(-Δ)^α/2g_≠=0,

and the g_≠ can be written as

    g_≠(t,x,y)=𝒮_tP_≠g_0(x,y).

Then we deduce by Theorem <ref> and (<ref>) that for any 0≤ s≤ t<∞, one has

    g_≠(t,·)^2_L^2+ν∫_s^tΛ^α/2g_≠(τ,·)^2_L^2dτ≲g_≠(s,·)^2_L^2≲ e^-2λ'_ν,α sg_0^2_L^2.

Thus, we have

    g_≠(t,·)^2_L^2≲ e^-2λ'_ν,α tg_0^2_L^2,    ν∫_s^tΛ^α/2𝒮_τ P_≠ g_0^2_L^2dτ≲ e^-2λ'_ν,α sg_0^2_L^2.

Considering the operator Λ^α/2𝒮_tP_≠, obviously,

    Λ^α/2𝒮_tP_≠ g_0_L^2≤Λ^α/2𝒮_tP_≠g_0_L^2,   Λ^α/2𝒮_tP_≠^2=sup_g_00Λ^α/2𝒮_t P_≠g_0^2/g_0_L^2^2,

then we have

    ν∫_s^tΛ^α/2𝒮_τP_≠^2dτ
    =νsup_g_0_L^20∫_s^tΛ^α/2𝒮_tP_≠ g_0^2_L^2g_0^-2_L^2dτ≲ e^-2λ'_ν,α s.

Similarly, one has

    ν∫_s^tΛ^α/2𝒮_τ^2dτ≲ 1.

This completes the proof of Lemma <ref>.




Let 𝒮_t be defined in (<ref>) and f_0∈ C^∞(𝕋^2), we have

    𝒮_t∂_yf_0=-Λ^α/2_y𝒮_tΛ^-α/2_y∂_yf_0
    -∫_0^te^-(t-τ)ℒ_ν,αℛ_f_0(τ,x,y)dτ,

where

    ℛ_f_0(τ,x,y)=C_α∑_k∈ℤP.V.∫_𝕋𝒦(y,y)
    Λ_x^α/2-1∂_x𝒮_τ(Λ^1-α/2_xΛ^-α/2_y∂_yf_0)dy,

and

    𝒦(y,y)=u(y)-u(y)/|y-y+k|^1+α/2.




In order to prove the Lemma <ref>, we consider the following linear equation

    ∂_tf+u(y)∂_x f+ν(-Δ)^α/2f=0,    f(0,x,y)=f_0(x,y).

If 𝔾(t,x_0,y_0,x,y) is fundamental solution of equation (<ref>) on 𝕋^2, which satisfies

    ∂_t𝔾+u(y)∂_x 𝔾+ν(-Δ)^α/2𝔾=0,
    lim_t→ 0𝔾(t,x_0,y_0,x,y)=δ(x-x_0,y-y_0),

where δ(x-x_0,y-y_0) denotes the point mass delta function at (x_0,y_0). Then the solution of (<ref>) can be written as

    f(t,x,y)=𝒮_t f_0=∫_𝕋^2𝔾(t,x_0,y_0,x,y)f_0(x_0,y_0)dx_0dy_0.

Combining Lemma <ref>, one gets

    𝒮_t ∂_yf_0   =∫_𝕋^2𝔾(t,x_0,y_0,x,y)∂_y_0f_0(x_0,y_0)dx_0dy_0
       =∫_𝕋^2Λ^α/2_y_0𝔾(t,x_0,y_0,x,y)Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0.

Applying the operator Λ^α/2_y_0 and Λ^α/2_y to (<ref>), then we have

    ∂_t (Λ^α/2_y𝔾+Λ^α/2_y_0𝔾)
    +u(y)∂_x (Λ^α/2_y𝔾+Λ^α/2_y_0𝔾)
                       +ν (-Δ)^α/2(Λ^α/2_y𝔾+Λ^α/2_y_0𝔾)+ℛ(t,x_0,y_0,x,y)=0,
    (Λ^α/2_y𝔾+Λ^α/2_y_0𝔾)
    (0,x_0,y_0,x,y)=0,

where

    ℛ(t,x_0,y_0,x,y)=C_α∑_k∈ℤ
    P.V.∫_𝕋(u(y)-u(y))∂_x𝔾
    (t,x_0,y_0,x,y)/|y-y+k|^1+α/2dy.



In (<ref>), we use the following equality as

    Λ^α/2_y(u(y)∂_x 𝔾)
    =   C_α∑_k∈ℤP.V.∫_𝕋u(y)∂_x 𝔾(t,x_0,y_0,x,y)-u(y)∂_x 𝔾(t,x_0,y_0,x,y)/|y-y+k|^1+α/2dy
    
    =   C_α∑_k∈ℤP.V.∫_𝕋u(y)∂_x 𝔾(t,x_0,y_0,x,y)-u(y)∂_x 𝔾(t,x_0,y_0,x,y)/|y-y+k|^1+α/2dy
       +C_α∑_k∈ℤP.V.∫_𝕋u(y)∂_x 𝔾(t,x_0,y_0,x,y)-u(y)∂_x 𝔾(t,x_0,y_0,x,y)/|y-y+k|^1+α/2dy
    
    =   u(y)C_α∑_k∈ℤP.V.∫_𝕋∂_x 𝔾(t,x_0,y_0,x,y)-∂_x 𝔾(t,x_0,y_0,x,y)/|y-y+k|^1+α/2dy
       +C_α∑_k∈ℤ
    P.V.∫_𝕋(u(y)-u(y))∂_x𝔾
    (t,x_0,y_0,x,y)/|y-y+k|^1+α/2dy
    
    =   u(y)∂_xΛ^α/2_y𝔾(t,x_0,y_o,x,y)+ℛ(t,x_0,y_0,x,y).





Since for any g∈ C^∞(𝕋^2), one has

    lim_t→ 0∫_𝕋^2Λ^α/2_y𝔾(t,x_0,y_0,x,y)g(x_0,y_0)dx_0dy_0
    =   Λ^α/2_y∫_𝕋^2δ(x-x_0,y-y_0)g(x_0,y_0)dx_0dy_0
    
    =   ∫_𝕋^2δ(x_0,y_0)Λ^α/2_yg(x-x_0,y-y_0)dx_0dy_0,

and

    lim_t→ 0∫_𝕋^2Λ^α/2_y_0𝔾(t,x_0,y_0,x,y)g(x_0,y_0)dx_0dy_0
    =   ∫_𝕋^2δ(x-x_0,y-y_0)Λ^α/2_y_0g(x_0,y_0)dx_0dy_0
    
    =   ∫_𝕋^2δ(x_0,y_0)Λ^α/2_y_0g(x-x_0,y-y_0)dx_0dy_0.

According to the definition of Λ^α/2_y and Λ^α/2_y_0 in (<ref>), one has

    Λ^α/2_yg(x-x_0,y-y_0)+Λ^α/2_y_0g(x-x_0,y-y_0)=0,

then we can get

    lim_t→ 0∫_𝕋^2(Λ^α/2_y𝔾+Λ^α/2_y_0𝔾)
    (t,x_0,y_0,x,y)g(x_0,y_0)dx_0dy_0=0.

Thus we have

    lim_t→ 0(Λ^α/2_y𝔾+Λ^α/2_y_0𝔾)
    (0,x_0,y_0,x,y)=0.





Combining Duhamel's principle, (<ref>) and (<ref>),we have

    Λ^α/2_y_0𝔾(t,x_0,y_0,x,y)=-Λ^α/2_y𝔾(t,x_0,y_0,x,y)
    -∫_0^te^-(t-τ)ℒ_ν,αℛ(τ,x_0,y_0,x,y)dτ.

By similar argument, we can easily get

    ∂_x𝔾(t,x_0,y_0,x,y)=-∂_x_0𝔾(t,x_0,y_0,x,y),


    Λ^α/2_x𝔾(t,x_0,y_0,x,y)=-Λ^α/2_x_0𝔾(t,x_0,y_0,x,y).

and

    Λ^-α/2_x∂_x 𝔾(t,x_0,y_0,x,y)=- Λ^-α/2_x_0∂_x_0𝔾(t,x_0,y_0,x,y),   0<α<2.

.05in

Next, we give the proof of Lemma <ref>.


Combining (<ref>) and (<ref>), one has

    𝒮_t∂_yf_0   =∫_𝕋^2Λ^α/2_y_0𝔾(t,x_0,y_0,x,y)
    Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
       =-∫_𝕋^2Λ^α/2_y𝔾(t,x_0,y_0,x,y)
    Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
          -∫_𝕋^2(∫_0^te^-(t-τ)ℒ_ν,αℛ(τ,x_0,y_0,x,y)dτ)
    Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0.

Then we deduce by (<ref>) that

    -∫_𝕋^2Λ^α/2_y𝔾(t,x_0,y_0,x,y)
    Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
    
    =   -Λ^α/2_y∫_𝕋^2𝔾(t,x_0,y_0,x,y)
    Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
    
    =   -Λ^α/2_y𝒮_tΛ^-α/2_y∂_yf_0(t,x,y).

Combining (<ref>), (<ref>) and (<ref>)-(<ref>), one has

    J=   -∫_𝕋^2(∫_0^te^-(t-τ)ℒ_ν,αℛ(τ,x_0,y_0,x,y)dτ)
    Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
    
    =   -C_α∑_k∈ℤ
    P.V.∫_𝕋^2(∫_0^te^-(t-τ)ℒ_ν,α(∫_𝕋𝒦(y,y)∂_x𝔾
    (τ,x_0,y_0,x,y)dy)dτ)
              ·Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
    
    =   C_α∑_k∈ℤ
    P.V.∫_𝕋^2Λ_x_0^α/2-1∂_x_0(∫_0^te^-(t-τ)ℒ_ν,α(∫_𝕋𝒦(y,y)𝔾
    (τ,x_0,y_0,x,y)dy)dτ)
              ·Λ^1-α/2_x_0Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0
    
    =   -C_α∑_k∈ℤ
    P.V.∫_𝕋^2(∫_0^te^-(t-τ)ℒ_ν,αΛ_x^α/2-1∂_x(∫_𝕋𝒦(y,y)𝔾
    (τ,x_0,y_0,x,y)dy)dτ)
              ·Λ^1-α/2_x_0Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0.

By exchanging the order of integrals, one has

    J=   -C_α∑_k∈ℤ
    P.V.∫_0^te^-(t-τ)ℒ_ν,αΛ_x^α/2-1∂_x∫_𝕋𝒦(y,y)
       ·(∫_𝕋^2𝔾
    (τ,x_0,y_0,x,y)Λ^1-α/2_x_0Λ^-α/2_y_0∂_y_0f_0(x_0,y_0)dx_0dy_0 )dydτ
    
    =   -C_α∑_k∈ℤP.V.∫_0^te^-(t-τ)ℒ_ν,α∫_𝕋𝒦(y,y)
    Λ_x^α/2-1∂_x𝒮_τ(Λ^1-α/2_xΛ^-α/2_y∂_yf_0)dydτ
    
    =   -∫_0^te^-(t-τ)ℒ_ν,αℛ_f_0(τ,x,y)dτ,

Combining (<ref>)-(<ref>), we have

    𝒮_t∂_yf_0=-Λ^α/2_y𝒮_tΛ^-α/2_y∂_yf_0
    -∫_0^te^-(t-τ)ℒ_ν,αℛ_f_0(τ,x,y)dτ.

This completes the proof of Lemma <ref>.




Acknowledgments.  The works of B.Niu and W.Wang were partially supported by the National Natural Science Foundation of China (Grant No.12271357,12161141004) and Shanghai Science and Technology Innovation Action Plan (Grant No.21JC1403600). The work of B.Shi was partially supported by the Jiangsu Funding Program for
Excellent Postdoctoral Talent (Grant No.2023ZB116). B.Shi also would like to thank Professor Xi Zhang for continued support and encouragement.



abbrv

