# Finite energy weak solutions of 2d Boussinesq Equations with diffusive
temperature

Tianwen Luo  School of Mathematical Sciences, Yau Mathematical Science center,
Tsinghua University, Beijing 100084, China<EMAIL_ADDRESS>,  Tao
Tao  School of Mathematics Sciences, Shandong University, jinan, 250100, China
<EMAIL_ADDRESS>and  Liqun Zhang  Academy of Mathematic and System Science ,
CAS Beijing 100190, China; and School of Mathematical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS>
###### Abstract.

We show the existence of finite kinetic energy solution with prescribed
kinetic energy to the 2d Boussinesq equations with diffusive temperature on
torus.

Keywords: Finite energy weak solution, Boussinesq equation, prescribed kinetic
energy

AMS Subject Classification (2000): 35Q30, 76D03

##  1\. Introduction

The Boussinesq equation was introduced for understanding the effect of
potentially large conversions between internal energy and mechanical energy in
fluids, and simulates many geophysical flows, such as atmospheric fronts and
ocean circulations (see, for example, [36],[41]). Moreover, it was used in
recent theoretical discussion of the energetics of horizontal convection and
the energetics of turbulent mixing in stratified fluids.

In this paper, we consider the following 2-dimensional Boussinesq system

$\begin{cases}\partial_{t}v+v\cdot\nabla v+\nabla p+(-\Delta)^{\alpha}v=\theta
e_{2},\quad{\rm in}\quad[0,1]\times{\rm T}^{2}\\\ \hbox{div}v=0,\quad{\rm
in}\quad[0,1]\times{\rm T}^{2}\\\
\partial_{t}\theta+v\cdot\nabla\theta-\triangle\theta=0,\quad{\rm
in}\quad[0,1]\times{\rm T}^{2}\end{cases}$ (1.1)

where $\alpha<1$ is a positive number, $e_{2}=(0,1)^{T}$ and ${\rm T}^{2}$ is
2d torus. Here, $v$ is the velocity vector, $p$ is the pressure, $\theta$
denotes the temperature which is a scalar function.

The global well-posedness have been established by many authors for the Cauchy
problem of (1.1) in 2d for regularity data(see, for example, [9], [25]). For
the 3-dimensional case, the global existence of smooth solution of (1.1)
remains open. To understand the turbulence phenomena in hydrodynamics, one
needs to go beyond classical solutions, and in this paper we are interested in
constructing weak solutions of (1.1) with bounded kinetic energy. The triple
$(v,p,\theta)$ on $[0,1]\times{\rm T}^{2}$ is called a weak solution of (1.1)
if they belong to $L^{\infty}(0,1;L^{2}({\rm T}^{2}))$ and solve (1.1) in the
following sense:

$\displaystyle\int_{0}^{1}\int_{{\rm
T}^{2}}(v\cdot\partial_{t}\varphi+v\otimes
v:\nabla\varphi-(-\Delta)^{\alpha}\varphi\cdot u+\theta
e_{2}\cdot\varphi)dxdt=0$

for all $\varphi\in C_{c}^{\infty}((0,1)\times{\rm T}^{2};R^{2})$ with ${\rm
div}\varphi(t,x)=0.$

$\displaystyle\int_{0}^{1}\int_{{\rm
T}^{2}}(\partial_{t}\phi\theta+v\cdot\nabla\phi\theta+\triangle\phi\theta)dxdt=0$

for all $\phi\in C_{c}^{\infty}((0,1)\times{\rm T}^{2};R)$ and

$\displaystyle\int_{{\rm T}^{2}}v(t,x)\cdot\nabla\psi(x)dx=0$

for all $\psi\in C^{\infty}({\rm T}^{2};R)$ and any $t\in[0,1]$.

The study of constructing non-unique or dissipative weak solution to fluid
system is very fashionable in recent years, and the construction is based on
convex integration method which pioneered by De Lellis-Székelyhidi Jr in [18,
21], where the author tackle the Onsager conjecture for the incompressible
Euler equation. So far, there are many important work about weak solution of
the incompressible Euler equation, see [10, 15, 19, 24, 26, 28, 40, 42, 43,
44, 46, 47]. The Onsager conjecture was proved by P.Isett in [29], based on a
series of process on this problem in [1, 3, 4, 5, 11, 16, 17, 22, 27], see
also [6] for the construction of admissible weak solution. Moreover, the idea
and method can be used to construct dissipative weak solution for other model,
see [7, 31, 34, 45, 48, 49, 50].

Recently, Buckmaster and Vicol establish the non-uniqueness of weak solution
to the 3D incompressible Navier-Stokes in [8] by introducing some new ideas.
The main idea is to use “intermittent” building blocks in the convex
integration scheme to control the dissipative term $\triangle v$, called
“intermittent Beltrami flow”, which are space inhomogeneous version of the
classical Beltrami flow, also see [2]. Compared with the homogenous case, the
“intermittent Beltrami flow” has different scaling in different $L^{p}$ norms.
In particular, one can ensure small $L^{p}$ norm for small $p>1$ which is key
to control the dissipative term. By choosing the parameter suitably, T.Luo and
E.S.Titi in [32] construct weak solution with compact support in time for
hyperviscous Navier-Stokes equation. For high dimension$(d\geq 4)$ stationary
Navier-Stokes equation, X. Luo in [35] show the non-uniqueness by constructing
the concentrated Mikado flows introduced in [37, 38]. Moreover, S.Modena and
Székelyhidi established the non-uniqueness for the linear transport equation
(and transport-diffusion equation) with divergence-free vector in some Sobolev
space, see [37, 38], where they use Mikado density and Mikado fields which is
highly concentrated such that the $L^{p}$ norm of Mikado field is small for
$p>1$ small.

Motivated by the above earlier works, we consider the 2d Boussinesq equations
(1.1) and want to know if the similar phenomena can also happen when adding
the temperature effects. Following the general scheme in the construction of
non-uniqueness to Navier-Stokes equation in [8], we obtain the following
existence result.

######  Theorem 1.1.

For any smooth function $e(t):[0,1]\rightarrow[1,+\infty)$ and
$\frac{1}{2}\leq\alpha<1$, there exist $v\in C([0,1],L^{2}({\rm
T}^{2})),\theta\in\bigcap_{2\leq p<\infty}C([0,1],L^{p}({\rm T}^{2}))\cap
L^{2}([0,1],H^{1}({\rm T}^{2}))$, which is weak solution of Boussinesq
equation (1.1) with

$\displaystyle\int_{{\rm T}^{2}}|v(x,t)|^{2}dx=e(t),\quad\forall t\in[0,1],$

and for any $t\in[0,1]$

$\displaystyle\frac{1}{2}\|\theta(t,\cdot)\|_{L^{2}}^{2}+\int_{0}^{t}\|\nabla\theta(s,\cdot)\|^{2}_{L^{2}}ds=\frac{1}{2}\|\theta(0,\cdot)\|_{L^{2}}^{2}.$

######  Remark 1.2.

For $0\leq\alpha<\frac{1}{2}$, we also can construct weak solution with
prescribed energy curve in the class $C([0,1],L^{2}(T^{2}))$ by the same
method. However, in a separate paper [33], we will construct Hölder continuous
solution for this case.

######  Remark 1.3.

The $\theta$ we construct in this paper is rather regular and satisfies energy
equality. When $\theta=0$, the equation (1.1) is 2d Navier-Stokes equation
with fractional diffusion, and our construction also work for this case.

The rest of the paper is organized as follows. In Section 2, we state the main
proposition and give a proof of Theorem 1.1. In Section 3, we collect some
technical tool which will be used frequently. In Section 4, we introduce the
intermittent plane wave which is the building block in our perturbation. In
Section 5 and 6, we construct velocity perturbations and temperature
perturbation, respectively. After the construction, we establish the related
estimates. In Section 7, we construct the Reynold-Stress and establish the
related estimates. Finally, in Section 8, we give a proof of Proposition 2.2.

##  2\. Main proposition and proof of main theorem

In this section, we state our main iterative proposition and give a proof of
theorem 1.1 by the help of main proposition.

######  Definition 2.1.

Assume that $(v_{0},p_{0},\theta_{0},\mathring{R}_{0})\in
C^{\infty}([0,1]\times{\rm T}^{2},R^{2}\times R\times R\times S_{0}^{2\times
2})$. We say that they solve the Boussinesq-Reynold equation if

$\left\\{\begin{array}[]{ll}\partial_{t}v_{0}+{\rm div}(v_{0}\otimes
v_{0})+\nabla p_{0}+(-\triangle)^{\alpha}v_{0}=\theta e_{2}+{\rm
div}\mathring{R}_{0},\quad\quad\mbox{in}\quad[0,1]\times{\rm T}^{2}\\\\[5.0pt]
{\rm div}v_{0}=0,\quad\quad\mbox{in}\quad[0,1]\times{\rm T}^{2}\\\\[5.0pt]
\partial_{t}\theta_{0}+{\rm
div}(v_{0}\theta_{0})-\triangle\theta_{0}=0,\quad\quad\mbox{in}\quad[0,1]\times{\rm
T}^{2}.\end{array}\right.$ (2.2)

Here and throughout the paper, $S_{0}^{2\times 2}$ is the set of trace-free
symmetric $2\times 2$ matrices.

We now state our main proposition, and Theorem 1.1 is a corollary.

######  Proposition 2.2.

Let $e(t),\alpha$ be as in Theorem 1.1 and $\varepsilon_{0}$ be a universal
constant from the Geometric Lemma 4.2. Then there exist universal constant $M$
and $\tilde{M}$ such that the following hold.

Let $\delta\leq 1$ be any positive number, $\theta^{0}\in C^{\infty}({\rm
T}^{2})$ be any function satisfies $\fint_{{\rm T}^{2}}\theta^{0}(x)dx=0$, and
$(v_{0},p_{0},\theta_{0},\mathring{R}_{0})$ is a solution of Boussinesq-
Reynold equation (2.2) with

$\displaystyle\|\mathring{R}_{0}\|_{L^{\infty}_{t}L^{1}_{x}}\leq\frac{\varepsilon_{0}\delta}{10000},\quad\fint_{{\rm
T}^{2}}\theta_{0}(t,x)dx=0,\quad\forall t\in[0,1]$ (2.3)

and

$\displaystyle\frac{3\delta}{4}e(t)\leq e(t)-\int_{{\rm
T}^{2}}|v_{0}(t,x)|^{2}dx\leq\frac{5\delta}{4}e(t),\quad\forall t\in[0,1].$
(2.4)

Then there exist another smooth functions
$(v_{1},p_{1},\theta_{1},\mathring{R}_{1})$ which is also a solution of
Boussinesq-Reynold equation (2.2), and for every $t\in[0,1]$,

$\displaystyle\|\mathring{R}_{1}(t,\cdot)\|_{L^{1}_{x}}$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon_{0}\delta}{20000},\quad\fint_{{\rm
T}^{2}}\theta_{1}(t,x)dx=0,$
$\displaystyle\|v_{1}(t,\cdot)-v_{0}(t,\cdot)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle M\sqrt{\delta},$ $\displaystyle\theta_{1}(0,x)=\theta^{0}(x),$
$\displaystyle\quad\|\theta_{1}(t,\cdot)\|_{L^{\infty}}\leq\tilde{M},$
$\displaystyle\|\theta_{1}(t,\cdot)-\theta_{0}(t,\cdot)\|^{2}_{L^{2}}$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\|\nabla\theta_{1}(s,\cdot)-\nabla\theta_{0}(s,\cdot)\|^{2}_{L^{2}}\leq\tilde{M}\delta,$
$\displaystyle\|\theta_{1}(t,\cdot)\|^{2}_{L^{2}}$ $\displaystyle+$
$\displaystyle
2\int_{0}^{t}\|\nabla\theta_{1}(s,\cdot)\|^{2}_{L^{2}}=\|\theta_{1}(0,\cdot)\|^{2}_{L^{2}},$
(2.5)

and

$\displaystyle\frac{3\delta}{8}e(t)\leq e(t)-\int_{{\rm
T}^{2}}|v_{1}(t,x)|^{2}dx\leq\frac{5\delta}{8}e(t).$ (2.6)

We will prove the above proposition in the next several sections. Here we
first give a proof of Theorem 1.1.

###### Proof.

We first fix $\delta=1$ and set

$\displaystyle v_{0}:=0,\quad\theta_{0}:=0,\quad
p_{0}:=0,\quad\mathring{R}_{0}:=0.$

Obviously, they solve Boussinesq-Reynolds system (2.2) and

$\displaystyle\frac{3\delta}{4}e(t)\leq e(t)-\int_{{\rm
T}^{2}}|v_{0}|^{2}(x,t)dx\leq\frac{5\delta}{4}e(t),\qquad\forall t\in[0,1]$
$\displaystyle\sup_{t\in[0,1]}\|\mathring{R}_{0}(t,\cdot)\|_{L^{1}_{x}}=0\leq\frac{\varepsilon_{0}\delta}{10000}.$

Then choosing $\theta^{0}\in C^{\infty}({\rm T}^{2})$ satisfies $\fint_{{\rm
T}^{2}}\theta^{0}(x)dx=0$, and using Proposition 2.2 iteratively, we can
construct a sequence $(v_{n},~{}p_{n},~{}\theta_{n},~{}\mathring{R_{n}})$,
which solve (2.2) and satisfy, for every $t\in[0,1]$

$\displaystyle\frac{3}{4}\frac{e(t)}{2^{n}}\leq e(t)-\int_{{\rm
T}^{2}}|v_{n}|^{2}(x,t)dx\leq$ $\displaystyle\frac{5}{4}\frac{e(t)}{2^{n}},$
(2.7) $\displaystyle\|\mathring{R_{n}}(t,\cdot)\|_{L^{1}_{x}}\leq$
$\displaystyle\frac{\varepsilon_{0}}{2^{n}\times 10000},$ (2.8)
$\displaystyle\|v_{n+1}(t,\cdot)-v_{n}(t,\cdot)\|_{L^{2}}\leq$ $\displaystyle
M\sqrt{\frac{1}{2^{n}}},$ (2.9)
$\displaystyle\|\theta_{n}(t,\cdot)\|_{L^{\infty}}\leq$
$\displaystyle\tilde{M},\quad\theta_{n}(0,x)=\theta^{0}(x),$ (2.10)
$\displaystyle\|\theta_{n+1}(t,\cdot)-\theta_{n}(t,\cdot)\|^{2}_{L^{2}}+$
$\displaystyle\int_{0}^{t}\|\nabla(\theta_{n+1}(s,\cdot)-\theta_{n}(s,\cdot))\|^{2}_{L^{2}}\leq\frac{\tilde{M}}{2^{n}},$
(2.11) $\displaystyle\|\theta_{n+1}(t,\cdot)\|^{2}_{L^{2}}+$ $\displaystyle
2\int_{0}^{t}\|\nabla\theta_{n+1}(s,\cdot)\|^{2}_{L^{2}}=\|\theta_{n+1}(0,\cdot)\|^{2}_{L^{2}}.$
(2.12)

From (2.8)-(2.12), we know that $(v_{n},~{}\theta_{n},~{}\mathring{R}_{n})$
are, respectively, Cauchy sequence in $C([0,1],L^{2}({\rm T}^{2}))$,
$\bigcap_{2\leq p<\infty}C([0,1],L^{p}({\rm T}^{2}))\cap
L^{2}([0,1],H^{1}({\rm T}^{2}))$ and $C([0,1],L^{1}({\rm T}^{2}))$, therefore
there exist

$\displaystyle v\in C([0,1],L^{2}({\rm T}^{2})),\quad\theta\in\bigcap_{2\leq
p<\infty}C([0,1],L^{p}({\rm T}^{2}))\cap L^{2}([0,1],H^{1}({\rm T}^{2}))$

such that

$\displaystyle v_{n}\rightarrow v\quad{\rm in}\quad C([0,1],L^{2}({\rm
T}^{2})),$ $\displaystyle\theta_{n}\rightarrow\theta\quad{\rm
in}\quad\bigcap_{2\leq p<\infty}C([0,1],L^{p}({\rm T}^{2}))\cap
L^{2}([0,1],H^{1}({\rm T}^{2})),$ $\displaystyle\mathring{R_{n}}\rightarrow
0\quad{\rm in}\quad C([0,1],L^{1}({\rm T}^{2}))$

as $n\rightarrow\infty$.
Passing into the limit in (2.2), we conclude that $(v,~{}\theta)$ solve (1.1)
in the sense of distribution. Moreover, by (2.7),

$e(t)=\int_{{\rm T}^{2}}|v|^{2}(x,t)dx,\qquad\forall t\in[0,1].$

Moreover, by (2.12), we deduce that the temperature $\theta$ satisfies the
energy equality: for every $t\in[0,1]$

$\displaystyle\|\theta(t,\cdot)\|^{2}_{L^{2}}+2\int_{0}^{t}\|\nabla\theta(s,\cdot)\|^{2}_{L^{2}}=\|\theta(0,\cdot)\|^{2}_{L^{2}}.$

This complete the proof of Theorem 1.1. MM $\square$

The rest of this paper will be dedicated to prove Proposition 2.2. First, we
add perturbations to $v_{0}$ and get new functions $v_{1}$ as following:

$\displaystyle v_{1}=$ $\displaystyle
v_{0}+w^{p}_{1}+w^{c}_{1}+w^{t}_{1}:=v_{0}+w_{1},$

where $w^{p}_{1},w^{c}_{1},w^{t}_{1}$ are smooth functions given by explicit
formulas. We introduce some parameters $\lambda_{1},\mu_{1},r_{1},\sigma_{1}$
satisfying the relation (5.24). After the construction of new velocity
$v_{1}$, we construction new temperature $\theta_{1}$ by solving the following
transport-diffusion equation: there exists $\theta_{1}\in
C^{\infty}([0,1]\times{\rm T}^{2},R)$ which solves

$\begin{cases}\partial_{t}\theta_{1}+v_{1}\cdot\nabla\theta_{1}-\triangle\theta_{1}=0,\\\\[3.0pt]
\theta_{1}|_{t=0}=\theta^{0},\end{cases}$ (2.13)

where $\theta^{0}$ is the function appeared in Proposition 2.2. After
construction of $v_{1},\theta_{1}$, we mainly focus on finding functions
$\mathring{R}_{1},p_{1}$ with the desired estimates and solving system (2.2).

##  3\. Technical tool

In this section, we collect some technical tools which will be frequently used
in the following.

###  3.1. Properties of fast oscillatory

In this subsection, we discuss some properties of fast oscillatory and the
proof can be found in [37], [38], which was inspired by [8]. More precisely,
we give an improved Hölder inequality which concern the $L^{p}$ norm of the
product of a slow oscillating function with a fast oscillating function, and a
mean value estimate which concern the mean value of the product of a slow
oscillating function with a fast oscillating function .

For a given function $f:{\rm T}^{2}\rightarrow R$, $\lambda\in{\rm N}$, we set

$\displaystyle f_{\lambda}(x):=f(\lambda x).$

######  Lemma 3.1.

Let $f,g:{\rm T}^{2}\rightarrow R$ be smooth functions, $\lambda\in{\rm N}$.
Then for every $p\in[1,+\infty]$, we have

$\displaystyle\|fg_{\lambda}\|_{L^{p}}\leq\|f\|_{L^{p}}\|g\|_{L^{p}}+\frac{C_{p}}{\lambda^{\frac{1}{p}}}\|f\|_{C^{1}}\|g\|_{L^{p}}.$

######  Lemma 3.2.

Let $f,g:{\rm T}^{2}\rightarrow R$ be smooth function with $\fint_{{\rm
T}^{2}}g(x)dx=0$, and $\lambda\in{\rm N}.$ Then there hold

$\displaystyle\Big{|}\fint_{{\rm
T}^{2}}fg_{\lambda}dx\Big{|}\leq\frac{\sqrt{2}\|f\|_{C^{1}}\|g\|_{L^{1}}}{\lambda}.$

###  3.2. Commutator for fast oscillation

######  Lemma 3.3.

Fix $\kappa\geq 1$. Let $a\in C^{2}(\mathbb{T}^{3})$. For $1<p<\infty$, and
any $f\in L^{p}(\mathbb{T}^{3})$, we have

$\displaystyle\||\nabla|^{-1}\mathbb{P}_{\neq 0}(a\mathbb{P}_{\geq
k}f)\|_{L^{p}}$ $\displaystyle\lesssim
k^{-1}(\|a\|_{L^{\infty}}+\|\nabla^{2}a\|_{L^{\infty}})\|f\|_{L^{p}}.$

The proof of this Lemma can be found in [8].

##  4\. Intermittent plane waves

In this section, we describe in detail the construction of the intermittent
plane waves which will form the building blocks of the convex integration
scheme.

We first recall the following stationary solution for the 2d Euler equation.
Our building block in this paper is the inhomogeneous version of it.

###  4.1. Stationary flows in 2D and Geometric Lemma

######  Proposition 4.1.

Let $\Lambda$ be a given finite symmetric subset of $S^{1}\cap Q^{2}$ with
$\lambda\Lambda\in Z^{2}.$ Then for any choice of coefficients
$a_{\bar{\xi}}\in{\rm C}$ with $\overline{a_{\bar{\xi}}}=a_{-\bar{\xi}}$, the
vector field

$\displaystyle
W(x)=\sum_{\bar{\xi}\in\Lambda}a_{\bar{\xi}}i\bar{\xi}^{\bot}e^{i\bar{\xi}\cdot
x},\quad\Psi(x)=\sum_{\bar{\xi}\in\Lambda}a_{\bar{\xi}}e^{i\bar{\xi}\cdot x}$

is real-valued and satisfies

$\displaystyle{\rm div}(W\otimes
W)=\nabla\Big{(}\frac{|W|^{2}}{2}+\frac{\Psi^{2}}{2}\Big{)},\quad
W(x)=\nabla^{\bot}\Psi(x).$

Here and throughout the paper, we denote
$\bar{\xi}^{\perp}=(-\bar{\xi}_{2},\bar{\xi}_{1})$ if
$\bar{\xi}=(\bar{\xi}_{1},\bar{\xi}_{2})$, and denote
$\nabla^{\bot}=(-\partial_{2},\partial_{1})$. Furthermore,

$\displaystyle\big{<}W\otimes W\big{>}:=\fint_{{\rm T}^{2}}W\otimes
W(x)dx=\sum_{\bar{\xi}\in\Lambda}|a_{\bar{\xi}}|^{2}({\rm
Id}-\bar{\xi}\otimes\bar{\xi}).$

The proof of this Proposition can be found in [11], and we omit it here.

Let

$\displaystyle\Lambda_{0}^{+}=\Big{\\{}e_{1},\frac{3}{5}e_{1}+\frac{4}{5}e_{2},\frac{3}{5}e_{1}-\frac{4}{5}e_{2}\Big{\\}},\quad\Lambda_{0}^{-}=-\Lambda_{0}^{+},\quad\Lambda_{0}=\Lambda_{0}^{+}\cup\Lambda_{0}^{-},$

and $\Lambda_{1}$ be given by the rotation of $\Lambda_{0}$ counter clock-wise
by $\pi/2$:

$\displaystyle\Lambda_{1}^{+}=\Big{\\{}e_{2},\frac{3}{5}e_{2}+\frac{4}{5}e_{1},\frac{3}{5}e_{2}-\frac{4}{5}e_{1}\Big{\\}},\quad\Lambda_{1}^{-}=-\Lambda_{1}^{+},\quad\Lambda_{1}=\Lambda_{1}^{+}\cup\Lambda_{1}^{-}.$

Clearly $\Lambda_{0},\Lambda_{1}\subseteq Q^{2}\cap S^{1}$ and we have the
representation

$\displaystyle\frac{25}{32}\Big{(}\Big{(}\frac{3}{5}e_{1}+\frac{4}{5}e_{2}\Big{)}\otimes\Big{(}\frac{3}{5}e_{1}+\frac{4}{5}e_{2}\Big{)}+\Big{(}\frac{3}{5}e_{1}-\frac{4}{5}e_{2}\Big{)}\otimes\Big{(}\frac{3}{5}e_{1}-\frac{4}{5}e_{2}\Big{)}\Big{)}+\frac{7}{16}e_{1}\otimes
e_{1}=\mathrm{Id}.$

In fact, such representation holds for $2\times 2$ symmetric matrices near
$\mathrm{Id}$.

######  Lemma 4.2 (Geometric Lemma).

There exists $\varepsilon_{0}>0$, and smooth positive functions
$\gamma_{\xi}$:

$\displaystyle\gamma_{\xi}\in C^{\infty}(B_{\varepsilon_{0}}({\rm Id}))$

such that for every $2\times 2$ symmetric matrix $R\in
B_{\varepsilon_{0}}({\rm Id})$, we have

$R=\sum_{\xi\in\Lambda_{0}^{+}}\gamma^{2}_{\xi}(R)\xi\otimes\xi.$

######  Remark 4.3.

By rotational symmetry, Geometric Lemma 4.2 also holds for
$\xi\in\Lambda_{1}^{+}$. It is convenient to introduce a small geometric
constant $c_{0}\in(0,1)$ such that

$\displaystyle|\xi+\xi^{\prime}|\geq 2c_{0}$

for all $\xi,\xi^{\prime}\in\Lambda_{0}\cap\Lambda_{1},\xi\neq-\xi^{\prime}$.
Moreover, for $\xi\in\Lambda_{k}^{-}$ with $k=0,1$, we set
$\gamma_{\xi}:=\gamma_{-\xi}$.

######  Remark 4.4.

When ${\rm Id}-\mathring{R}\in B_{\varepsilon_{0}}({\rm Id})$, by Geometric
Lemma, we have

$\displaystyle{\rm
Id}-\mathring{R}=\sum_{\xi\in\Lambda_{0}^{+}}\gamma_{\xi}^{2}\big{(}{\rm
Id}-\mathring{R}\big{)}\xi\otimes\xi.$

Thus, taking trace in both side, we obtain

$\displaystyle\sum_{\xi\in\Lambda_{0}^{+}}\gamma_{\xi}^{2}\big{(}{\rm
Id}-\mathring{R}\big{)}=2.$ (4.14)

###  4.2. Intermittent plane flow

The Dirichlet kernel $\tilde{D}_{r}$ is defined as

$\displaystyle\tilde{D}_{r}(x):=\sum_{\xi=-r}^{r}e^{ix\cdot\xi}=\frac{{\rm
sin}((r+\frac{1}{2})x)}{{\rm sin}(\frac{x}{2})},$

and it obeys the estimates: for any $p>1$,

$\displaystyle\|\tilde{D}_{r}\|_{L^{p}}\sim r^{1-\frac{1}{p}},$

where the implicit constant only depend only on $p$. Define a 2d square

$\displaystyle\Omega_{r}:=\\{(j,k):j,k\in\\{-r,...,r\\}\\}$

and normalizing to unit size in $L^{2}$, we obtain a kernel

$\displaystyle
D_{r}(x):=\frac{1}{2r+1}\sum_{\xi\in\Omega_{r}}e^{ix\cdot\xi}=\frac{1}{2r+1}\sum_{(j,k)\in\Omega_{r}}e^{i(jx_{1}+kx_{2})}$

which has the property: for $1<p\leq\infty$

$\displaystyle\|D_{r}\|_{L^{p}}\lesssim
r^{1-\frac{2}{p}},\quad\|D_{r}\|_{L^{2}}\thickapprox 1,$

where the implicit constant only depend on $p$. This computation is very easy
due to the fact:

$\displaystyle\sum_{\xi\in\Omega_{r}}e^{ix\cdot\xi}=\Big{(}\sum_{i=-r}^{r}e^{ijx_{1}}\Big{)}\Big{(}\sum_{i=-r}^{r}e^{ikx_{2}}\Big{)}.$

We first fixed a large parameter $\lambda\in{\rm N}$, then introduce a
parameter $\sigma$ such that $\lambda\sigma\in{\rm N}$ which parameterizes the
spacing between frequencies. We assume that

$\displaystyle\sigma r\leq\frac{c_{0}}{50N},$

where $c_{0}$ is the constant in Remark 4.3, and $N$ is a fixed integer(for
example, we can take $N=5$). Furthermore, we introduce a parameter
$\mu\in(0,\lambda)$, which describes temporal oscillation in the building
blocks.

As in [8], for $\xi\in\Lambda_{j}^{+}$, we define a directed and rescaled
$(\frac{2\pi}{\lambda\sigma})^{2}$-periodic Dirichlet kernel by

$\displaystyle\eta_{(\xi)}(x,t):=\eta_{\xi,\lambda,\sigma,r,\mu}(x,t)$
$\displaystyle=$ $\displaystyle D_{r}\big{(}\lambda\sigma N(\xi\cdot x+\mu
t),\lambda\sigma N\xi^{\bot}\cdot x\big{)}$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{j=-r}^{r}e^{ij\lambda\sigma N(\xi\cdot x+\mu
t)}\Big{)}\Big{(}\sum_{k=-r}^{r}e^{ik\lambda\sigma N\xi^{\bot}\cdot x}\Big{)}$

and set $\eta_{(\xi)}(x,t)=\eta_{(-\xi)}(x,t)$ for $\xi\in\Lambda_{j}^{-}$,
where $N$ is a integer(we can set $N=5$ duo to our construction of
$\Lambda_{0},\Lambda_{1}$). Observe that $\eta_{(\xi)}(x,t)$ satisfies the
following important identity:

$\displaystyle\frac{1}{\mu}\partial_{t}\eta_{(\xi)}(x,t)=\xi\cdot\nabla\eta_{(\xi)}(x,t)$
(4.15)

for every $\xi\in\Lambda_{j}^{+}$, $j=0,1$.

A change of variable gives that

$\displaystyle\fint_{{\rm
T}^{2}}\eta^{2}_{(\xi)}(x,t)dx=1,\quad\|\eta_{(\xi)}(t)\|_{L^{p}}\leq
C_{p}r^{1-\frac{2}{p}}$ (4.16)

for all $1<p\leq\infty$, for any $t$.

Let $W_{(\xi)}(x)$ be the stationary wave at frequency $\lambda$, namely

$\displaystyle W_{(\xi)}(x):=W_{\xi,\lambda}=i\xi e^{i\lambda\xi^{\bot}\cdot
x}.$

We define the intermittent plane waves $\textbf{W}_{(\xi)}$ as

$\displaystyle\textbf{W}_{(\xi)}(x,t)=\textbf{W}_{\xi,\lambda,\sigma,r,\mu}(x,t)=\eta_{\xi,\lambda,\sigma,r,\mu}(x,t)W_{\xi,\lambda}(x)=\eta_{(\xi)}(x,t)W_{(\xi)}(x).$

######  Remark 4.5.

The explicit representation of $\textbf{W}_{(\xi)}(x,t)$ is as following:

$\displaystyle\textbf{W}_{(\xi)}(x,t)$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{j=-r}^{r}e^{ij\lambda\sigma N(\xi\cdot x+\mu
t)}\Big{)}\Big{(}\sum_{k=-r}^{r}e^{ik\lambda\sigma N\xi^{\bot}\cdot
x}\Big{)}i\xi e^{i\lambda\xi^{\bot}\cdot x}.$

Some facts about the frequency support of $\eta_{(\xi)}$ and
$\textbf{W}_{(\xi)}$:

######  Lemma 4.6.

We have the following frequency support property:

$\displaystyle{\rm P}_{\leq 2\lambda\sigma
rN}\eta_{(\xi)}=\eta_{(\xi)},\quad{\rm P}_{\leq 2\lambda}{\rm
P}_{\geq\frac{\lambda}{2}}\textbf{W}_{(\xi)}=\textbf{W}_{(\xi)}.$

For $\xi+\xi^{{}^{\prime}}\neq 0$, by the definition of $c_{0}$, we have

$\displaystyle{\rm P}_{\leq 4\lambda}{\rm P}_{\geq
c_{0}\lambda}\big{(}\textbf{W}_{(\xi)}\otimes\textbf{W}_{(\xi^{{}^{\prime}})}\big{)}=\textbf{W}_{(\xi)}\otimes\textbf{W}_{(\xi^{{}^{\prime}})}.$

These facts can be obtained directly from the definition. In fact, the
frequency support of $\eta_{(\xi)}$ is obvious. Then, using the fact that
$2\lambda\sigma rN\leq\frac{c_{0}\lambda}{5}$, it’s easy to obtain the
frequency support of $\textbf{W}_{(\xi)}$. Finally, a direct computation gives
that

$\displaystyle\textbf{W}_{(\xi)}\otimes\textbf{W}_{(\xi^{{}^{\prime}})}=-\eta_{(\xi)}\eta_{(\xi^{{}^{\prime}})}\xi\otimes\xi^{{}^{\prime}}e^{i\lambda(\xi+\xi^{{}^{\prime}})^{\bot}\cdot
x}.$

Due to

$\displaystyle{\rm P}_{\leq 4\lambda\sigma
rN}(\eta_{(\xi)}\eta_{(\xi^{{}^{\prime}})})=\eta_{(\xi)}\eta_{(\xi^{{}^{\prime}})},\quad|\xi+\xi^{{}^{\prime}}|\geq
2c_{0}$

and the fact

$\displaystyle 4\lambda\sigma rN\leq\frac{2\lambda c_{0}}{5},$

we obtain the frequency support of
$\textbf{W}_{(\xi)}\otimes\textbf{W}_{(\xi^{{}^{\prime}})}$ for
$\xi+\xi^{{}^{\prime}}\neq 0$.

From these frequency support properties (from which we can use the Berstein
inequality) and the estimates for Dirichlet kernel, we have the following
estimates.

######  Proposition 4.7.

Let $\textbf{W}_{(\xi)}$ be defined as above. Then

$\displaystyle\|\nabla^{N}\partial_{t}^{K}\textbf{W}_{(\xi)}\|_{L^{p}}\leq
C(N,K,p)\lambda^{N}(\lambda\sigma\mu r)^{K}r^{1-\frac{2}{p}},$
$\displaystyle\|\nabla^{N}\partial_{t}^{K}\eta_{(\xi)}\|_{L^{p}}\leq
C(N,K,p)(\lambda\sigma r)^{N}(\lambda\sigma\mu r)^{K}r^{1-\frac{2}{p}},$

for any $1<p\leq\infty$, and $N,K\geq 0$ are integer.

These estimates are direct, and we omit the proof here.

##  5\. The velocity perturbation: Construction and Estimates

In this section, we construct the perturbation of velocity and give some
estimates for it.

###  5.1. Construction of the velocity perturbation

In this subsection, we give the detailed construction of velocity
perturbation.

####  5.1.1. Definition of amplitude

Choose two smooth cutoff functions $\tilde{\chi}_{0},\tilde{\chi}_{1}$ such
that

$\displaystyle{\rm supp}\tilde{\chi}_{0}\subseteq[0,4],\quad{\rm
supp}\tilde{\chi}_{1}\subseteq\Big{[}\frac{1}{4},4\Big{]}$

and

$\displaystyle\tilde{\chi}_{0}^{2}(y)+\sum_{j\geq 1}\tilde{\chi}_{j}^{2}(y)=1$

for any $y>0$, where $\tilde{\chi}_{j}(y)=\tilde{\chi}(4^{-j}y)$. We then
define

$\displaystyle\chi_{j}(t,x):=\tilde{\chi}_{j}\Big{(}\Big{<}\frac{50\mathring{R}_{0}(t,x)}{\varepsilon_{0}\delta}\Big{>}\Big{)}$
(5.17)

for all $j\geq 0.$ Here and throughout the paper we use the notation
$\big{<}A\big{>}=(1+|A|^{2})^{\frac{1}{2}}$ where $|A|$ denotes the standard
norm of the matrix $A$. By the definition of the cutoff functions, we have

$\displaystyle\sum_{j\geq
0}\chi_{j}^{2}(t,x)=1,\quad\chi_{j_{1}}\chi_{j_{2}}(t,x)=0\quad{\rm
if}\quad|j_{1}-j_{2}|\geq 2.$ (5.18)

Moreover, it’s obvious that there exists an index
$j_{max}=j_{max}(\mathring{R}_{0},\delta)$ such that $\chi_{j}(t,x)=0$ for all
$j\geq j_{max}.$ Another important fact is

$\displaystyle\int_{{\rm T}^{2}}\chi^{2}_{0}(t,x)dx>0,\quad\forall t\in[0,1].$
(5.19)

In fact,

$\displaystyle\chi_{0}(t,x)=0\Leftrightarrow\Big{<}\frac{50\mathring{R}_{0}(t,x)}{\varepsilon_{0}\delta}\Big{>}\geq
4.$

Thus, for fixed $t\in[0,1]$, we have

$\displaystyle\chi_{0}(t,x)=0\Rightarrow 50|\mathring{R}_{0}(t,x)|\geq
3\varepsilon_{0}\delta.$

Hence, for fixed $t\in[0,1]$, we have

$\displaystyle\int_{{\rm
T}^{2}}|\mathring{R}_{0}(t,x)|dx\geq\frac{3}{50}\varepsilon_{0}\delta.$

Thus, the assumption (2.3) tells us that for every $t\in[0,1]$, $\int_{{\rm
T}^{2}}\chi^{2}_{0}(t,x)dx>0.$

For $j\in{\rm N}$, denote

$\displaystyle\Lambda_{(j)}=\Lambda_{j~{}\text{mod}~{}2},\quad\Lambda_{(j)}^{\pm}=\Lambda_{j~{}\text{mod}~{}2}^{\pm}.$

For $\xi\in\Lambda_{(j)}^{+}$, define the coefficient function $a_{(\xi,j)}$
by

$\displaystyle
a_{(\xi,j)}(t,x):=\sqrt{\rho_{j}}\chi_{j}(t,x)\gamma_{\xi}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}$ (5.20)

where $\rho_{j}$, $j\geq 1$, are defined by

$\displaystyle\rho_{j}:=4^{j}\delta,$ (5.21)

and $\rho_{0}$ is defined later.

We first claim that $a_{(\xi,j)}(t,x)$ is well-defined for $j\geq 1$. We only
need to show that ${\rm Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\in
B_{\varepsilon_{0}}({\rm Id})$. In fact, when $\chi_{j}\neq 0$, there holds

$\displaystyle\Big{<}\frac{50\mathring{R}_{0}(t,x)}{\varepsilon_{0}\delta}\Big{>}\leq
4^{j+1},$

which implies

$\displaystyle\frac{50|\mathring{R}_{0}(t,x)|}{\varepsilon_{0}\delta}\leq
4^{j+1}.$

Thus,

$\displaystyle\frac{|\mathring{R}_{0}(t,x)|}{\rho_{j}}=\frac{|\mathring{R}_{0}(t,x)|}{4^{j}\delta}\leq\varepsilon_{0},$

thus $a_{(\xi,j)}(t,x)$ is well-defined for $j\geq 1$. We define $\rho_{0}(t)$
as following:

$\displaystyle\rho_{0}(t):=\frac{1}{2}\Big{(}\int_{{\rm
T}^{2}}\chi_{0}^{2}(t,x)dx\Big{)}^{-1}\Big{[}e(t)\Big{(}1-\frac{\delta}{2}\Big{)}-\int_{{\rm
T}^{2}}|v_{0}(t,x)|^{2}dx\Big{]}.$ (5.22)

Due to (5.19), we deduce that $\rho_{0}(t)$ is well-defined. Next, we show
that

$\displaystyle\Big{\|}\frac{\mathring{R}_{0}}{\rho_{0}}\Big{\|}_{L^{\infty}({\rm
supp\chi_{0})}}\leq\varepsilon_{0}.$

In fact, using the assumption (2.4), we know that

$\displaystyle\frac{\delta}{4}e(t)\leq
e(t)\Big{(}1-\frac{\delta}{2}\Big{)}-\int_{{\rm
T}^{2}}|v_{0}(t,x)|^{2}dx\leq\frac{3\delta}{4}e(t),\quad\forall t\in[0,1].$
(5.23)

On the support of $\chi_{0}$, there holds

$\displaystyle 50|\mathring{R}_{0}(t,x)|\leq 4\varepsilon_{0}\delta.$

Thus, on the support of $\chi_{0}$, for any $t\in[0,1]$, there holds

$\displaystyle\Big{|}\frac{\mathring{R}_{0}(t,x)}{\rho_{0}(t)}\Big{|}\leq\frac{4\varepsilon_{0}\delta}{50}\frac{8\int_{{\rm
T}^{2}}\chi_{0}^{2}(t,x)dx}{\delta e(t)}\leq\varepsilon_{0}.$

Thus, $a_{(\xi,0)}$ is well-defined.

####  5.1.2. Construction of velocity perturbation

Let us fix $\lambda_{1},\sigma_{1},r_{1},\mu_{1}$ such that
$\lambda_{1}\sigma_{1}\in{\rm N}$ and the integer $r_{1}$, the parameter
$\sigma_{1}$ and $\mu_{1}$ are defined by

$\displaystyle
r_{1}=[\lambda_{1}^{\alpha}],\quad\sigma_{1}=\lambda_{1}^{-\frac{1+\alpha}{2}},\quad\mu_{1}=[\lambda_{1}^{\frac{3\alpha+1}{4}}].$
(5.24)

The principle part of perturbation $w^{p}_{1}$ will be defined as

$\displaystyle
w_{1}^{p}=\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a_{(\xi,j)}\eta_{(\xi)}(W_{(\xi)}+W_{(-\xi)})=\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}a_{(\xi,j)}\eta_{(\xi)}W_{(\xi)}$
(5.25)

where $0\leq j\leq j_{max}$. Here and throughout the paper,
$\eta_{(\xi)},W_{(\xi)}$ denote, respectively,

$\displaystyle\eta_{(\xi)}=\eta_{\xi,\lambda_{1},\sigma_{1},r_{1},\mu_{1}},\quad
W_{(\xi)}=W_{\xi,\lambda_{1}},$

where $\xi\in\Lambda_{(j)}$.

Then we define an incompressibility corrector

$\displaystyle
w^{c}_{1}:=-\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\nabla^{\bot}\big{(}a_{(\xi,j)}\eta_{(\xi)}\big{)}\frac{e^{i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-i\lambda_{1}\xi^{\bot}\cdot x}}{\lambda_{1}}.$ (5.26)

Here and throughout the paper, we denote $\nabla^{\bot}$ as
$\nabla^{\bot}=(-\partial_{2},\partial_{1}).$ Thus, we have

$\displaystyle
w^{p}_{1}+w^{c}_{1}=-\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\nabla^{\bot}\Big{(}a_{(\xi,j)}\eta_{(\xi)}\frac{e^{i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-i\lambda_{1}\xi^{\bot}\cdot x}}{\lambda}\Big{)}$

and

$\displaystyle{\rm div}(w^{p}_{1}+w^{c}_{1})=0.$

As in paper [8], in addition to the incompressibility corrector $w_{1}^{c}$,
we introduce a temporal corrector $w_{1}^{t}$, which is defined by

$\displaystyle
w_{1}^{t}:=-\frac{1}{\mu}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}P_{H}P_{\neq
0}\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\xi\big{)}.$ (5.27)

Here $P_{\neq 0}f=f-\fint_{{\rm T}^{2}}fdx$ and
$P_{H}f=f-\nabla\triangle^{-1}{\rm div}f.$ Finally, we define the velocity
increment $w_{1}$ by

$\displaystyle w_{1}=w^{p}_{1}+w^{c}_{1}+w^{t}_{1}.$

It’s obvious that

$\displaystyle{\rm div}w_{1}=0,\quad\fint_{{\rm T}^{2}}w_{1}(t,x)dx=0.$

After the construction of $w_{1}$, we define the new velocity field $v_{1}$ as

$\displaystyle v_{1}:=v_{0}+w_{1}.$

###  5.2. Estimate of the perturbation

In this subsection, we establish some estimates for the velocity perturbation.

Firstly, we collect some estimates concerning the cutoffs function
$\chi_{j}(t,x)$.

######  Lemma 5.1.

There exists a $j_{max}=j_{max}(\mathring{R}_{0},\delta)$ such that

$\displaystyle\chi_{j}(t,x)=0,\quad{\rm for\quad all}\quad j>j_{max}.$

Moreover, for all $0\leq j\leq j_{max}$, there holds

$\displaystyle\rho_{j}\leq 4^{j_{max}}.$

###### Proof.

For $j\geq 1$,

$\displaystyle\chi_{j}(t,x)\neq 0\Leftrightarrow
4^{j-1}\leq\Big{<}\frac{50\mathring{R}_{0}(t,x)}{\varepsilon_{0}\delta}\Big{>}\leq
4^{j+1}.$

Thus, $\chi_{j}(t,x)\neq 0$ implies

$\displaystyle 4^{j-2}\leq\frac{50|\mathring{R}_{0}|}{\varepsilon_{0}\delta}.$

Thus, there exists $j_{max}=j_{max}(\mathring{R}_{0},\delta)$ such that

$\displaystyle\chi_{j}(t,x)=0,\quad{\rm for\quad all}\quad j>j_{max}.$

More precisely, we have

$\displaystyle
4^{j_{max}}\leq\frac{800|\mathring{R}_{0}|}{\varepsilon_{0}\delta}.$ (5.28)

MM $\square$

######  Lemma 5.2.

Let $0\leq j\leq j_{max}$. There holds

$\displaystyle\|\chi_{j}\|_{C^{L}_{t,x}}\leq C(\mathring{R}_{0},\delta,L),$

where $L$ is integer, and the constant $C$ also depend on $\varepsilon_{0}$,
but $\varepsilon_{0}$ is a universal constant and we omit it.

###### Proof.

Direct computation gives that

$\displaystyle\partial_{l}(\big{<}A\big{>})=\frac{A:\partial_{l}A}{\big{<}A\big{>}},\quad\partial_{ll}(\big{<}A\big{>})=\frac{\partial_{l}A:\partial_{l}A+A:\partial_{ll}A}{\big{<}A\big{>}}-\frac{(A:\partial_{l}A)^{2}}{\big{<}A\big{>}^{2}}$

hence

$\displaystyle|\partial_{l}(\big{<}A\big{>})|\leq|\partial_{l}A|.$

Since

$\displaystyle\partial_{l}(\chi_{j}(t,x))=\tilde{\chi}^{\prime}\Big{(}\frac{1}{4^{j}}\Big{<}\frac{\mathring{R}_{0}}{\varepsilon_{0}\delta}\Big{>}\Big{)}\frac{1}{4^{j}}\partial_{l}\Big{<}\frac{\mathring{R}_{0}}{\varepsilon_{0}\delta}\Big{>},$

thus

$\displaystyle|\partial_{l}(\chi_{j}(t,x))|\leq\frac{C}{4^{j}}\frac{|\partial_{l}\mathring{R}_{0}|}{\varepsilon_{0}\delta}\leq
C(\varepsilon_{0},\mathring{R}_{0},\delta),\quad|\partial_{ll}(\chi_{j}(t,x))|\leq
C(\varepsilon_{0},\mathring{R}_{0},\delta).$

By the inequality

$\displaystyle\|f\circ g\|_{C^{L}}\leq
C_{L}\big{(}\|f\|_{C^{L}}\|g\|_{C^{1}}^{L}+\|f\|_{C^{1}}\|g\|_{C^{L}}\big{)},$

we know that

$\displaystyle\|\chi_{j}\|_{C^{L}_{t,x}}\leq
C(\varepsilon_{0},\mathring{R}_{0},\delta,L).$

MM $\square$

######  Lemma 5.3 (Estimate on the amplitude).

For $0\leq j\leq j_{max}$, we have

$\displaystyle\|a_{(\xi,j)}\|_{L^{\infty}}$ $\displaystyle\leq$
$\displaystyle\sqrt{\rho_{j}}\leq 2^{j}\sqrt{\delta},$
$\displaystyle\|a_{(\xi,j)}\|_{C^{L}_{t,x}}$ $\displaystyle\leq$
$\displaystyle C(\mathring{R}_{0},\delta,L).$

###### Proof.

Recall that

$\displaystyle
a_{(\xi,j)}(t,x)=\sqrt{\rho_{j}}\chi_{j}(t,x)\gamma_{\xi}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}.$

By Lemma 5.2, the estimate on $\|a_{(\xi,j)}\|_{L^{\infty}}$ and
$\|a_{(\xi,j)}\|_{C^{L}_{t,x}}$ is obvious. In fact, noticing (5.28), we have

$\displaystyle\|a_{(\xi,j)}\|_{C^{L}_{t,x}}$ $\displaystyle=$
$\displaystyle\sqrt{\rho_{j}}\Big{\|}\chi_{j}\gamma_{\xi}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}\Big{\|}_{C^{L}_{t,x}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\rho_{j}}\|\chi_{j}\|_{C^{L}_{t,x}}\Big{\|}\gamma_{\xi}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}\Big{\|}_{C^{L}_{t,x}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,L)\sqrt{\delta}2^{j_{max}}\leq
C(\mathring{R}_{0},\delta,L).$

MM $\square$

######  Proposition 5.1 (Estimate on the perturbation).

For the velocity perturbation, we have the following bound: for every
$t\in[0,1]$
1\. $L^{2}$ estimate:

$\displaystyle\|w^{p}_{1}(t)\|_{L^{2}}\leq\frac{M\sqrt{\delta}}{16}+C(\mathring{R}_{0},\delta)(\lambda_{1}\sigma_{1})^{-\frac{1}{2}},$
$\displaystyle\|w^{c}_{1}(t)\|_{L^{2}}\leq
C(\mathring{R}_{0},\delta)\sigma_{1}r_{1},$
$\displaystyle\|w^{t}_{1}(t)\|_{L^{2}}\leq
C(\mathring{R}_{0},\delta)r_{1}\mu^{-1}_{1},$

where $M$ is a universal constant.
2\. $L^{p}$ estimate: for $p>1$, there holds

$\displaystyle\|w^{p}_{1}(t)\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)r_{1}^{1-\frac{2}{p}},$
$\displaystyle\|w^{c}_{1}(t)\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{2-\frac{2}{p}},$
$\displaystyle\|w^{t}_{1}(t)\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)r_{1}^{2-\frac{2}{p}}\mu_{1}^{-1}.$

3\. $W^{1,p}$ estimate: for $p>1$, there holds

$\displaystyle\|w^{p}_{1}(t)\|_{W^{1,p}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\lambda_{1}r_{1}^{1-\frac{2}{p}},$
$\displaystyle\|w^{c}_{1}(t)\|_{W^{1,p}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\lambda_{1}\sigma_{1}r_{1}^{2-\frac{2}{p}},$
$\displaystyle\|w^{t}_{1}(t)\|_{W^{1,p}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\mu_{1}^{-1}\lambda_{1}\sigma_{1}r_{1}^{3-\frac{2}{p}}.$

In particular, we have

$\displaystyle\|w^{p}_{1}(t)\|_{C^{L}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,L)\lambda_{1}^{L}r_{1},$
$\displaystyle\|w^{c}_{1}(t)\|_{C^{L}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,L)\lambda_{1}^{L}\sigma_{1}r_{1}^{2},$
$\displaystyle\|w^{t}_{1}(t)\|_{C^{L}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,L)\mu_{1}^{-1}(\lambda_{1}\sigma_{1}r_{1})^{L}r_{1}^{2}.$

4\. Time derivative estimate: for $p>1$, there holds

$\displaystyle\|\partial_{t}w^{p}_{1}(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle
C(\mathring{R}_{0},\delta,p)\lambda_{1}\sigma_{1}\mu_{1}r_{1}^{2-\frac{2}{p}}.$

###### Proof.

Step 1: $L^{2}$ estimate. Recall the definition (5.25) of $w^{p}_{1}$, using
the support property (5.18) of cutoff function $\chi_{i}$, Proposition 4.7,
Lemma 5.3 and Lemma 3.1, we have

$\displaystyle\frac{1}{2}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\int_{{\rm
T}^{2}}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\Big{|}W_{(\xi)}+W_{(-\xi)}\Big{|}^{2}dx$
$\displaystyle\leq$ $\displaystyle
2\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\int_{{\rm
T}^{2}}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}dx$ $\displaystyle\leq$ $\displaystyle
2\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\Big{(}\|a_{(\xi,j)}\|^{2}_{L^{2}}\|\eta_{(\xi)}\|^{2}_{L^{2}}+\frac{C}{\lambda_{1}\sigma_{1}}\|a_{(\xi,j)}\|^{2}_{C^{1}}\|\eta_{(\xi)}\|^{2}_{L^{2}}\Big{)}$
$\displaystyle\leq$ $\displaystyle 6\sum_{j}\Big{(}4^{j}\delta\int_{{\rm
T}^{2}}\chi_{j}^{2}(t,x)dx+\frac{C(\mathring{R}_{0},\delta)}{\lambda_{1}\sigma_{1}}\Big{)}.$

Moreover, for any $t\in[0,1]$, there hold

$\displaystyle\sum_{j\geq 1}4^{j}\delta\int_{{\rm T}^{2}}\chi_{j}^{2}(t,x)dx$
$\displaystyle\leq$ $\displaystyle\delta\sum_{j\geq 1}4^{j}\int_{\\{x\in{\rm
T}^{2}:4^{j-1}\varepsilon_{0}\delta\leq|\mathring{R}_{0}(t,x)|\leq
4^{j+1}\varepsilon_{0}\delta\\}}dx$ $\displaystyle\leq$
$\displaystyle\delta\sum_{j\geq 1}4^{j}\int_{\\{x\in{\rm
T}^{2}:4^{j-1}\varepsilon_{0}\delta\leq|\mathring{R}_{0}(t,x)|\leq
4^{j+1}\varepsilon_{0}\delta\\}}\frac{|\mathring{R}_{0}(t,x)|}{4^{j-1}\varepsilon_{0}\delta}dx$
$\displaystyle\leq$ $\displaystyle 4\varepsilon_{0}^{-1}\sum_{j\geq
1}\int_{\\{x\in{\rm T}^{2}:4^{j-1}\delta\leq|\mathring{R}_{0}|\leq
4^{j+1}\delta\\}}|\mathring{R}(t,x)|dx$ $\displaystyle\leq$ $\displaystyle
4\varepsilon_{0}^{-1}\int_{{\rm
T}^{2}}|\mathring{R}_{0}(t,x)|dx\leq\frac{\delta}{500}.$

Thus, we obtain

$\displaystyle\frac{1}{2}\sum_{j\geq 1}\sum_{\xi\in\Lambda_{(j)}}\int_{{\rm
T}^{2}}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\Big{|}W_{(\xi)}+W_{(-\xi)}\Big{|}^{2}dx\leq\frac{\delta}{80}+\frac{C(\mathring{R}_{0},\delta)}{\lambda_{1}\sigma_{1}}.$
(5.29)

Finally, due to the definition of $\rho_{0}$ and estimate (5.23), we deduce
that

$\displaystyle\frac{1}{2}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}}{\rho_{0}}\Big{)}\eta^{2}_{(\xi)}\Big{|}W_{(\xi)}+W_{(-\xi)}\Big{|}^{2}dx\leq
C_{0}\delta,$

where $C_{0}$ is a universal constant. Taking $M$ to be a universal constant,
we obtain

$\displaystyle\|w^{p}_{1}\|_{L^{2}}(t)\leq\frac{M\sqrt{\delta}}{16}+C(\mathring{R},\delta)\frac{1}{\sqrt{\lambda_{1}\sigma_{1}}}.$

From the definition (5.26) of $w^{c}_{1}$ , Proposition 4.7 and lemma 5.3, and
noticing the fact (5.28)(we use this fact frequently below), we deduce that

$\displaystyle\|w^{c}_{1}(t)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\lambda_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\big{(}\|a_{(\xi,j)}\|_{C^{1}_{x}}\|\eta_{(\xi)}\|_{L^{2}}+\|a_{(\xi,j)}\|_{L^{\infty}_{x}}\|\nabla\eta_{(\xi)}\|_{L^{2}_{x}}\big{)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\big{(}C(\mathring{R}_{0},\delta)+2^{j}\sqrt{\delta}\lambda_{1}\sigma_{1}r_{1}\big{)}$
$\displaystyle\leq$ $\displaystyle C(\mathring{R}_{0},\delta)\sigma_{1}r_{1}.$

By (5.27), Proposition 4.7 and Lemma 5.3, we obtain

$\displaystyle\|w^{t}_{1}(t)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\|a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\|a_{(\xi,j)}\|^{2}_{L^{\infty}}\|\eta_{(\xi)}\|^{2}_{L^{4}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}4^{j}r_{1}\leq
C(\mathring{R}_{0},\delta)\frac{r_{1}}{\mu_{1}}.$

Step 2: $L^{p}$ estimate. By Lemma 3.1, proposition 4.7 and Lemma 5.3, we
obtain

$\displaystyle\|w^{p}_{1}(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\Big{(}\|a_{(\xi,j)}(t)\|_{L^{p}}\|\eta_{(\xi)}(t)\|_{L^{p}}+C_{p}(\lambda_{1}\sigma_{1})^{-\frac{1}{p}}\|a_{(\xi,j)}(t)\|_{C^{1}_{x}}\|\eta_{(\xi)}(t)\|_{L^{p}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\sum_{j}\sum_{\xi\in\Lambda_{(j)}}2^{j}r_{1}^{1-\frac{2}{p}}\leq
C(\mathring{R}_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

For $\|w^{c}_{1}(t)\|_{L^{p}}$ and $\|w^{t}_{1}\|_{L^{p}}$, there hold

$\displaystyle\|w^{c}_{1}(t)\|_{L^{p}}$
$\displaystyle\leq\frac{C}{\lambda_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\big{(}\|a_{(\xi,j)}\|_{C^{1}_{x}}\|\eta_{(\xi)}\|_{L^{p}}+\|a_{(\xi,j)}\|_{L^{\infty}_{x}}\|\nabla\eta_{(\xi)}\|_{L^{p}_{x}}\big{)}$
$\displaystyle\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{2-\frac{2}{p}}$

and

$\displaystyle\|w^{t}_{1}(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\frac{C}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\|a_{(\xi,j)}\|^{2}_{L^{\infty}_{x}}\|\eta_{(\xi)}\|^{2}_{L^{2p}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)r_{1}^{2-\frac{2}{p}}\mu_{1}^{-1}.$

Step 3: $W^{1,p}$ estimate. Recalling (5.25), a direct computation gives that

$\displaystyle\partial_{l}w^{p}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\Big{(}\partial_{l}\big{(}a_{(\xi,j)}\eta_{(\xi)}\big{)}\big{(}W_{(\xi)}+W_{(-\xi)}\big{)}+a_{(\xi,j)}\eta_{(\xi)}\big{(}\partial_{l}W_{(\xi)}+\partial_{l}W_{(-\xi)}\big{)}\Big{)},$

thus by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we have

$\displaystyle\|\nabla w^{p}_{1}\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\Big{(}\|\nabla
a_{(\xi,j)}\|_{L^{p}}\|\eta_{(\xi)}\|_{L^{p}}+\|a_{(\xi,j)}\|_{L^{p}}\|\nabla\eta_{(\xi)}\|_{L^{p}}$
$\displaystyle\quad+C_{p}(\lambda_{1}\sigma_{1})^{-\frac{1}{p}}\|a_{(\xi,j)}\|_{C^{2}_{x}}\|\eta_{(\xi)}\|_{W^{1,p}}\Big{)}$
$\displaystyle+\lambda_{1}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\Big{(}\|a_{(\xi,j)}\|_{L^{p}}\|\eta_{(\xi)}\|_{L^{p}}+C_{p}(\lambda_{1}\sigma_{1})^{-\frac{1}{p}}\|a_{(\xi)}\|_{C^{1}_{x}}\|\eta_{(\xi)}\|_{L^{p}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\lambda_{1}r_{1}^{1-\frac{2}{p}},$

and

$\displaystyle\|\nabla w^{p}_{1}\|_{L^{\infty}}\leq
C(\mathring{R}_{0},\delta)\lambda_{1}r_{1}.$

Recalling (5.26), there holds

$\displaystyle\partial_{l}w^{c}_{1}:$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\nabla^{\bot}\partial_{l}\big{(}a_{(\xi,j)}\eta_{(\xi)}\big{)}\frac{e^{i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-i\lambda_{1}\xi^{\bot}\cdot x}}{\lambda_{1}}$
$\displaystyle-\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\nabla^{\bot}\big{(}a_{(\xi,j)}\eta_{(\xi)}\big{)}\partial_{l}\Big{(}\frac{e^{i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-i\lambda_{1}\xi^{\bot}\cdot x}}{\lambda_{1}}\Big{)}.$

Thus, by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we get

$\displaystyle\|\nabla w^{c}_{1}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\lambda_{1}\sigma_{1}r_{1}^{2-\frac{2}{p}}.$

Recalling (5.27), we have

$\displaystyle\partial_{l}w_{1}^{t}:=-\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}P_{H}P_{\neq
0}\partial_{l}\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\xi\big{)}.$

Thus, by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we get

$\displaystyle\|\nabla w^{t}_{1}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\mu_{1}^{-1}\lambda_{1}\sigma_{1}r_{1}^{3-\frac{2}{p}}.$

The same argument gives that

$\displaystyle\|w^{p}_{1}(t)\|_{C^{L}}\leq
C(\mathring{R}_{0},\delta)\lambda^{L}r_{1}.$

The estimate for $\|w^{c}_{1}(t)\|_{C^{L}},\|w^{t}_{1}(t)\|_{C^{L}}$ is
similar, and we omit the detail here.

Step 4: Time derivative estimate. A direct computation gives that

$\displaystyle\partial_{t}w^{p}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\partial_{t}\big{(}a_{(\xi,j)}\eta_{(\xi)}\big{)}\big{(}W_{(\xi)}+W_{(-\xi)}\big{)},$

thus by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we deduce that

$\displaystyle\|\partial_{t}w^{p}_{1}\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}}\Big{(}\|\partial_{t}a_{(\xi,j)}\|_{L^{p}}\|\eta_{(\xi)}\|_{L^{p}}+\|a_{(\xi,j)}\|_{L^{p}}\|\partial_{t}\eta_{(\xi)}\|_{L^{p}}$
$\displaystyle\quad+C_{p}(\lambda_{1}\sigma_{1})^{-\frac{1}{p}}\|a_{(\xi,j)}\|_{C^{2}_{t,x}}\|\partial_{t}\eta_{(\xi)}\|_{L^{p}}\Big{)}\leq
C(\mathring{R}_{0},\delta,p)\lambda_{1}\sigma_{1}\mu_{1}r_{1}^{2-\frac{2}{p}}.$

MM $\square$

######  Corollary 5.4.

For all $1<p<2$, by taking $\lambda_{1}$ large enough, we have

$\displaystyle\|w_{1}\|_{L^{2}}\leq\frac{M\sqrt{\delta}}{10},\quad\quad\|w_{1}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta)r_{1}^{1-\frac{2}{p}}.$

###### Proof.

From the definition of $w_{1}$, by the $L^{2}$ estimate in Proposition 5.1, we
deduce that

$\displaystyle\|w_{1}\|_{L^{2}}\leq\frac{M\sqrt{\delta}}{16}+C(\mathring{R}_{0},\delta)\big{(}(\lambda_{1}\sigma_{1})^{-\frac{1}{2}}+\sigma_{1}r_{1}+r_{1}\mu_{1}^{-1}\big{)}.$

Using the relationship of parameter (5.24) and taking $\lambda_{1}$ large
enough, we obtain

$\displaystyle\|w_{1}\|_{L^{2}}\leq\frac{M\sqrt{\delta}}{10}.$

Similarly, we get

$\displaystyle\|w_{1}(t)\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\Big{(}r_{1}^{1-\frac{2}{p}}+\sigma_{1}r_{1}^{2-\frac{2}{p}}+r_{1}^{2-\frac{2}{p}}\mu_{1}^{-1}\Big{)}.$

Then, the parameter relationship gives

$\displaystyle\|w_{1}(t)\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

MM $\square$

##  6\. Construction and Estimate on temperature perturbation

After the construction of new velocity $v_{1}$, we construct new temperature
$\theta_{1}$ as following. Consider the transport-diffusion equation:

$\begin{cases}\partial_{t}\theta_{1}+v_{1}\cdot\nabla\theta_{1}-\triangle\theta_{1}=0,\\\\[3.0pt]
\theta_{1}|_{t=0}=\theta^{0},\end{cases}$ (6.30)

where $\theta^{0}(x)$ is the function in Proposition 2.2. From the standard
theory, we know that there exists a unique solution $\theta_{1}\in
C^{\infty}([0,1]\times{\rm T}^{2})$ and it obeys the following estimates:

$\displaystyle\|\theta_{1}\|_{L^{\infty}_{t,x}}\leq\|\theta^{0}\|_{L^{\infty}},\quad\|\theta_{1}(t)\|^{2}_{L^{2}}+2\int_{0}^{t}\|\nabla\theta_{1}(s)\|^{2}_{L^{2}}ds=\|\theta^{0}\|^{2}_{L^{2}}$

and

$\begin{cases}\partial_{t}(\theta_{1}-\theta_{0})+v_{1}\cdot\nabla(\theta_{1}-\theta_{0})+(v_{1}-v_{0})\cdot\nabla\theta_{0}-\triangle(\theta_{1}-\theta_{0})=0,\\\\[3.0pt]
(\theta_{1}-\theta_{0})|_{t=0}=0.\end{cases}$ (6.31)

Direct energy estimate gives that

$\displaystyle\frac{d}{dt}\|\theta_{1}-\theta_{0}\|^{2}_{L^{2}}+2\|\nabla(\theta_{1}-\theta_{0})\|^{2}_{L^{2}}$
$\displaystyle=$ $\displaystyle 2\int_{{\rm
T}^{2}}\theta_{0}(v_{1}-v_{0})\cdot\nabla(\theta_{1}-\theta_{0})dx$
$\displaystyle\leq$
$\displaystyle\|\nabla(\theta_{1}-\theta_{0})\|^{2}_{L^{2}}+\parallel\theta_{0}\parallel_{\infty}\|v_{1}-v_{0}\|^{2}_{L^{2}}.$

which implies

$\displaystyle\sup_{t\in[0,1]}\|\theta_{1}-\theta_{0}\|^{2}_{L^{2}}(t)+\int_{0}^{1}\|\nabla(\theta_{1}-\theta_{0})\|^{2}_{L^{2}}(t)dt\leq\parallel\theta^{0}\parallel_{\infty}M^{2}\delta.$

##  7\. Reynold Stress: Construction and Estimate

###  7.1. Anti-divergence operator

We first recall the anti-divergence operator:

######  Lemma 7.1.

There exists an operator $\mathcal{R}$ satisfying the following property:

  * •

For any $v\in C^{\infty}({\rm T}^{2};R^{2})$, $\mathcal{R}v(x)$ is a symmetric
trace-free matrix for each $x\in{\rm T}^{2}$ and ${\rm
div}\mathcal{R}v(x)=v(x)-\fint_{{\rm T}^{2}}v(x)dx.$

  * •

The following estimates hold: $\||\nabla|\mathcal{R}\|_{L^{p}\rightarrow
L^{p}}\leq C_{p}$, $\|\mathcal{R}\|_{L^{p}\rightarrow L^{p}}\leq
C_{p},\quad\|\mathcal{R}\|_{C^{0}\rightarrow C^{0}}\leq C.$

###### Proof.

Let $u\in C_{0}^{\infty}({\rm T}^{2})$ be a solution to

$\displaystyle\triangle u=v-\fint_{{\rm T}^{2}}v(x)dx,$

where $C_{0}^{\infty}({\rm T}^{2})=\\{f\in C^{\infty}({\rm T}^{2}):\fint
f(x)dx=0\\}.$ Then set

$\displaystyle\mathcal{R}v(x):=\nabla u+(\nabla u)^{T}-({\rm div}u){\rm Id}.$

Then $\mathcal{R}$ satisfies the above property. MM $\square$

###  7.2. Construction of new error $\mathring{R}_{1}$

In this subsection, we define the new error $\mathring{R}_{1}$. We first
compute the interaction $w^{p}_{1}\otimes w^{p}_{1}$ of principle
perturbation. Recalling the definition (5.25) of $w^{p}_{1},$ we have

$\displaystyle w^{p}_{1}\otimes w^{p}_{1}=T_{self}+T_{inter},$

where

$\displaystyle T_{self}$ $\displaystyle:=$
$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\Big{(}\xi\otimes\xi-\frac{1}{2}\xi\otimes\xi\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\xi\otimes\xi+\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi$
$\displaystyle-\frac{1}{2}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\xi\otimes\xi\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)},$

and $T_{inter}=w^{p}_{1}\otimes w^{p}_{1}-T_{self}$. Recalling (5.20), we
deduce

$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\xi\otimes\xi=\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\rho_{j}\chi_{j}^{2}\gamma^{2}_{\xi}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}\xi\otimes\xi.$

However, by Geometric Lemma 4.2,

$\displaystyle\mathring{R}_{0}=\sum_{j}\chi_{j}^{2}\mathring{R}_{0}$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{j}\rho_{j}\chi_{j}^{2}\Big{)}{\rm
Id}-\sum_{j}\rho_{j}\chi_{j}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{j}\rho_{j}\chi_{j}^{2}\Big{)}{\rm
Id}-\sum_{j}\rho_{j}\chi_{j}^{2}\Big{[}\sum_{\xi\in\Lambda_{(j)}^{+}}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{j}}\Big{)}\xi\otimes\xi\Big{]}.$

Thus, there holds

$\displaystyle\mathring{R}_{0}+\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\xi\otimes\xi=\Big{(}\sum_{j}\rho_{j}\chi_{j}^{2}\Big{)}{\rm
Id}.$

Furthermore, by (4.15) and using the identity ${\rm div}f={\rm P}_{\neq 0}{\rm
div}f$, there hold

$\displaystyle{\rm
div}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\Big{)}$
$\displaystyle=$ $\displaystyle{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\Big{)}+{\rm
P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\xi\otimes\xi\nabla\big{(}\eta^{2}_{(\xi)}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\Big{)}+{\rm
P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\frac{\xi}{\mu_{1}}\partial_{t}\big{(}\eta^{2}_{(\xi)}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\Big{)}+{\rm
P}_{\neq
0}\Big{(}\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\partial_{t}\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\xi\big{)}\Big{)}$
$\displaystyle-{\rm P}_{\neq
0}\Big{(}\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\partial_{t}\big{(}a^{2}_{(\xi,j)}\big{)}\eta^{2}_{(\xi)}\xi\Big{)}$

and

$\displaystyle{\rm
div}\Big{(}\frac{1}{2}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\xi\otimes\xi\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\big{)}\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}.$

Using (5.18), we have

$\displaystyle T_{inter}$
$\displaystyle=\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})$
$\displaystyle\quad+\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})}).$

It follows from Proposition 4.1 that

$\displaystyle\mathrm{div}(W_{\xi,\xi^{\prime}}\otimes
W_{\xi,\xi^{\prime}})=2\nabla\tilde{P}_{\xi,\xi^{\prime}},$

where

$\displaystyle W_{\xi,\xi^{\prime}}$
$\displaystyle=W_{(\xi)}+W_{(-\xi)}+W_{(\xi^{\prime})}+W_{(-\xi^{\prime})},$
$\displaystyle\tilde{P}_{\xi,\xi^{\prime}}$
$\displaystyle=\left|\xi\sin(\lambda\xi^{\perp}\cdot
x)+\xi^{\prime}\sin(\lambda{\xi^{\prime}}^{\perp}\cdot
x)\right|^{2}+\left(\cos(\lambda\xi^{\perp}\cdot
x)+\cos(\lambda{\xi^{\prime}}^{\perp}\cdot x)\right)^{2}$
$\displaystyle=2+2\cos(\lambda{\xi}^{\perp}\cdot
x)\cos(\lambda{\xi^{\prime}}^{\perp}\cdot
x)+2\xi\cdot\xi^{\prime}\sin(\lambda\xi^{\perp}\cdot
x)\sin(\lambda{\xi^{\prime}}^{\perp}\cdot x).$

Set

$\displaystyle P_{\xi,\xi^{\prime}}=2\cos(\lambda{\xi}^{\perp}\cdot
x)\cos(\lambda{\xi^{\prime}}^{\perp}\cdot
x)+2\xi\cdot\xi^{\prime}\sin(\lambda\xi^{\perp}\cdot
x)\sin(\lambda{\xi^{\prime}}^{\perp}\cdot x),$ (7.32)

then

$\displaystyle\mathrm{div}(W_{\xi,\xi^{\prime}}\otimes
W_{\xi,\xi^{\prime}})=2\nabla P_{\xi,\xi^{\prime}}.$

Thus, we obtain

$\displaystyle\mathrm{div}T_{inter}=$
$\displaystyle\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}\Big{(}(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})\nabla(a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})})$
$\displaystyle+\nabla\big{(}a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}P_{\xi,\xi^{\prime}}\big{)}-P_{\xi,\xi^{\prime}}\nabla(a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})})\Big{)}$
$\displaystyle+$
$\displaystyle\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}\Big{(}(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})\nabla(a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})})$
$\displaystyle+\nabla\big{(}a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}P_{\xi,\xi^{\prime}}\big{)}-P_{\xi,\xi^{\prime}}\nabla(a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})})\Big{)}.$

Finally, by combining the definition (5.27) of $w^{t}_{1}$, we obtain

$\displaystyle{\rm div}T_{self}+{\rm div}T_{inter}+{\rm
div}\mathring{R}_{0}+\partial_{t}w_{1}^{t}=-\nabla(p_{1}-p_{0})+T_{1,{\rm
osc}},$ (7.33)

where we define the new pressure $p_{1}$ such that

$\displaystyle p_{1}-p_{0}$ $\displaystyle=-\Big{(}$
$\displaystyle\sum_{j}\rho_{j}\chi_{j}^{2}+\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\triangle^{-1}{\rm
div}\partial_{t}\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\xi\big{)}$
$\displaystyle+\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}P_{\xi,\xi^{\prime}}$
$\displaystyle+\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}P_{\xi,\xi^{\prime}}\Big{)},$

and oscillatory term

$\displaystyle T_{1,{\rm osc}}:$ $\displaystyle=$ $\displaystyle{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\Big{)}$
(7.34) $\displaystyle-{\rm P}_{\neq
0}\Big{(}\frac{1}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\partial_{t}\big{(}a^{2}_{(\xi,j)}\big{)}\eta^{2}_{(\xi)}\xi\Big{)}$
$\displaystyle-\frac{1}{2}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\big{)}\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}$ $\displaystyle+{\rm P}_{\neq
0}\Big{[}\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}\Big{(}-P_{\xi,\xi^{\prime}}\nabla(a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})})$
$\displaystyle\quad\quad+(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})\nabla\left(a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}\right)\Big{)}\Big{]}$
$\displaystyle+{\rm P}_{\neq
0}\Big{[}\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}\Big{(}-P_{\xi,\xi^{\prime}}\nabla(a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})})$
$\displaystyle\quad\quad+(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})\nabla\left(a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}\right)\Big{)}\Big{]}.$

From (7.33), we know that $\fint_{{\rm T}^{2}}T_{1,{\rm osc}}(t,x)dx=0.$

After the computation of interaction of principle perturbation, we define
$\mathring{R}_{1}$ as follows:

$\displaystyle\mathring{R}_{1}$ $\displaystyle=$
$\displaystyle\underbrace{\mathcal{R}\big{(}\partial_{t}(w^{p}_{1}+w^{c}_{1})\big{)}+\mathcal{R}\big{(}{\rm
div}(v_{0}\otimes w_{1}+w_{1}\otimes
v_{0})\big{)}+\mathcal{R}((-\Delta)^{\alpha}w_{1})}_{R_{linear}}$
$\displaystyle+\underbrace{\mathcal{R}\big{(}{\rm
div}(w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})+(w^{c}_{1}+w^{t}_{1})\otimes
w_{1})\big{)}}_{R_{cor}}$
$\displaystyle+\underbrace{\mathcal{R}\big{(}T_{1,{\rm
osc}}\big{)}}_{R_{osc}}-\underbrace{\mathcal{R}((\theta_{1}-\theta_{0})e_{2})}_{R_{tem}}.$

Since $\fint_{{\rm T}^{2}}\theta^{0}(x)dx=0,$ thus $\fint_{{\rm
T}^{2}}\theta_{1}(t,x)dx=0$ for every $t$, hence $\fint_{{\rm
T}^{2}}(\theta_{1}-\theta_{0})(t,x)dx=0$ for every $t$. Recalling the
construction of $v_{1},p_{1},\theta_{1},\mathring{R}_{1}$ and (7.33), a direct
computation gives that

$\displaystyle{\rm div}\mathring{R}_{1}$ $\displaystyle=$
$\displaystyle\partial_{t}(w^{p}_{1}+w^{c}_{1})+{\rm div}(v_{0}\otimes
w_{1}+w_{1}\otimes v_{0})+(-\Delta)^{\alpha}w_{1}$ (7.35) $\displaystyle+{\rm
div}(w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})+(w^{c}_{1}+w^{t}_{1})\otimes
w_{1})\big{)}$ $\displaystyle+T_{1,{\rm osc}}-(\theta_{1}-\theta_{0})e_{2}$
$\displaystyle=$ $\displaystyle\partial_{t}(w^{p}_{1}+w^{c}_{1})+{\rm
div}(v_{0}\otimes w_{1}+w_{1}\otimes v_{0})+(-\Delta)^{\alpha}w_{1}$
$\displaystyle+{\rm
div}(w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})+(w^{c}_{1}+w^{t}_{1})\otimes
w_{1})\big{)}$ $\displaystyle+{\rm div}\big{(}w^{p}_{1}\otimes
w^{p}_{1}\big{)}+{\rm
div}\mathring{R}_{0}+\partial_{t}w_{1}^{t}+\nabla(p_{1}-p_{0})-(\theta_{1}-\theta_{0})e_{2}$
$\displaystyle=$ $\displaystyle\partial_{t}w_{1}+{\rm div}(v_{0}\otimes
w_{1}+w_{1}\otimes v_{0})+(-\Delta)^{\alpha}w_{1}+{\rm div}\big{(}w_{1}\otimes
w_{1}\big{)}+{\rm div}\mathring{R}_{0}$
$\displaystyle+\nabla(p_{1}-p_{0})-(\theta_{1}-\theta_{0})e_{2}$
$\displaystyle=$ $\displaystyle\partial_{t}v_{1}+{\rm div}\big{(}v_{1}\otimes
v_{1}\big{)}+\nabla p_{1}+(-\Delta)^{\alpha}v_{1}-\theta_{1}e_{2}.$

Thus, the new function $(v_{1},p_{1},\theta_{1},\mathring{R}_{1})$ satisfies
Boussinesq-Reynold equation (2.2). Next, we prove that the error
$\|\mathring{R}_{1}\|_{L^{\infty}_{t}L^{1}_{x}}$ is very small.

###  7.3. Estimate on $\mathring{R}_{1}$

In this subsection, we estimate $\mathring{R}_{1}$. We deal with it term by
term.

####  7.3.1.  Estimate on linear term $R_{linear}$:

For $1<p<2$,

$\displaystyle\|\mathcal{R}\big{(}\partial_{t}(w^{p}_{1}+w^{c}_{1})\big{)}\|_{L^{p}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\sigma_{1}\mu_{1}r_{1}^{2-\frac{2}{p}},$
$\displaystyle\|\mathcal{R}((-\Delta)^{\alpha}w_{1})\|_{L^{p}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\lambda_{1}^{2\alpha-1}r_{1}^{1-\frac{2}{p}},$
$\displaystyle\|\mathcal{R}\big{(}{\rm div}(v_{0}\otimes w_{1}+w_{1}\otimes
v_{0})\big{)}\|_{L^{p}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},v_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

###### Proof.

Due to the $L^{p}$ estimate, $W^{1,p}$ estimate and time derivative estimate
in Proposition 5.1, we deduce

$\displaystyle\|\mathcal{R}\big{(}\partial_{t}(w^{p}_{1}+w^{c}_{1})\big{)}\|_{L^{p}}=\frac{1}{\lambda_{1}}\|\mathcal{R}\nabla^{\bot}(\partial_{t}w^{p}_{1}\big{)}\|_{L^{p}}\leq\frac{C_{p}}{\lambda_{1}}\|\partial_{t}w^{p}_{1}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}\mu_{1}r_{1}^{2-\frac{2}{p}},$
$\displaystyle\|\mathcal{R}((-\Delta)^{\alpha}w_{1})\|_{L^{p}}\leq
C\|\mathcal{R}|\nabla|^{2\alpha}(w_{1})\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\lambda_{1}^{2\alpha-1}r_{1}^{1-\frac{2}{p}},$
$\displaystyle\|\mathcal{R}\big{(}{\rm div}(v_{0}\otimes w_{1}+w_{1}\otimes
v_{0})\big{)}\|_{L^{p}}\leq C_{p}\|v_{0}\otimes w_{1}+w_{1}\otimes
v_{0}\|_{L^{p}}\leq C(\mathring{R}_{0},v_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

MM $\square$

####  7.3.2.  Estimate on corrector term $R_{cor}$:

For $1<p<2$,

$\displaystyle\|\mathcal{R}\big{(}{\rm
div}(w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})+(w^{c}_{1}+w^{t}_{1})\otimes
w_{1})\big{)}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)(\sigma_{1}+\mu_{1}^{-1})r_{1}^{3(1-\frac{1}{p})}.$

###### Proof.

By Proposition 5.1 and parameter relationship (5.24), we get

$\displaystyle\|\mathcal{R}\big{(}{\rm
div}(w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})+(w^{c}_{1}+w^{t}_{1})\otimes
w_{1})\big{)}\|_{L^{p}}$ $\displaystyle\leq$ $\displaystyle
C_{p}\|w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})+(w^{c}_{1}+w^{t}_{1})\otimes
w_{1}\|_{L^{p}}$ $\displaystyle\leq$ $\displaystyle
C_{p}\Big{(}\|w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})\|_{L^{1}}^{\frac{1}{p}}\|w^{p}_{1}\otimes(w^{c}_{1}+w^{t}_{1})\|_{L^{\infty}}^{1-\frac{1}{p}}$
$\displaystyle+\|(w^{c}_{1}+w^{t}_{1})\otimes
w_{1}\|_{L^{1}}^{\frac{1}{p}}\|(w^{c}_{1}+w^{t}_{1})\otimes
w_{1}\|_{L^{\infty}}^{1-\frac{1}{p}}\Big{)}$ $\displaystyle\leq$
$\displaystyle
C(\mathring{R}_{0},\delta,p)(\sigma_{1}+\mu_{1}^{-1})r_{1}^{3(1-\frac{1}{p})}.$

MM $\square$

####  7.3.3. Estimate on temperature term $R_{tem}$:

For $1<p<2$,

$\displaystyle\|\mathcal{R}\big{(}(\theta_{1}-\theta_{0})e_{2}\big{)}\|_{p}\leq
C(\theta_{0},\mathring{R}_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

###### Proof.

For $1<p<2$, we have

$\displaystyle\|\mathcal{R}\big{(}(\theta_{1}-\theta_{0})e_{2}\big{)}\|_{p}\leq
C_{p}\|\theta_{1}-\theta_{0}\|_{p}.$

From the equation (6.31), we try $L^{p}$ estimates: Multiplying
$|\theta_{1}-\theta_{0}|^{p-2}(\theta_{1}-\theta_{0})$, we arrive at

$\displaystyle\frac{1}{p}\frac{d}{dt}\|\theta_{1}-\theta_{0}\|^{p}_{p}+\int_{{\rm
T}^{2}}\nabla(\theta_{1}-\theta_{0})\cdot\nabla\big{(}|\theta_{1}-\theta_{0}|^{p-2}(\theta_{1}-\theta_{0})\big{)}dx$
$\displaystyle\leq$
$\displaystyle\|v_{1}-v_{0}\|_{p}\|\nabla\theta_{0}\|_{\infty}\|\theta_{1}-\theta_{0}\|^{p-1}_{p}.$

A direct computation gives

$\displaystyle\int_{{\rm
T}^{2}}\nabla(\theta_{1}-\theta_{0})\cdot\nabla\big{(}|\theta_{1}-\theta_{0}|^{p-2}(\theta_{1}-\theta_{0})\big{)}dx=(p-1)\int_{{\rm
T}^{2}}|\nabla(\theta_{1}-\theta_{0})|^{2}|\theta_{1}-\theta_{0}|^{p-2}.$

Thus, we obtain

$\displaystyle\|\theta_{1}-\theta_{0}\|_{p}(t)$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\|\nabla\theta_{0}\|_{\infty}(s)\|v_{1}-v_{0}\|_{p}(s)ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\|\nabla\theta_{0}\|_{\infty}(s)\|v_{1}-v_{0}\|_{p}(s)ds,$

then by Corollary 5.4

$\displaystyle\sup_{t\in[0,1]}\|\theta_{1}-\theta_{0}\|_{p}\leq\sup_{t\in[0,1]}\|\nabla\theta_{0}\|_{L^{\infty}_{x}}\sup_{t\in[0,1]}\|v_{1}-v_{0}\|_{p}\leq
C(\theta_{0},\mathring{R}_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

MM $\square$

####  7.3.4. Estimate on the oscillatory term $\mathcal{R}_{osc}$:

For every $1<p<2$,

$\displaystyle\|\mathcal{R}_{osc}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)(\lambda_{1}\sigma_{1})^{-1}r_{1}^{2-\frac{2}{p}}.$

###### Proof.

Recall (7.34). By Lemma 3.3 and Proposition 4.7

$\displaystyle\Big{\|}\mathcal{R}{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq$
$\displaystyle\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\Big{\|}|\nabla|^{-1}{\rm
P}_{\neq
0}\Big{(}\big{(}\eta^{2}_{(\xi)}-1\big{)}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\Big{(}1+\|a^{2}_{(\xi,j)}\|_{C^{3}}\Big{)}\frac{\|\eta^{2}_{(\xi)}-1\|_{L^{p}}}{\lambda_{1}\sigma_{1}}\leq
C(\mathring{R}_{0},\delta,p)\frac{r_{1}^{2-\frac{2}{p}}}{\lambda_{1}\sigma_{1}}.$

Similarly, we have

$\displaystyle\Big{\|}\mathcal{R}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\eta^{2}_{(\xi)}\big{)}\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq$
$\displaystyle\Big{\|}|\nabla|^{-1}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\xi\otimes\xi\nabla\big{(}a^{2}_{(\xi,j)}\big{)}\eta^{2}_{(\xi)}\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle+\Big{\|}|\nabla|^{-1}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\xi\otimes\xi
a^{2}_{(\xi,j)}\nabla\big{(}\eta^{2}_{(\xi)}\big{)}\big{(}e^{2i\lambda_{1}\xi^{\bot}\cdot
x}+e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\big{)}\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\Big{(}1+\|a^{2}_{(\xi,j)}\|_{C^{3}}\Big{)}\frac{\|\eta^{2}_{(\xi)}\|_{L^{p}}}{\lambda_{1}}+C(\mathring{R}_{0},\delta,p)\Big{(}1+\|a^{2}_{(\xi,j)}\|_{C^{3}}\Big{)}\frac{\|\nabla(\eta^{2}_{(\xi)})\|_{L^{p}}}{\lambda_{1}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{3-\frac{2}{p}}.$

Recalling Remark 4.3 and (7.32), we have

$\displaystyle\quad\Big{\|}\mathcal{R}{\rm P}_{\neq
0}\Big{(}\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}P_{\xi,\xi^{\prime}}\nabla\left(a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}\right)\Big{)}\Big{\|}_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{3-\frac{2}{p}},$
$\displaystyle\quad\Big{\|}\mathcal{R}{\rm P}_{\neq
0}\Big{(}\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})\nabla\left(a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}\right)\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{3-\frac{2}{p}}.$

Similarly,

$\displaystyle\quad\Big{\|}\mathcal{R}{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}P_{\xi,\xi^{\prime}}\nabla\left(a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}\right)\Big{)}\Big{\|}_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{3-\frac{2}{p}},$
$\displaystyle\quad\Big{\|}\mathcal{R}{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}(W_{(\xi)}+W_{(-\xi)})\otimes(W_{(\xi^{\prime})}+W_{(-\xi^{\prime})})\nabla\left(a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}\right)\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq
C(\mathring{R}_{0},\delta,p)\sigma_{1}r_{1}^{3-\frac{2}{p}}.$

By Proposition 4.7 and Lemma 5.3, there hold

$\displaystyle\Big{\|}\frac{1}{\mu_{1}}\mathcal{R}{\rm P}_{\neq
0}\Big{(}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\partial_{t}\big{(}a^{2}_{(\xi,j)}\big{)}\eta^{2}_{(\xi)}\xi\Big{)}\Big{\|}_{L^{p}}$
$\displaystyle\leq$
$\displaystyle\frac{C_{p}}{\mu_{1}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}\|\partial_{t}\big{(}a^{2}_{(\xi,j)}\big{)}\|_{L^{\infty}}\|\eta^{2}_{(\xi)}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,p)r_{1}^{2-\frac{2}{p}}\mu_{1}^{-1}.$

Summing the parts and using the parameter relationship (5.24), we complete the
proof. MM $\square$

Finally, collecting these term together and noticing that $\alpha<1$, we
obtain the estimate on the error term $\mathring{R}_{1}$:

$\displaystyle\|\mathring{R}_{1}\|_{L^{p}}$ $\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,v_{0},\theta_{0},p)\Big{(}(\lambda_{1}\sigma_{1})^{-1}r_{1}^{2-\frac{2}{p}}+r_{1}^{1-\frac{2}{p}}+\sigma_{1}\mu_{1}r_{1}^{2-\frac{2}{p}}+\sigma_{1}r_{1}^{3-\frac{2}{p}}+\mu_{1}^{-1}r_{1}^{3(1-\frac{1}{p})}+\lambda_{1}^{2\alpha-1}r_{1}^{1-\frac{2}{p}}\Big{)}.$

Using the parameter relationship (5.24), taking
$1<p<\frac{2\alpha}{3\alpha-1}$ and noticing $\frac{1}{2}\leq\alpha<1$, we
obtain

$\displaystyle\|\mathring{R}_{1}\|_{L^{p}}\leq
C(\mathring{R}_{0},\delta,v_{0},\theta_{0},p)\lambda_{1}^{3\alpha-1-\frac{2\alpha}{p}}.$

In summary: we have constructed smooth function
$(v_{1},p_{1},\theta_{1},R_{1})\in C^{\infty}([0,1]\times{\rm
T}^{2},R^{2}\times R\times R\times S^{2\times 2})$, they satisfies Boussinesq-
Reynold equation (2.2). By taking $\lambda_{1}$ large enough, the following
estimate hold: for any $t\in[0,1]$

$\displaystyle\|v_{1}-v_{0}\|_{L^{2}}(t)$ $\displaystyle\leq$
$\displaystyle\frac{M\sqrt{\delta}}{10},\quad\|\theta_{1}\|_{L^{\infty}_{t,x}}\leq\|\theta^{0}\|_{\infty},$
$\displaystyle\|\theta_{1}\|_{L^{2}}(t)$ $\displaystyle+$ $\displaystyle
2\int_{0}^{t}\|\nabla\theta_{1}\|_{2}^{2}(s)ds=\|\theta^{0}\|_{L^{2}}(t),$
$\displaystyle\|\theta_{1}-\theta_{0}\|^{2}_{L^{2}}(t)$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\|\nabla(\theta_{1}-\theta_{0})\|_{2}^{2}(s)ds\leq\|\theta^{0}\|_{\infty}M^{2}\delta,$
$\displaystyle\|\mathring{R}_{1}\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta,v_{0},\theta_{0},p)\lambda_{1}^{3\alpha-1-\frac{2\alpha}{p}}.$

##  8\. Proof of main Proposition

In this section, we give a proof of Proposition 2.2 by combining the above
construction and estimate.

Noticing that $3\alpha-1-\frac{2\alpha}{p}<1$ for
$1<p<\frac{2\alpha}{3\alpha-1}$, thus we first take $\lambda_{1}$ to be a
integer, large enough such that

$\displaystyle
C(\mathring{R}_{0},\delta,v_{0},\theta_{0},p)\lambda_{1}^{3\alpha-1-\frac{2\alpha}{p}}\leq\frac{\varepsilon_{0}\delta}{60000},$

Thus, there holds

$\displaystyle\|\mathring{R}_{1}\|_{L^{\infty}_{t}L^{1}_{x}}\leq\frac{\varepsilon_{0}\delta}{20000}.$

To complete the proof of Proposition 2.2, we only need to estimate the energy
difference between $e(t)$ and $\int_{{\rm T}^{2}}|v_{1}(t,x)|^{2}dx.$

Direct computation gives that

$\displaystyle\int_{{\rm T}^{2}}|v_{1}(t,x)|^{2}dx$ $\displaystyle=$
$\displaystyle\int_{{\rm T}^{2}}|v_{0}(t,x)+w_{1}(t,x)|^{2}dx$
$\displaystyle=$ $\displaystyle\int_{{\rm
T}^{2}}\Big{(}|v_{0}(t,x)|^{2}+|w_{1}(t,x)|^{2}\Big{)}dx+2\int_{{\rm
T}^{2}}v_{0}(t,x)\cdot w_{1}(t,x)dx.$

From the definition of $w_{1}^{p}$, we deduce

$\displaystyle\int_{{\rm
T}^{2}}|w_{1}^{p}(t,x)|^{2}dx=\underbrace{\frac{1}{2}\int_{{\rm
T}^{2}}\sum_{j}\sum_{\xi\in\Lambda_{(j)}^{+}}a_{(\xi,j)}^{2}\eta_{(\xi)}^{2}\Big{|}W_{(\xi)}+W_{(-\xi)}\Big{|}^{2}dx}_{I}$
$\displaystyle+\underbrace{\frac{1}{2}\sum_{|j-j^{\prime}|=1}\sum_{\xi\in\Lambda_{(j)}^{+},\xi^{\prime}\in\Lambda_{(j^{\prime})}^{+}}\int_{{\rm
T}^{2}}a_{(\xi,j)}a_{(\xi^{\prime},j^{\prime})}\eta_{(\xi)}\eta_{(\xi^{\prime})}\big{(}W_{(\xi)}+W_{(-\xi)}\big{)}\cdot\big{(}W_{(\xi^{\prime})}+W_{(-\xi^{\prime})}\big{)}dx}_{II}$
$\displaystyle+\underbrace{\frac{1}{2}\sum_{j}\sum_{\xi,\xi^{\prime}\in\Lambda_{(j)}^{+},\xi\neq\xi^{\prime}}\int_{{\rm
T}^{2}}a_{(\xi,j)}a_{(\xi^{\prime},j)}\eta_{(\xi)}\eta_{(\xi^{\prime})}\big{(}W_{(\xi)}+W_{(-\xi)}\big{)}\cdot\big{(}W_{(\xi^{\prime})}+W_{(-\xi^{\prime})}\big{)}dx}_{II}.$

Recalling (5.25) and (5.18), it’s easy to deduce that

$\displaystyle I$ $\displaystyle=$
$\displaystyle\sum_{\xi\in\Lambda_{0}^{+}}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{0}}\Big{)}dx+\sum_{\xi\in\Lambda_{0}^{+}}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{0}}\Big{)}\big{(}\eta^{2}_{(\xi)}-1\big{)}dx$
$\displaystyle+\frac{1}{2}\sum_{\xi\in\Lambda_{0}^{+}}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{0}}\Big{)}\eta^{2}_{(\xi)}\Big{(}-e^{2i\lambda_{1}\xi^{\bot}\cdot
x}-e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\Big{)}dx$
$\displaystyle+\frac{1}{2}\int_{{\rm T}^{2}}\sum_{j\geq
1}\sum_{\xi\in\Lambda_{(j)}^{+}}a_{(\xi,j)}^{2}\eta_{(\xi)}^{2}\Big{|}W_{(\xi)}+W_{(-\xi)}\Big{|}^{2}dx.$

Recalling (4.14), there holds

$\displaystyle\sum_{\xi\in\Lambda_{0}^{+}}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma^{2}_{\xi}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{0}}\Big{)}dx=2\rho_{0}\int_{{\rm
T}^{2}}\chi_{0}^{2}dx.$

Set

$\displaystyle E(t)_{err}:$ $\displaystyle=$
$\displaystyle\underbrace{2\int_{{\rm T}^{2}}v_{0}(t,x)\cdot
w_{1}dx}_{(1)}+\underbrace{\int_{{\rm
T}^{2}}\Big{(}2w_{1}^{p}\cdot\big{(}w^{c}_{1}+w^{t}_{1}\big{)}+\big{|}w^{c}_{1}+w^{t}_{1}\big{|}^{2}\Big{)}dx+II}_{(2)}$
$\displaystyle+\underbrace{\sum_{\xi\in\Lambda_{0}^{+}}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{0}}\Big{)}\big{(}\eta^{2}_{(\xi)}-1\big{)}dx}_{(3)}$
$\displaystyle+\underbrace{\frac{1}{2}\sum_{\xi\in\Lambda_{0}^{+}}\int_{{\rm
T}^{2}}\rho_{0}\chi_{0}^{2}\gamma_{\xi}^{2}\Big{(}{\rm
Id}-\frac{\mathring{R}_{0}}{\rho_{0}}\Big{)}\eta^{2}_{(\xi)}\Big{(}-e^{2i\lambda_{1}\xi^{\bot}\cdot
x}-e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\Big{)}dx}_{(4)}$
$\displaystyle+\underbrace{\frac{1}{2}\int_{{\rm T}^{2}}\sum_{j\geq
1}\sum_{\xi\in\Lambda_{(j)}^{+}}a_{(\xi,j)}^{2}\eta_{(\xi)}^{2}\Big{|}W_{(\xi)}+W_{(-\xi)}\Big{|}^{2}dx}_{(5)}.$

Thus, combing the definition (5.22) of $\rho_{0}$, we obtain

$\displaystyle\int_{{\rm T}^{2}}|v_{1}(t,x)|^{2}dx$ $\displaystyle=$
$\displaystyle\int_{{\rm T}^{2}}|v_{0}(t,x)|^{2}dx+2\rho_{0}\int_{{\rm
T}^{2}}\chi_{0}^{2}dx+E(t)_{err}$ $\displaystyle=$ $\displaystyle
e(t)\Big{(}1-\frac{\delta}{2}\Big{)}+E(t)_{err}.$

Next, we will show that by choosing the parameter $\lambda_{1}$ sufficiently
large, there holds

$\displaystyle\big{|}E(t)_{err}\big{|}\leq\frac{\delta e(t)}{8},\quad\forall
t\in[0,1],$

thus, we obtain

$\displaystyle\Big{|}e(t)\Big{(}1-\frac{\delta}{2}\Big{)}-\int_{{\rm
T}^{2}}|v_{1}(t,x)|^{2}dx\Big{|}\leq\frac{\delta e(t)}{8},\quad\forall
t\in[0,1],$

which give (2.6).

Estimate on $E(t)_{err}$: We estimate $E(t)_{err}$ term by term.

Estimate on (1): By Lemma 3.1 and the $L^{p}$ estimate in Proposition 5.1, we
deduce that for every $t\in[0,1]$ and $p>1$

$\displaystyle|(1)|\leq C(v_{0})\|w_{1}(t)\|_{L^{1}}\leq
C(\mathring{R}_{0},v_{0},\delta,p)r_{1}^{1-\frac{2}{p}}.$

Estimate on (2): By Proposition 5.1, it’s direct to get

$\displaystyle\int_{{\rm T}^{2}}\big{|}w^{c}_{1}+w^{t}_{1}\big{|}^{2}dx$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta)\big{(}(r_{1}\sigma_{1})^{2}+(r_{1}\mu_{1}^{-1})^{2}\big{)},$
$\displaystyle\Big{|}\int_{{\rm
T}^{2}}w_{1}^{p}\cdot\big{(}w^{c}_{1}+w^{t}_{1}\big{)}dx\Big{|}$
$\displaystyle\leq$ $\displaystyle
C(\mathring{R}_{0},\delta)\big{(}r_{1}\sigma_{1}+r_{1}\mu_{1}^{-1}\big{)}.$

Then, by Lemma 3.2 and Proposition 5.1, we deduce that for any $t\in[0,1]$ and
$p>1$

$\displaystyle|II|$ $\displaystyle\leq
C(\mathring{R}_{0},\delta,p)\lambda_{1}^{-1}\|w_{1}\|_{L^{2p}}^{2}\leq
C(\mathring{R}_{0},\delta,p)\lambda_{1}^{-1}r_{1}^{2(1-\frac{1}{p})}.$

Summing these term, we obtain

$\displaystyle|(2)|\leq
C(\mathring{R}_{0},\delta,p)\Big{(}r_{1}\sigma_{1}+r_{1}\mu_{1}^{-1}+\lambda_{1}^{-1}r_{1}^{2(1-\frac{1}{p})}\Big{)}.$

Estimate on (3): Notice that

$\displaystyle\fint_{{\rm T}^{2}}\big{(}\eta^{2}_{(\xi)}-1\big{)}=0,$

thus,

$\displaystyle{\rm
P}_{\geq\frac{\lambda_{1}\sigma_{1}}{2}}\big{(}\eta^{2}_{(\xi)}-1\big{)}=\big{(}\eta^{2}_{(\xi)}-1\big{)}.$

Lemma 3.2, Lemma 5.1 and Lemma 5.2 implies

$\displaystyle|(3)|\leq
C(\mathring{R}_{0},\delta)\frac{1}{\lambda_{1}\sigma_{1}}.$

Estimate on (4): Notice the fact

$\displaystyle{\rm
P}_{\geq\frac{\lambda_{1}}{2}}\Big{(}\eta^{2}_{(\xi)}\Big{(}-e^{2i\lambda_{1}\xi^{\bot}\cdot
x}-e^{-2i\lambda_{1}\xi^{\bot}\cdot
x}\Big{)}\Big{)}=\eta^{2}_{(\xi)}\Big{(}-e^{2i\lambda_{1}\xi^{\bot}\cdot
x}-e^{-2i\lambda_{1}\xi^{\bot}\cdot x}\Big{)}.$

Thus, Lemma 3.2, Lemma 5.1 and Lemma 5.2 gives

$\displaystyle|(4)|\leq\frac{C(\mathring{R}_{0},\delta)}{\lambda_{1}}.$

Estimate on (5): Recalling (5.29), we obtain

$\displaystyle|(5)|\leq\frac{\delta}{20}+C(\mathring{R}_{0},\delta)\frac{1}{\lambda_{1}\sigma_{1}}\leq\frac{\delta
e(t)}{20}+C(\mathring{R}_{0},\delta)\frac{1}{\lambda_{1}\sigma_{1}}.$

Finally, collecting estimate (1)-(5), noticing the parameter relationship
(5.24), taking $p$ sufficiently close to 1 and parameter $\lambda_{1}$
sufficiently large, we arrive at

$\displaystyle E_{err}(t)\leq\frac{\delta e(t)}{8}.$

This completes the proof.

Acknowledgments. The first author is supported in part by NSFC Grants
11601258.6. The second author is supported by the fundamental research funds
of Shandong university under Grant 11140078614006. The third author is
partially supported by the Chinese NSF under Grant 11471320 and 11631008.

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