H\"{o}lder continuous weak solution of 2d Boussinesq equation with diffusive temperature

TL;DR

This paper proves the existence of Hölder continuous weak solutions to the 2D Boussinesq equations with diffusive temperature, which match prescribed kinetic energy profiles, demonstrating non-uniqueness and regularity properties.

Contribution

It constructs Hölder continuous weak solutions to the 2D Boussinesq equations with diffusive temperature that realize any smooth kinetic energy profile.

Findings

Existence of Hölder continuous weak solutions with prescribed energy.

Solutions satisfy the Boussinesq equations in the distributional sense.

Solutions exhibit specific regularity properties in space and time.

Abstract

We show the existence of H\"{o}lder continuous periodic weak solutions of the 2d Boussinesq equation with diffusive temperature which satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,1]\rightarrow R_+$ and $\varepsilon\in (0, \frac{1}{10})$, there exist $v\in C^{\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2), \theta\in C_t^{1,\frac{1}{20}-\frac{\varepsilon}{2}}C_x^{2,\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2)$ which solve boussinesq equation in the sense of distribution and satisfy e(t)=\int_{{\rm T}^2}|v(t,x)|^2dx, \quad \forall t\in [0,1].

Equations

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

$$\{\begin{array}{l}{\partial }_{t}v+v\cdot \nabla v+\nabla p=\theta e_2,\text{mbox}in{\mathrm{T}}^2\times [0,1]\\ divv=0,\\ {\partial }_t\theta +v\cdot \nabla \theta -△\theta =0,\text{mbox}in{\mathrm{T}}^2\times [0,1],\end{array}$$

$${\int }_0^1{\int }_{{\mathrm{T}}^2}({\partial }_t\phi \cdot v+\nabla \phi :v\otimes v+p\mathrm{div}\phi +\theta e_2\cdot \phi )dxdt=0$$

$${\int }_0^1{\int }_{{\mathrm{T}}^2}({\partial }_t\varphi \theta +v\cdot \nabla \varphi \theta -\nabla \theta \cdot \nabla \varphi )dxdt=0$$

$${\int }_0^1{\int }_{{\mathrm{T}}^2}v\cdot \nabla \psi dxdt=0$$

$$(v,p)\in C([0,1]\times {\mathrm{T}}^2),\theta \in C([0,1]\times {\mathrm{T}}^2)\cap L^2(0,1;H^1({\mathrm{T}}^2))$$

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

$$v\in C_{t,x}^{\frac{1}{10}-\epsilon },p\in C_{t,x}^{\frac{1}{5}-\epsilon },\theta \in C_t^{1,\frac{1}{20}-\frac{\epsilon }{2}}{C}_x^{2,\frac{1}{10}-\epsilon }.$$

$$\frac{1}{2}\Vert \theta (t,\cdot ){\Vert }_{L^2}^2+{\int }_0^t\Vert \nabla \theta (s,\cdot ){\Vert }_{L^2}^{2}ds=\frac{1}{2}\Vert {\theta }^0(\cdot ){\Vert }_{L^2}^2,\forall t\in [0,1].$$

$$\{\begin{array}{l}{\partial }_{t}v+v\cdot \nabla v+\nabla p+(-△{)}^{\alpha }v=\theta e_2,\text{mbox}in{\mathrm{T}}^2\times [0,1]\\ \mathrm{div}v=0,\\ {\partial }_t\theta +v\cdot \nabla \theta -△\theta =0,\text{mbox}in{\mathrm{T}}^2\times [0,1].\end{array}$$

$$\{\begin{array}{l}{\partial }_{t}v+v\cdot \nabla v+\nabla p=\theta e_2+\mathrm{div}\mathring{R},\\ \mathrm{div}v=0,\\ {\partial }_t\theta +v\cdot \nabla \theta -△\theta =0,\\ \theta (0,x)={\theta }^0(x),\end{array}$$

$$[f{]}_{m+\alpha }=\underset{|\beta |=m}{\max }\underset{x\ne y,t}{\sup }\frac{|{\nabla }^{\beta }f(t,x)-{\nabla }^{\beta }f(t,y)|}{|x-y{|}^{\alpha }}$$

$$[f{]}_m=\sum _{|\beta |=m}\Vert {\nabla }^{\beta }f{\Vert }_0,$$

$$\Vert v{\Vert }_0:=\underset{t,x}{\sup }|v(t,x)|,\Vert v(t,\cdot ){\Vert }_0:=\underset{x}{\sup }|v(t,\cdot )|,$$

$$\Vert f{\Vert }_m=\sum _{k\le m}[f{]}_k,\Vert f(t,\cdot ){\Vert }_m=\sum _{k\le m}[f(t,\cdot ){]}_k$$

$$\Vert f{\Vert }_{m+\alpha }=\Vert f{\Vert }_m+[f{]}_{m+\alpha }.$$

$${\delta }_q=a^{-b^q},a^{c{b}^{q+1}}\le {\lambda }_q\le 2a^{c{b}^{q+1}},$$

$$\Vert {\mathring{R}}_q{\Vert }_0\le \eta {\delta }_{q+1},\Vert {\mathring{R}}_q{\Vert }_1\le \eta {\delta }_{q+1}{\lambda }_q,$$

$$\Vert v_{q+1}-v_q{\Vert }_0\le M{\delta }_{q+1}^{\frac{1}{2}},\Vert v_{q+1}-v_q{\Vert }_m\le C(m){\delta }_{q+1}^{\frac{1}{2}}{\lambda }_{q+1}^m,m\ge 1,$$

$$\Vert p_{q+1}-p_q{\Vert }_0\le M^2{\delta }_{q+1},\Vert p_{q+1}-p_q{\Vert }_1\le M^2{\delta }_{q+1}{\lambda }_{q+1},$$

$$\frac{1}{2}\Vert {\theta }_q(t,\cdot ){\Vert }_{L^2}^2+{\int }_0^t\Vert \nabla {\theta }_q(s,\cdot ){\Vert }_{L^2}^{2}ds=\frac{1}{2}\Vert {\theta }^0(\cdot ){\Vert }_{L^2}^2,\forall t\in [0,1],$$

$$\Vert ({\theta }_{q+1}-{\theta }_q)(t,\cdot ){\Vert }_{L^2}^2+{\int }_0^t\Vert \nabla ({\theta }_{q+1}-{\theta }_q)(s,\cdot ){\Vert }_{L^2}^{2}ds\le 4M^2\Vert {\theta }^0{\Vert }_0^2{\delta }_{q+1},\forall t\in [0,1],$$

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

$$\Vert {\partial }_t(v_{q+1}-v_q){\Vert }_0\le C{\delta }_{q+1}^{\frac{1}{2}}{\lambda }_{q+1},\Vert {\partial }_t(p_{q+1}-p_q){\Vert }_0\le C{\delta }_{q+1}{\lambda }_{q+1}.$$

$$(v,p)\in C([0,1]\times {\mathrm{T}}^2),\theta \in L^{\infty }(0,1;L^2({\mathrm{T}}^2))\cap L^2(0,1;H^1({\mathrm{T}}^2))$$

$$(v_n,p_n)\to (v,p)$$

$${\theta }_n\to \theta$$

$$\mathring{R_n}\to 0$$

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

$$\Vert v_{q+1}-v_q{\Vert }_{\alpha }\le \Vert v_{q+1}-v_q{\Vert }_0^{1-\alpha }\Vert v_{q+1}-v_q{\Vert }_1^{\alpha }\le M{\delta }_{q+1}^{\frac{1}{2}}{\lambda }_{q+1}^{\alpha }\le 2M{a}^{(-\frac{1}{2}+\alpha bc)b^{q+1}},$$

$$\Vert p_{q+1}-p_q{\Vert }_{\alpha }\le \Vert p_{q+1}-p_q{\Vert }_0^{1-\alpha }\Vert p_{q+1}-p_q{\Vert }_1^{\alpha }\le M^2{\delta }_{q+1}{\lambda }_{q+1}^{\alpha }\le 2M{a}^{(-1+\alpha bc)b^{q+1}}.$$

$$\Vert \theta (t,\cdot ){\Vert }_{L^2}^2+2{\int }_0^t\Vert \nabla \theta (s,\cdot ){\Vert }_{L^2}^2=\Vert {\theta }^0(\cdot ){\Vert }_{L^2}^2.$$

$$v_{q+1}=$$

$$\{\begin{array}{l}{\partial }_t{\theta }_{q+1}+v_{q+1}\cdot \nabla {\theta }_{q+1}-△{\theta }_{q+1}=0,\\ {\theta }_{q+1}{|}_{t=0}={\theta }^0,\end{array}$$

Full text

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Keywords: 2d Boussinesq equation with diffusive temperature, Hölder continuous periodic weak solutions, Prescribed kinetic energy

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

1. Introduction

In this paper, we consider the following 2d Boussinesq equation

[TABLE]

where ${\rm T}^{2}$ denotes the 2-dimensional torus, and $e_{2}=\binom{0}{1}$. Here in our notations, $v$ is the velocity vector, $p$ is the pressure, and $\theta$ denotes the temperature or density which is a scalar function. The Boussinesq equation was introduced to model many geophysical flows, such as atmospheric fronts and ocean circulations (see, for example, [36],[43]).

The global well-posedness of strong solution has been established by many authors for the Cauchy problem of (1.1) in 2d with regularity data (see, for example, [9], [25]). For the 3-dimensional case, the global existence of smooth solution of (1.1) remains open.

Moreover, the study of weak solutions in fluid dynamics, including those which fail to conserve energy, is quite natural in the context of turbulent flow, and has been conducted by many people in the past two decades (see [44], [45, 46],[18, 19]). The triplet $(v,p,\theta)$ on $[0,1]\times{\rm T}^{2}$ is called a weak solution of (1.1) if $\theta\in L^{2}_{loc}((0,1)\times{\rm T}^{2}),~{}p\in L^{2}_{loc}((0,1)\times{\rm T}^{2}),~{}\theta\in L^{2}_{loc}((0,1);H^{1}({\rm T}^{2}))$ and solve (1.1) in the following sense:

[TABLE]

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

[TABLE]

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

[TABLE]

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

The study of constructing non-unique or dissipative weak solution to fluid systems is very fashionable in recent years. The construction is based on the convex integration method pioneered by De Lellis-Székelyhidi Jr in [18, 21], where the authors tackle the Onsager conjecture for the incompressible Euler equation, showing that the incompressible Euler equation admits Hölder continuous solutions which dissipate kinetic energy. More precisely, the Onsager conjecture on incompressible Euler equation can be stated as follows:

1. (1)

$C^{0,\alpha}$ weak solutions are energy conservative when $\alpha>\frac{1}{3}$. 2. (2)

For any $\alpha<\frac{1}{3}$, there exist dissipative solutions with $C^{0,\alpha}$ regularity .

For this conjecture, the part (a) has been proved by P. Constantin, E, Weinan and E. Titi in [15]. Slightly weaker assumptions on the solution were subsequently shown to be sufficient for energy conservation by P. Constantin, etc. in [10, 24], also see [48]. More recently, P. Isett and Sung-jin Oh gave a proof to this part of Onsager’s conjecture for the Euler equations on manifolds by the heat flow method in [26].

Part (b) was proved by P. Isett in [29], based on a series of progress on this problem in [1, 3, 4, 5, 11, 16, 17, 22, 27], see also [6] for the construction of admissible weak solutions. Moreover, the idea and method can be used to construct dissipative weak solutions for other models, see [7, 31, 34, 41, 47, 50, 51, 52].

Recently, Buckmaster and Vicol established the non-uniqueness of weak solution to the 3D incompressible Navier-Stokes in [8] by introducing some new ideas. Furthermore, the method was used to construct weak solution in [2, 32, 33, 35, 37, 38, 39].

Motivated by the study of Onsager’s conjecture of the incompressible Euler equation in earlier works and non-uniqueness of weak solutions to the Navier-Stokes equation, we consider the Boussinesq equation with diffusive temperature and want to know if the anomalous dissipation of kinetic energy can also happened when considering the temperature effects. The difference is that there are conversions between internal energy and mechanical energy, and we need to overcome the difficulty of interactions between velocity and temperature. To this end, we consider the existence of Hölder continuous periodic solutions of the 2d Boussinesq equations with diffusive temperature which satisfy the prescribed kinetic energy. Following the general scheme in the construction of Hölder continuous weak solution of the incompressible Euler equation and introducing some new ideas, we obtain the following result:

Theorem 1.1**.**

Assume that $e(t):[0,1]\rightarrow R$ is a given positive smooth function and let $\theta^{0}(x)\in X:={\rm span}\{\sin(x_{2}),\cos(x_{2})\}$. Then there exist

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such that they solve the system (1.1) in the sense of distribution and

[TABLE]

Furthermore, for any $\varepsilon\in(0,\frac{1}{10})$, there holds

[TABLE]

Moreover, $\theta$ satisfies the following identity:

[TABLE]

Remark 1.1**.**

We consider the following 2d Boussinesq-equation with fractional dissipation in velocity:

[TABLE]

For $\alpha<\frac{1}{2}$, we also can construct Hölder continuous weak solutions for (1.2) by the argument in this paper. For $\frac{1}{2}\leq\alpha<1$, we have constructed finite energy weak solutions for (1.2) in [33].

2. Proof of main result and Plan of the paper

As in [3], the proof of Theorem 1.1 will be achieved through an iteration procedure. In this and subsequent sections, $\mathcal{S}_{0}^{2\times 2}$ denotes the vector space of trace-free symmetric $2\times 2$ matrices.

Definition 2.1**.**

Assume $(v,p,\theta,\mathring{R})$ are smooth functions on $[0,1]\times{\rm T}^{2}$ taking values, respectively, in $R^{2},R,R,\mathcal{S}_{0}^{2\times 2}$. We say that they solve the Boussinesq-Reynold system if

[TABLE]

where $\theta^{0}(x)\in X$.

We now state the main proposition of this paper, by which Theorem 1.1 is implied directly. For the statement of the main proposition, we introduce some notations.

In the following, $m=0,1,\cdot\cdot\cdot$, $\alpha\in(0,1)$ and $\beta$ is a multi-index. We define the Hölder semi-norms as

[TABLE]

and semi-norms

[TABLE]

where

[TABLE]

and $\nabla^{\beta}$ are spatial derivatives. Then, define norms

[TABLE]

and Hölder norms

[TABLE]

Moreover, we introduce two parameters:

[TABLE]

where $a,b,c>1$ and $a$ is a integer.

Proposition 2.1**.**

Let $e(t),\theta^{0}(x)$ be as in Theorem 1.1. Then we can choose two positive constants $\eta$ and $M$ only dependent of $e(t)$ such that the following holds. For any small $\gamma>0$ and $b=\frac{6+\gamma}{4},c=\frac{4(5+\gamma)}{6+\gamma}$, if $a$ is sufficiently large, then there exists a sequence of functions $(v_{q},~{}p_{q},~{}\theta_{q},~{}\mathring{R}_{q})\in C^{\infty}([0,1]\times{\rm T}^{2})$ starting with $(v_{0},~{}p_{0},~{}\theta_{0},~{}\mathring{R}_{0})$, solving the Boussinesq-Stress system (2.1) and satisfying the following estimates

[TABLE]

Moreover, there holds

[TABLE]

We will prove Proposition 2.1 in the subsequent sections. Here we first give a proof of Theorem 1.1 by using this proposition.

Proof of Theorem 1.1.

From (2.2)-(2.6), we know that $(v_{n},~{}p_{n},~{}\mathring{R}_{n})$ are Cauchy sequences in $C([0,1]\times{\rm T}^{2})$, and $\theta_{n}$ is a Cauchy sequence in $L^{\infty}(0,1;L^{2}({\rm T}^{2}))\cap L^{2}(0,1;H^{1}({\rm T}^{2}))$, therefore there exist

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such that

[TABLE]

as $n\rightarrow\infty$.

Passing into limit in (2.1), we conclude that $(v,p,\theta)$ solve (1.1) in the sense of distribution. Moreover, (2.7) implies

[TABLE]

From (2.3), (2.4) and interpolation, we conclude

[TABLE]

Thus, for any small $\gamma>0$ and every $\alpha<\frac{1}{2bc}=\frac{1}{2(5+\gamma)}$, $v_{q}$ is a Cauchy sequence in $C_{t,x}^{\alpha}$ and $p_{q}$ is a Cauchy sequence in $C_{t,x}^{2\alpha}$. Hence, for every $\alpha<\frac{1}{10}$, there eixts $(v,p,\theta)$ with $v\in C^{\alpha}_{t,x},~{}p\in C^{2\alpha}_{t,x}$ and they solve the Boussinesq equation. By the Schauder estimate of linear parabolic equations, we deduce that $\theta\in C^{1,\frac{\alpha}{2}}_{t}C^{2,\alpha}_{x}$.

Furthermore, by (2.5), we deduce that the temperature $\theta$ satisfies the energy equality: for every $t\in[0,1]$

[TABLE]

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

2.1. Outline of the proof of Propositions 2.1

The rest of this paper will be dedicated to prove Proposition 2.1. We perform a inductive procedure, and construct $v_{q+1}$ from $v_{q}$ by adding some perturbations as follows:

[TABLE]

where $w_{o},w_{c}$ are smooth functions given by explicit formulas which depend on $(e(t),v_{q},\mathring{R}_{q})$. After the construction of the new velocity $v_{q+1}$, we construct the new temperature $\theta_{q+1}$ by solving the following transport-diffusion equation: there exists a $\theta_{q+1}\in C^{\infty}([0,1]\times{\rm T}^{2},R)$ which solves

[TABLE]

where $\theta^{0}$ is the function appeared in Proposition 2.1. After the construction of $v_{q+1},\theta_{q+1}$, we mainly focus on finding functions $\mathring{R}_{q+1},p_{q+1}$ with the desired estimates and solving the system (2.1).

The rest of paper is organized as follows. In Section 3, we do some preliminaries. We introduce the Geometric Lemma in [21], stationary solutions of 2d Euler equation, anti-divergence operator and estimates of transport-diffusion equation with highly oscillatory forces. In Section 4, we first define the new velocity $v_{q+1}$ by constructing velocity perturbations $w_{o},w_{c}$, and then construct the new temperature $\theta_{q+1}$ by solving the transport-diffusion equation. In Section 5 and Section 6, we establish various estimates for the perturbation. Finally, in Section 7, we give a proof of Proposition 2.1 by using the estimate which was established in Section 5 and 6.

3. Some preliminaries

We first recall the following stationary solution for the 2d Euler equation which is the building block in our iterative scheme.

3.1. Stationary flows in 2D

Proposition 3.1**.**

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

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is real-valued and satisfies

[TABLE]

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

[TABLE]

The proof of this proposition can be found in [11], and for completeness, we give a direct proof here. We first prove the following lemma.

Lemma 3.2**.**

Let $\vec{f}_{1}(x)=\binom{-b}{a}e^{i(a,b)\cdot x},\quad\vec{f}_{2}(x)=\binom{-d}{c}e^{i(c,d)\cdot x}$ with $a^{2}+b^{2}=c^{2}+d^{2}$. There holds

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Here and below, we denote

[TABLE]

Proof.

A computation gives

[TABLE]

where we used $a^{2}+b^{2}=c^{2}+d^{2}$ in the penultimate line. MM $\square$

Proof of Proposition 3.1

Proof.

We only prove the first identity in (3.1). The others are obvious. It’s direct to obtain

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Thus, using the above lemma, we obtain

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[TABLE]

and hence

[TABLE]

MM $\square$

3.2. Geometric Lemma

Let

[TABLE]

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

[TABLE]

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

[TABLE]

Moreover, notice that $\mathcal{S}^{2\times 2}$($2\times 2$ symmetric matric) is a 3-dimensional linear space, thus by the above representation and uniqueness, we know that such representation holds for $2\times 2$ symmetric matrices near $\mathrm{Id}$.

Lemma 3.3** (Geometric Lemma).**

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

[TABLE]

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

[TABLE]

Remark 3.4**.**

By rotational symmetry, Geometric Lemma 3.3 also holds for $k\in\Lambda_{1}^{+}$. It is convenient to introduce a small geometric constant $c_{0}\in(0,1)$ such that

[TABLE]

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

3.3. Anti-divergence operator

We recall the anti-divergence operator in this subsection.

Lemma 3.5**.**

(Anti-divergence operator) 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: for any $a\in C^{\infty}({\rm T}^{2})$ and any $m\in{\rm N}_{+},~{}\alpha\in(0,1)$, there holds

[TABLE]

Proof.

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

[TABLE]

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

[TABLE]

Then $\mathcal{R}$ satisfies the above property. The detail can be found in [11], and we omit it here. MM $\square$

Moreover, $\mathcal{R}$ satisfies the following property:

Lemma 3.6**.**

For any $s>0$, there holds

[TABLE]

Here and below,

[TABLE]

Proof.

In fact, we can give a explicit formula for $\mathcal{R}$ by the Fourier series expansion. Let

[TABLE]

where $v_{k}\in{\rm C}^{2}$ is the Fourier coefficient of the vector function $v(x)$. Then

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Thus, due to $u\in C_{0}^{\infty}({\rm T}^{2})$ and $\triangle u=v$, we deduce

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Thus, there hold

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Summing them together, we obtain

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hence for any $s>0$, there hold

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where we used

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is convergent for any $s>0$. In fact,

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MM $\square$

3.4. Transport-diffusion equation with oscillatory force

We consider the transport-diffusion equation with oscillatory force

[TABLE]

where $k\in S^{1}\cap Q^{2}$ is a vector and $\lambda k\in Z^{2},\lambda\neq 0$.

To simplify the formulas, we introduce the following notation: for any $n\in{\rm N}_{+}$,

[TABLE]

where $\|\cdot\|_{X}$ is a norm.

Lemma 3.7**.**

Let $\theta(t,x)$ be a solution of (3.16). Then there hold

[TABLE]

Proof.

Estimate on $\|\theta\|_{L^{\infty}L^{2}}$: A direct energy estimate gives

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By integration by parts, for any integer $M$, there holds

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thus, we deduce that

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Finally, using interpolation inequality and Young inequality, we know

[TABLE]

which gives the estimate on $\|\theta\|_{L^{\infty}L^{2}_{x}}$ in (3.7).

Estimate on $\|\theta\|_{L^{\infty}\dot{H}^{N}}$($N\geq 1$): For $N\geq 1$, acting $\nabla^{N}$ on both sides of (3.16), taking $L^{2}$ inner products with $\nabla^{N}\theta$ and integrating by parts, we obtain

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Using interpolation inequality

[TABLE]

and Young inequality, we deduce that

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Moreover, it’s easy to obtain

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Putting these estimates into (3.4), we get

[TABLE]

thus, we deduce that

[TABLE]

this completes the proof of this lemma. MM $\square$

Remark 3.8**.**

Similarly, we consider

[TABLE]

where $k,\lambda$ are as above. Then, the following fact still holds: Let $\theta(t,x)$ be a solution of (3.22), then the estimate (3.7) is valid for $\theta$.

Generally, if we consider the transport-diffusion equation

[TABLE]

where $a_{k}(t,x)$ complex-valued functions with $\bar{a}_{k}(t,x)=a_{-k}(t,x)$. Then, there exists a unique solution $\theta(t,x)$ for (3.26) and it satisfies the following estimate

[TABLE]

4. Construction of $(v_{q+1},p_{q+1},\theta_{q+1},\mathring{R}_{q+1})$

In this section, we perform the inductive procedure which allows us to construct $(v_{q+1},p_{q+1},\theta_{q+1},\\ \mathring{R}_{q+1})$ from $(v_{q},p_{q},\theta_{q},\mathring{R}_{q})$. Recalling the choice of the sequence $\{\delta_{q}\}_{q\in N}$ and $\{\lambda_{q}\}_{q\in N}$, for sufficiently large $a$, we have, for any $q\geq 1$,

[TABLE]

As in [3], we write $(v,p,\theta,\mathring{R})$ instead of $(v_{q},p_{q},\theta_{q},\mathring{R}_{q})$ and $(v_{1},p_{1},\theta_{1},\mathring{R}_{1})$ instead of $(v_{q+1},p_{q+1},\theta_{q+1},\\ \mathring{R}_{q+1})$. Thus, the following estimates hold:

[TABLE]

4.1. Space-time regularization of $v,\mathring{R}$

Let $\psi\in C_{c}^{\infty}(R^{3}\times R)$ be a radial symmetry nonnegative function supported in $[-1,1]^{4}$ and $\ell$ be a small parameter. Set

[TABLE]

Standard estimates on convolutions give

[TABLE]

and for any $N\geq 1$, there exists a constant $C=C(N)$ such that

[TABLE]

4.2. Partition of unity on time

Fix a smooth function $\chi\in C_{c}^{\infty}((-\frac{3}{4},\frac{3}{4}))$ such that

[TABLE]

and a large parameter $\mu\in{\rm N}_{+}$ which will be determined later.

For any $l\in[0,\mu]$, set

[TABLE]

Due to (2.7), we deduce that there exists a universal constant $C_{0}$ such that

[TABLE]

4.3. Construction of velocity perturbation $w$

Firstly, for any integer $l\in[0,\mu]$, we construct a smooth functions $\Phi_{l}(t,x):{\rm T}^{2}\times[0,1]\rightarrow{\rm T}^{2}$ by solving the following transport equations:

[TABLE]

Next, for any $k\in\Lambda_{0}\cup\Lambda_{1}$ and any integer $l\in[0,\mu]$, we set

[TABLE]

Here and throughout the paper, $R_{\ell,l}(t,x):=\rho_{l}{\rm Id}-\mathring{R}_{\ell}(t,x)$.

Due to (4.2) and (4.6), we deduce that there exists an $\eta=\eta(e,r_{0}):=C_{0}^{-1}r_{0}\min_{t\in[0,1]}e(t)$ such that

[TABLE]

Hence $a_{kl}$ in (4.3) is well-defined.

We define the principle part $w_{o}$ of the velocity perturbation $w$

[TABLE]

where

[TABLE]

Moreover, as in [3], we set

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Then

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Then, set the incompressibility corrector $w_{c}$ as

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Here and throughout the paper, $\widetilde{{\rm Id}}$ denotes

[TABLE]

Finally, we set

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Obviously,

[TABLE]

hence ${\rm div}w(t,x)=0.$

Set

[TABLE]

thus the perturbation $w$ can also be written as

[TABLE]

After the construction of the perturbation $w$, we choose the constant $M$. By (4.6) and the support property of $\chi_{l}(t)$, it’s direct to obtain

[TABLE]

Then, we set

[TABLE]

thus, there holds

[TABLE]

4.4. Construction of new velocity $v_{1}$ and new temperature $\theta_{1}$

After the construction of the velocity perturbation $w$, we define the new velocity as follows:

[TABLE]

Thus there holds ${\rm div}v_{1}(t,x)=0$. Then, we define the new temperature by solving the following transport-diffusion equation

[TABLE]

By the maximum principle, we deduce that

[TABLE]

Moreover, the basic energy estimate gives

[TABLE]

Finally, it’s obvious that

[TABLE]

4.5. The new pressure $p_{1}$ and new Reynolds stress $\mathring{R}_{1}$

We first compute ${\rm div}(w_{o}\otimes w_{o})$. Recalling (4.11), we deduce

[TABLE]

where we used the following fact

[TABLE]

Here we used Geometric Lemma 3.3 in the last step.

Furthermore,

[TABLE]

thus, by Proposition 3.1, we deduce that

[TABLE]

where

[TABLE]

Moreover, set

[TABLE]

From (4.13) and the fact $f_{klk^{\prime}l^{\prime}}=f_{k^{\prime}l^{\prime}kl}$, we deduce that

[TABLE]

Thus, by Proposition 3.1, there hold

[TABLE]

where

[TABLE]

Set

[TABLE]

thus, by (4.5)-(4.5), we deduce

[TABLE]

After the computation of ${\rm div}(w_{o}\otimes w_{o})$, we set

[TABLE]

Obviously, there holds ${\rm tr}\mathring{R}_{1}=0$. Finally, put

[TABLE]

A direct computation gives

[TABLE]

In fact,

[TABLE]

where we used

[TABLE]

Thus, the new functions $(v_{1},p_{1},\theta_{1},\mathring{R}_{1})$ solves the Boussinesq-Reynold (2.1) system.

5. Estimate on the perturbation

5.1. Some elementary inequalities

Recall the following elementary inequalities:

[TABLE]

5.2. Condition on the parameter

To simplify the computation, as in [3], we assume the following conditions on $\mu,\ell\leq 1,\lambda_{q+1}\geq 1$:

[TABLE]

where $\beta>0$ is a number which will be determined later. These conditions imply

[TABLE]

5.3. Estimate on velocity perturbation

In this subsection, we collect some estimates on the velocity perturbation.

Lemma 5.1**.**

Assume (5.2) holds. For any $l\in Z$ and $t$ in the range $|\mu t-l|<1$, we have

[TABLE]

Moreover, there hold

[TABLE]

Consequently, there hold

[TABLE]

Proof.

Estimate on $\Phi_{l}$: The first estimate in (5.1) can be directly obtained by using (A.5), the second and third estimates in (5.1) can be directly obtained by using (A.7).

Estimates on $a_{kl}$: Firstly, by (4.6), it’s easy to obtain

[TABLE]

Recalling (4.2), (4.4), (4.6), (4.3), (5.1) and (5.3), it’s easy to obtain

[TABLE]

for any $N\geq 1$.

Estimates on $L_{kl}$: Recalling (4.19), by (5.1) and (5.2), we deduce that

[TABLE]

Notice that

[TABLE]

Thus, we obtain

[TABLE]

Moreover,

[TABLE]

thus by (5.1) and the parameter assumption (5.3), we deduce that

[TABLE]

Hence by (4.1), (5.1) and (5.2), we deduce that

[TABLE]

Estimate on $\phi_{kl}$: By (5.1) and (5.1), we obtain

[TABLE]

Furthermore, by (5.2), we arrive at

[TABLE]

Estimate on $w_{o},w_{c}$: Recalling (4.13), by (5.1), we deduce that

[TABLE]

Recalling (4.3) and (5.2), we know that

[TABLE]

MM $\square$

5.4. Estimate on temperature perturbation

Next, we estimate the difference of temperature $\theta_{1}-\theta$. A direct computation gives that

[TABLE]

Taking the $L^{2}$ inner product with $\theta_{1}-\theta$ and integrating by parts, we arrive at

[TABLE]

Thus, by (4.25), there holds

[TABLE]

Moreover,

[TABLE]

By (3.8), to establish decay estimate for $\|\theta_{1}-\theta\|_{L^{\infty}L^{2}}$, we need the estimates $\|\nabla^{N}\theta\|_{L^{\infty}L^{2}_{x}}$. To this end, we establish the following lemma.

Lemma 5.1**.**

Let $\theta$ be a smooth solution of the following transport-diffusion equation

[TABLE]

Then, there holds

[TABLE]

Proof.

Step 1: Acting $\partial_{j}$ on both sides of (5.14), we obtain

[TABLE]

Taking $L^{2}$ inner products with $\partial_{j}\theta$ and integrating by parts, we deduce that

[TABLE]

Summing about $j=1,2$, we arrive at

[TABLE]

Moreover, by the interpolation inequality $\|\nabla\theta(t,\cdot)\|_{L^{2}}\leq\|\theta(t,\cdot)\|^{\frac{1}{2}}_{L^{2}}\|\nabla^{2}\theta(t,\cdot)\|^{\frac{1}{2}}_{L^{2}}$, we arrive at

[TABLE]

Thus, by the Hölder inequality and noticing the fact $\|\theta(t,\cdot)\|_{L^{2}}\leq\|\theta^{0}\|_{2}$, we obtain

[TABLE]

which gives the first estimate in (5.1).

Step 2: For $N\geq 2$, acting $\nabla^{N}$ on both sides of (5.16), taking $L^{2}$ inner products with $\nabla^{N}\theta$ and integrating by parts, we obtain

[TABLE]

Integrating by parts and using interpolation inequality

[TABLE]

we deduce that

[TABLE]

Thus, by Young inequality, we obtain

[TABLE]

Combining the estimate (5.17), we obtain

[TABLE]

which is the second estimate in (5.1). MM $\square$

Corollary 5.2**.**

Under the assumption (4.1), for any $N\geq 2$, there holds

[TABLE]

Proof.

By (4.1) and (5.1), we deduce that

[TABLE]

MM $\square$

Proposition 5.3**.**

There holds

[TABLE]

Furthermore, there holds

[TABLE]

Proof.

Estimate on $\|\theta_{1}-\theta\|_{L^{\infty}L^{2}_{x}}$: Recalling $b_{kl}=\chi_{l}(t)\phi_{kl}L_{kl}\cdot\nabla\theta$, thus by (4.1), (5.1), (5.1), (5.18) and the parameter assumption (5.2), we deduce that for any $l\in Z$ and $k\in\Lambda_{0}\cup\Lambda_{1}$, there holds

[TABLE]

Thus, by (3.8) and the parameter assumption (5.2), we obtain

[TABLE]

Taking $N$ large enough such that $N\beta\geq 1$, we arrive at

[TABLE]

Estimate on $\|\nabla^{N}(\theta_{1}-\theta)\|_{L^{\infty}L^{2}_{x}}$($N\geq 1$): Recalling (4.1), we deduce that

[TABLE]

By (5.4) and the parameter assumption (5.2), we deduce that

[TABLE]

Thus, by (3.8) and the parameter assumption (5.2), we obtain

[TABLE]

this completes the proof. MM $\square$

6. Estimate on the Reynold stress

In this section, we prove the following estimate for the Reynold stress.

Proposition 6.1**.**

The Reynold stress $\mathring{R}_{1}$ defined in (4.5) satisfies the estimate

[TABLE]

Proof.

We deal with it term by term.

Estimate on $\mathcal{R}^{0}$: A direct computation gives

[TABLE]

Set

[TABLE]

thus, by $|\chi_{l}^{\prime}(t)|\leq C\mu$ and (5.1), we deduce that for any $l\in Z$ and $k\in\Lambda_{0}\cup\Lambda_{1}$, $|\mu t-l|<1$, there hold

[TABLE]

By (3.11), we deduce that

[TABLE]

Combining the parameter assumption (5.2) and taking $N$ large enough such that $N\beta>1$, we arrive at

[TABLE]

The same argument gives

[TABLE]

Estimate on $\mathcal{R}^{1}$: A direct computation gives

[TABLE]

Set

[TABLE]

thus, (5.1) and (5.1) implies that for any $l\in Z$ and $k\in\Lambda_{0}\cup\Lambda_{1}$, $|\mu t-l|<1$,

[TABLE]

Combining the parameter assumption (5.2), we obtain

[TABLE]

By taking $N$ sufficiently large, a similar argument as for $\mathcal{R}^{0}$ gives

[TABLE]

Estimate on $\mathcal{R}^{2}$: By Lemma 3.5 and (5.19), we deduce that

[TABLE]

Estimate on $\mathcal{R}^{3}$: Recalling (4.35), we know

[TABLE]

We first deal with $T^{1}_{osc}$. By (5.1) and the parameter assumption (5.3), we deduce that for any $l\in Z$ and $k\in\Lambda_{0}\cup\Lambda_{1}$, $|\mu t-l|<1$, there holds

[TABLE]

Recalling (4.5), by (3.11), choosing $N$ sufficiently large, we deduce that

[TABLE]

Similarly, we can obtain

[TABLE]

Next, we deal with the second term $T^{2}_{osc}$. From (4.31) and (5.1), we deduce that

[TABLE]

Thus, the same argument as above gives that

[TABLE]

Summing the two parts, we obtain

[TABLE]

Estimate on $\mathcal{R}^{4}$: By (5.1), we deduce

[TABLE]

Estimate on $\mathcal{R}^{5}$: By (5.1), (4.4), (4.3) and the parameter assumption (5.3), we deduce

[TABLE]

Estimate on $\mathcal{R}^{6}$: By (4.3), we deduce

[TABLE]

Finally, summing estimates (6.2)-(6), we obtain (6.1). MM $\square$

7. Proof of Proposition 2.1

Proof.

Step 1: Choice of the parameters $\mu,\ell$. Recalling that the sequence $\{\delta_{q}\}_{q\in{\rm N}}$ and $\{\lambda_{q}\}_{q\in{\rm N}}$ are chosen to satisfy

[TABLE]

where $b=\frac{6+\gamma}{4},c=\frac{4(5+\gamma)}{6+\gamma}$ and $a>1$ which will be determined later. Take

[TABLE]

We check that $\mu,\ell,\delta_{q},\lambda_{q}$ satisfies (5.2). Firstly, by (7.1), it’s easy to obtain

[TABLE]

Taking $a$ large enough, a direct computation gives that

[TABLE]

where we fix $\beta=\frac{1}{8}$. This justifies the parameter assumption (5.2).

Step 2: Proof of (2.2)-(2.6). Firstly, (6.1) and (7.1) imply

[TABLE]

A direct computation gives

[TABLE]

where we take $a$ sufficiently large and $\varepsilon=\varepsilon(\gamma)$ sufficiently small. This gives (2.2).

By (5.1), we deduce

[TABLE]

Noticing $\lambda_{q}\geq\lambda_{1}\geq a^{cb^{2}}$, thus we obtain (2.3) by taking $a$ sufficiently large.

Recalling (4.5), by (5.1) and the support property of $\chi$, we obtain

[TABLE]

Thus, from (4.37), (5.1) and (7.1), we deduce

[TABLE]

this give (2.4).

Moreover, (4.26) implies (2.5). Using the above estimate on $\|v_{q+1}-v_{q}\|_{0}$, we easily get (2.6) from (5.10).

Step 3: Proof of (2.8). Recall that $v_{q+1}=v_{q}+w$ and

[TABLE]

From (5.1), (5.1) and the parameter assumption (5.2), we deduce that

[TABLE]

hence

[TABLE]

Similarly, there holds

[TABLE]

this completes the proof of (2.8).

Step 4: Proof of (2.7). Finally, to complete the proof of this proposition, we only need to justify (2.7). A direct computation gives

[TABLE]

We first deal with $Err_{2}$. From (5.1), it’s easy to obtain

[TABLE]

Moreover,

[TABLE]

Thus, by (5.1) and (B.8), we deduce that

[TABLE]

Combining the two parts, we obtain

[TABLE]

For $Err_{1}$, we first compute $\int_{{\rm T}^{2}}|w_{o}|^{2}dx$. From (4.5), (4.5) and (4.5), we deduce that

[TABLE]

Recalling (4.5), we obtain

[TABLE]

It’s easy to obtain that

[TABLE]

As in [3], using the equation (2.1), we can deduce

[TABLE]

thus, by (4.1) and (4.2), for any $l\in Z$ and $t$ in the range $|\mu t-l|<1$, there hold

[TABLE]

Moreover, by (6) and (B.8), we deduce

[TABLE]

Thus, collecting these estimate together, we obtain

[TABLE]

Combining (7.2) and (7.3), we deduce that

[TABLE]

Recalling (7.1), we obtain

[TABLE]

which implies (2.7) by taking $a$ sufficiently large.

Step 5: Choice of the $(v_{0},p_{0},\theta_{0},\mathring{R}_{0})$. Set

[TABLE]

where $\theta^{0}\in X$ as Theorem 1.1.

It’s direct to justify that $(v_{0},p_{0},\theta_{0},\mathring{R}_{0})$ satisfies the Boussinesq-Reynold system (2.1) and

[TABLE]

Moreover, by taking $\lambda_{0}$ sufficiently large, we obtain

[TABLE]

Finally, staring with $(v_{0},p_{0},\theta_{0},\mathring{R}_{0})$, we can construct sequence $(v_{q},p_{q},\theta_{q},\mathring{R}_{q})$ inductively which solve Boussinesq-Reynold system (2.1) and satisfy (2.2)-(2.8). Thus, we complete the proof of Proposition 2.1.

MM $\square$

Appendix A Estimate for transport equation

In this section we give some well known estimates for the smooth solution of transport equation:

[TABLE]

where $v=v(t,x)$ is a given smooth vector field. The proof can be found in [3].

Proposition A.1**.**

Assume $|t|\|v\|_{1}\leq 1$. Then any solution of (A.3) satisfies the following estimates:

[TABLE]

for all $0<\alpha\leq 1$. Generally, for any $N\geq 1$ and $0\leq\alpha\leq 1$, there hold

[TABLE]

Let $\Phi(t,\cdot)$ be the inverse of the flux $X$ of $v$ staring at time $t_{0}$ as the identity. Under the above assumption $|t-t_{0}|\|v\|_{1}\leq 1$ we have

[TABLE]

Appendix B Stationary phase estimate

In the section, we recall the following simple fact, and the proof can also be found in [3].

Lemma B.1**.**

Let $k\in Z^{2},k\neq 0$ and $\lambda\geq 1$ be given. For any $a\in C^{\infty}({\rm T}^{2})$ and $m\in{\rm N}$, there hold

[TABLE]

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.