The Non-Uniqueness of Weak Solutions of the Boussinesq System in the Periodic Domain $\mathbb{T}$³

Abstract

In this article, we prove the non-uniqueness of very weak solutions (or wild solutions) of the incompressible Boussinesq system in the three-dimensional periodic domain ${\mathbb{T}}^3$ for any positive viscosity $\nu >0$ and non-negative heat conductivity $\mu \ge 0$. The proof is based on a convex integration scheme, with building blocks of intermittent Mikado flows to tackle the temperature equation, combined with a glued step to decrease the regularity of the spatial variable.

1. Introduction

The Boussinesq equation was introduced to model the interaction between thermodynamics and fluid dynamics. In this paper, we investigate the non-uniqueness of weak solutions of the incompressible Boussinesq system

$$\left\{\begin{array}{c}{\partial }_{t}u-\nu \Delta u+u·\nabla u+\nabla p=-\theta e_3,\hfill \\ {\partial }_t\theta +u·\nabla \theta -\mu \Delta \theta =0,\hfill \\ \mathrm{div}u=0,\hfill \end{array}\right.$$

(1)

in the periodic domain ${\mathbb{T}}^3={\mathbb{R}}^3/2\pi {\mathbb{Z}}^3$. Here, $u:[0,T]\times {\mathbb{T}}^3\to {\mathbb{R}}^3$ denotes the unknown divergence-free velocity field, $p:[0,T]\times {\mathbb{T}}^3\to \mathbb{R}$ is the scalar pressure, and $\theta :[0,T]\times {\mathbb{T}}^3\to \mathbb{R}$ represents the scalar temperature. The parameter $\mu \ge 0$ represents the heat conductivity, while the parameter $\nu$ represents the fluid’s viscosity. The term $\theta e_3$ models the effect of the buoyancy force.

It is well known that the global well-posedness of smooth solutions to the Cauchy problem for the 3D Boussinesq equations remains an open question. In recent research, Feng [1] constructed nontrivial weak solutions in dimensions $D\ge 4$ using a convex integration scheme with intermittent Mikado flows as building blocks, showing the non-uniqueness of weak solutions in a very weak sense.

The well-posedness of weak solutions in fluid dynamic systems has attracted the attention of many scholars. Leray first established the existence of weak solutions in $L_t^{\infty }{L}_x^2\cap L_t^2{H}_x^1$, but the regularity and stability of the weak solutions remained undetermined. In recent years, Buckmaster established the non-uniqueness of weak solutions in $L_t^{\infty }{H}_x^{\beta }$ for the Navier–Stokes equation, although not Leray–Hopf weak solutions. Since then, the method of convex integration schemes has been widely used to ascertain the non-uniqueness of weak solutions in fluid dynamic systems with viscosity; see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and the references therein. Convex integration schemes commonly use building blocks such as Beltrami flows, jet flows, shear flows and Mikado flows; see, for example, [2,4,5,6,7,8,9,14]. Beltrami flows are primarily used to construct the non-unique weak solutions in $L_t^{\infty }{H}_x^{\beta }$, for sufficiently small $\beta >0$, of the Navier–Stokes equations, where the weak solutions have a reasonable regularity that almost fulfills the conditions of the Millennium Problem. Mikado flows are often used to construct the non-unique weak solutions of transport equations; see, for example, Cheskidov [8].

The intermittency dimension is another technique to perform high-frequency perturbation in the construction of weak solutions; intermittent flow is used to control the convergence of Reynolds stress in this article. It was first introduced by Buckmaster and Vicol [7] using the Dirichlet kernel in their groundbreaking work on the non-uniqueness of weak solutions of the 3D Navier–Stokes equations.

In the context of thermal equations, if one uses Beltrami flows or jet flows, the drawback is that the nonlinear term leaves a coupling term that must be absorbed by the new pressure term $\nabla p$, which is missing in the thermal equations. The intermittency dimension of shear flows is not sufficient. Therefore, in a previous paper [1], we used Mikado flows as building blocks to establish the non-uniqueness of weak solutions; the lack of one-dimensional intermittency led us to a result of non-uniqueness in dimension $d\ge 4$, leaving $d=3$ unknown.

Inspired by the study of the Navier–Stokes equations by Buckmaster [5] and Cheskidov [9], we establish the non-uniqueness of very weak solutions of the Boussinesq system on ${\mathbb{T}}^3$ using a convex integration scheme with a glued step and temporal intermittency. The Serrin criterion states that if the weak solution $u\in L_t^p{L}_x^q$ of the Navier–Stokes equation satisfies the condition

$${\displaystyle \frac{2}{p}}+{\displaystyle \frac{3}{q}}\le 1,p<\infty ,q>3,$$

then the Leray–Hopf solution with the same initial data is unique. The established non-uniqueness of the weak solution in this paper is sharp to some extent, although the result may or may not be a Leray–Hopf solution. The glued step is introduced to ensure that the constructed weak solution has reasonable regularity in dimension 3 through the regularity of the temporal variable.

Now, we give the definition of the very weak solutions of (1).

Definition 1.

(Very weak solution.) Given any divergence-free initial data $u_0\in L^2$ and ${\theta }_0\in L^2$, we can say that $(u,\theta )\in L_{t,x}^2$ is a very weak solution of (1) with initial data $u_0,{\theta }_0$ if the following statements are true:

(1). 

u is weakly divergence free, i.e., for any test function $\phi \in C_c^{\infty }\left({\mathbb{T}}^3\right)$,

$${\int }_{{\mathbb{T}}^3}u(t,x)·\nabla \phi (x)\mathrm{d}x=0,\forall t\in [0,T];$$

(2). 

For any divergence-free test function $v\in C_c^{\infty }\left([0,T)\times {\mathbb{T}}^3\right)$,

$${\int }_{{\mathbb{T}}^3}{u}_0(x)v(0,x)\mathrm{d}x+{\int }_0^T{\int }_{{\mathbb{T}}^3}u·\left({\partial }_{t}v+\nu \Delta v+(u·\nabla )v\right)\mathrm{d}x\mathrm{d}t={\int }_0^T{\int }_{{\mathbb{T}}^3}v·\theta e_3\mathrm{d}x\mathrm{d}t;$$

(3). 

For any test function $\vartheta \in C_c^{\infty }\left([0,T)\times {\mathbb{T}}^3\right)$,

$${\int }_{{\mathbb{T}}^3}\theta (0,x)\vartheta (x)\mathrm{d}x+{\int }_0^T{\int }_{{\mathbb{T}}^3}\theta \left({\partial }_t\vartheta +u·\nabla \vartheta +\mu \Delta \vartheta \right)\mathrm{d}x\mathrm{d}t=0.$$

The pressure p can be uniquely determined up to a constant, i.e., p has a zero mean on ${\mathbb{T}}^3$, through

$$\Delta p=-\mathrm{div}\mathrm{div}(u\otimes u)-{\partial }_3\theta .$$

(2)

In this paper, we establish the non-uniqueness of weak solutions of the 3D Boussinesq Equation (1) with viscosity $\nu >0$ and thermal conductivity $\mu \ge 0$.

Theorem 1.

Suppose that $\nu >0$, $\mu \ge 0$, $0<T_1<T_2<T$, and $1\le s<2$. Let $\left(u^{(1)},{\theta }^{(1)}\right)$ and $\left(u^{(2)},{\theta }^{(2)}\right)$ be two classical solutions in $C^{\infty }\left([0,T]\times {\mathbb{T}}^3\right)$ of Equation (1), where $u^{(1)}$ and $u^{(2)}$ are divergence free, and suppose that

$${\int }_{{\mathbb{T}}^3}{\theta }^{(1)}-{\theta }^{(2)}\mathrm{d}x=0.$$

(3)

Then, for any $\epsilon >0$, there exists a weak solution $\left(u,\theta \right)\in L_{t,x}^2\cap L_t^s{L}_x^{\infty }$ to the Cauchy problem for (1) on $[0,T]$ according to Definition 1. Moreover, $\left(u,\theta \right)$ coincides with $(u^{(1)},{\theta }^{(1)})$ on $[0,T_1]$, coincides with $(u^{(2)},{\theta }^{(2)})$ on $[T_2,T]$, and is smooth except on a singular set in space–time with a Hausdorff dimension less than ε.

The condition (3) ensures that the inverse divergence operator $\mathcal{R}({\theta }^{(1)}-{\theta }^{(2)})$ is well defined and that it is also a necessary compatibility condition in incompressible flows. In particular, we obtain the following corollary:

Corollar 1.

Let $0<T_1<T_2<T$ and $1\le s<2$. If there exist two smooth velocity fields $u^{(1)},u^{(2)}\in C^{\infty }\left([0,T]\times {\mathbb{T}}^3\right)$ such that the smooth scalar fields ${\rho }^{(1)},{\rho }^{(2)}\in C^{\infty }\left([0,T]\times {\mathbb{T}}^3\right)$ satisfy the transport equation with velocity $u^{(1)}$ and $u^{(2)}$, respectively, and

$${\int }_{{\mathbb{T}}^3}{\rho }^{(1)}\mathrm{d}x\equiv {\int }_{{\mathbb{T}}^3}{\rho }^{(2)}\mathrm{d}x,$$

then there exists $\left(u,\rho \right)\in L_{t,x}^2\cap L_t^s{L}_x^{\infty }$ on $[0,T]$ that connects these two solutions, where ρ satisfies the transport equation with velocity u and fulfills $\rho ={\rho }^{(1)}$ on $t\in [0,T_1]$ and $\rho ={\rho }^{(2)}$ on $t\in [T_2,T]$.

2. Elementary Tools

We introduce the Reynolds stress tensors $R^u$ and $R^{\theta }$ associated with the velocity and temperature fields. Suppose that the smooth functions $u,\theta ,R^u,R^{\theta }$, and p satisfy the following system:

$$\left\{\begin{array}{c}{\partial }_{t}u+u·\nabla u+\nabla p-\nu \Delta u=-\theta e_3+\mathrm{div}{R}^u,\hfill \\ {\partial }_t\theta +u·\nabla \theta -\mu \Delta \theta =\mathrm{div}{R}^{\theta },\hfill \\ \mathrm{div}u=0,\hfill \end{array}\right.$$

(4)

where $\nu >0$, $\mu \ge 0$, $R^u$ is a traceless symmetric tensor, and $R^{\theta }$ is a vector. In the equation above, the pressure p can be uniquely determined by

$$\Delta p=\mathrm{div}\mathrm{div}{R}^u-\mathrm{div}\mathrm{div}(u\otimes u)-{\partial }_3\theta .$$

(5)

The proof of Theorem 1 consists mainly of two steps. The first step is to construct a “glued solution”, denoted as $\overline{u}$ and $\overline{\theta }$, that satisfies system (4) with suitable Reynolds terms $\overline{R^u}$ and $\overline{R^{\theta }}$. Then, we use the method of convex integration with Mikado flows as building blocks.

2.1. Geometric Lemma

The geometric lemma is used to eliminate the main part of Reynolds stress; the proofs can be found in [2,4,5,10].

Lemma 1.

There exists a finite set ${\Lambda }_u\subseteq {\mathbb{S}}^{d-1}\cap {\mathbb{Q}}^d$, $\epsilon >0$, such that there are smooth positive functions

$${\gamma }_{\xi }\in C^{\infty }\left(B_{\epsilon }(\mathrm{Id})\right),\xi \in {\Lambda }_u,$$

where $B_{\epsilon }(\mathrm{Id})$ is the ball of $d\times d$ symmetric matrices, centered at $\mathrm{Id}$ of radius ε, such that for any $R\in B_{\epsilon }(\mathrm{Id})$, we have the identity

$$R=\sum _{\xi \in {\Lambda }_u}{\gamma }_{\xi }^2(R)\xi \otimes \xi .$$

In addition, one can choose an orthogonal basis $\left\{\xi ,{\xi }_1,{\xi }_2\right\}\subseteq {\mathbb{S}}^2\cap {\mathbb{Q}}^3$ for each $\xi \in {\Lambda }_u$.

To eliminate the vector-valued Reynolds term $\overline{R^{\theta }}$, we must apply similar tools, as follows.

Lemma 2.

There exists $\epsilon >0$ and a finite set ${\Lambda }_{\theta }\subset {\mathbb{S}}^{d-1}\cap {\mathbb{Q}}^d$ such that for any $\xi \in {\Lambda }_{\theta }$, there are smooth functions ${\gamma }_{\xi }:B_{\epsilon }(0)\to \mathbb{R}$, where $B_{\epsilon }(0)$ is the ball of radius ε centered at 0 in ${\mathbb{R}}^d$, such that for any $R\in B_{\epsilon }(0)$ we have

$$R=\sum _{\xi \in {\Lambda }_{\theta }}{\gamma }_{\xi }^2(R)\xi .$$

(6)

Remark 1.

The sets ${\Lambda }_u$ and ${\Lambda }_{\theta }$ can be chosen to have no common elements. Note that a fixed value can be chosen for $\epsilon >0$. Let $\Lambda ={\Lambda }_u\cup {\Lambda }_{\theta }$; as Λ is a finite subset of ${\mathbb{Q}}^3$, there exists $N_{\Lambda }\in {\mathbb{N}}_{+}$ such that $N_{\Lambda }\Lambda \subseteq {\mathbb{Z}}^3$.

2.2. The Anti-Divergence Operator

In order to rebuild the new Reynolds stress, the following anti-divergence operator is needed:

Definition 2

([10]). Let $v\in C^{\infty }\left({\mathbb{T}}^3;{\mathbb{R}}^3\right)$ be a smooth vector field, and define ${\mathcal{R}}^{u}v$ as the matrix-valued periodic function

$${\mathcal{R}}^{u}v:={\displaystyle \frac{1}{4}}\left(\nabla \mathbb{P}u+{(\nabla \mathbb{P}u)}^{\mathsf{T}}\right)+{\displaystyle \frac{3}{4}}\left(\nabla u+{(\nabla u)}^{\mathsf{T}}\right)-{\displaystyle \frac{1}{2}}(\mathrm{div}u)\mathrm{Id},$$

where $\mathbb{P}$ is the Leray projection operator and $u\in C^{\infty }({\mathbb{T}}^3;{\mathbb{R}}^3)$ is the solution of

$$\left\{\begin{array}{c}\Delta u=v-f_{{\mathbb{T}}^3}v\hfill \\ f_{{\mathbb{T}}^3}u=0.\hfill \end{array}\right.$$

By Definition 2, the anti-divergence operator has the following property:

Lemma 3.

For any $v\in C^{\infty }({\mathbb{T}}^3;{\mathbb{R}}^3)$, the following statements are true:

• ${\mathcal{R}}^u\Delta v(x)=\nabla v+\nabla v^{\mathsf{T}}$ for any divergence-free vector v;
• $\mathrm{div}{\mathcal{R}}^{u}v=v-f_{{\mathbb{T}}^3}v;$
• If we suppose $1\le q\le \infty$, then we have

  $${∥{\mathcal{R}}^{u}v∥}_{L^q({\mathbb{T}}^3)}\lesssim {∥v∥}_{L^q({\mathbb{T}}^3)};$$
• If we suppose $1\le q\le \infty$, then, for any $v\in C_0^{\infty }({\mathbb{T}}^3;{\mathbb{R}}^3)$,

  $${∥{\mathcal{R}}^{u}v(\sigma ·)∥}_{L^q({\mathbb{T}}^3)}\lesssim {\sigma }^{-1}{∥v∥}_{L^q({\mathbb{T}}^3)},\forall \sigma \in {\mathbb{N}}_{+}.$$

For a scalar-based anti-divergence operator, if we let ${\mathcal{R}}^{\theta }\nabla {\Delta }^{-1}:C_0^{\infty }({\mathbb{T}}^3)\to C^{\infty }({\mathbb{T}}^3;{\mathbb{R}}^3)$, then we have the following lemma, as in Modena [14]:

Lemma 4.

For any $k\in \mathbb{N}$, $q\in \left[1,\infty \right]$,

$${∥D^k{\mathcal{R}}^{\theta }g∥}_{L^q}\le C_{k,q}{∥D^{k}g∥}_{L^q}.$$

3. The Glued Step

Suppose the points ${\left\{t_i\right\}}_{i=0}^n$ divide the interval $[T_1,T_2]\subseteq [0,T]$ into equal subintervals, where $t_0=T_1$ and $t_n=T_2$. Let ${\tau }^{\epsilon }=t_{i+1}-t_i$ for a given $\epsilon >0$; we will determine the value of $\tau$ later. The parameter $\epsilon$ bounds the Hausdorff dimension of the temporal singular set.

Assume the smooth solution $\left(u,\theta ,R^u,R^{\theta },p\right)$ satisfies (4). Consider the system

$$\left\{\begin{array}{c}{\partial }_t{v}_i+\mathrm{div}\left(v_i\otimes v_i+u\otimes v_i+v_i\otimes u\right)+\nabla q_i-\nu \Delta v_i=-{\phi }_i{e}_3-\mathrm{div}{R}^u,\hfill \\ {\partial }_t{\phi }_i+\mathrm{div}\left(u·{\phi }_i+v_i·\theta +v_i·{\phi }_i\right)-\mu \Delta {\phi }_i=-\mathrm{div}{R}^{\theta },\hfill \\ v_i(t_{i-1})=0,{\phi }_i(t_{i-1})=0,\mathrm{div}{v}_i=0,\hfill \end{array}\right.$$

(7)

which evolves from time $t=t_{i-1}$ for $i=1,\cdots ,n$. There exists a unique exact solution $(v_i,{\phi }_i)$ on $[t_i,t_i+T^{*})$, where $T^{*}>0$ depends only on $\nu$, $\mu$ and $u,\theta ,R^u,R^{\theta }$. Furthermore, $T^{*}$ is independent of i; see, for example, Larios [18] or Majda [19]. The formulation of Equation (7) ensures that its superposition with Equation (4) recovers the Boussinesq system (1), yet it cancels out for most times $t\in [t_i+{\tau }^{\epsilon },t_{i+1}-{\tau }^{\epsilon }]$. Thereby, the resulting Reynolds stress is concentrated in a brief time window. A key principle of the glued step is to ensure that ${∥\overline{R}∥}_{L_t^1{W}_x^{k,q}}$ does not increase uncontrollably, as illustrated by comparing Equations (15) and (16) for $p=1$ and for $p>1$.

Define a smooth cutoff function ${\chi }_i:[0,T]\to [0,1]$ supported on $[t_{i-1},t_i]$. Specifically, ${\chi }_i(t)\equiv 1$ on $[t_{i-1}+\tau ,t_i-\tau ]$ for $1<i<n$, on $[t_0,t_1-\tau ]$ for $i=1$, and on $[t_{n-1}+\tau ,t_n]$ for $i=n$. The construction is within the bounds ${∥{\partial }_t{\chi }_i∥}_{L^{\infty }}\le {\displaystyle \frac{2}{\tau }}$. Define the glued solution $(\overline{u},\overline{\theta })$ as

$$\left\{\begin{array}{c}\overline{u}=u+\sum _{i=1}^n{\chi }_i{v}_i,\hfill \\ \overline{\theta }=\theta +\sum _{i=1}^n{\chi }_i{\phi }_i.\hfill \end{array}\right.$$

(8)

Then, by (7), the solution $(\overline{u},\overline{\theta })$ satisfies

$$\left\{\begin{array}{c}{\partial }_t\overline{u}-\nu \Delta \overline{u}+\mathrm{div}\left(\overline{u}\otimes \overline{u}\right)+\nabla \overline{p}=-\overline{\theta }{e}_3+\mathrm{div}\overline{R^u},\hfill \\ {\partial }_t\overline{\theta }+\mathrm{div}\left(\overline{u}·\overline{\theta }\right)-\mu \Delta \overline{\theta }=\mathrm{div}\overline{R^{\theta }},\hfill \end{array}\right.$$

where

$$\begin{array}{cc}\hfill \overline{R^u}& :=\left(1-\sum _{i=1}^n{\chi }_i\right)R^u+\sum _{i=1}^n{\partial }_t{\chi }_i·{\mathcal{R}}^u{v}_i-\sum _{i=1}^n{\chi }_i(1-{\chi }_i)v_i\mathring{\otimes }{v}_i,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \overline{R^{\theta }}& :=\left(1-\sum _{i=1}^n{\chi }_i\right)R^{\theta }+\sum _{i=1}^n{\partial }_t{\chi }_i·{\mathcal{R}}^{\theta }{\phi }_i-\sum _{i=1}^n{\chi }_i(1-{\chi }_i)(v_i·{\phi }_i),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \overline{p}& :=p+\sum _{i=1}^n{\chi }_i{q}_i+{\displaystyle \frac{1}{3}}\sum _{i=1}^n{\chi }_i\left(1-{\chi }_i\right)·{\left|v_i\right|}^2.\hfill \end{array}$$

(11)

As $|1-{\sum }_{i=1}^n{\chi }_i|\le 1$, we can determine from (9) that

$$\begin{array}{cc}\hfill {∥\overline{R^u}∥}_{L_t^p{W}_x^{k,q}}& \le {∥R^u∥}_{L_t^p{W}_x^{k,q}}+\sum _{i=1}^n{∥{\partial }_t{\chi }_i∥}_{L_t^p}·{∥{\mathcal{R}}^u{v}_i∥}_{L_t^{\infty }{W}_x^{k,q}}\hfill \\ \hfill & +\sum _{i=1}^n{∥{\chi }_i(1-{\chi }_i)∥}_{L_t^p}·{∥v_i\mathring{\otimes }{v}_i∥}_{L_t^{\infty }{W}_x^{k,q}},\hfill \end{array}$$

where $1\le p\le \infty$, $1<q<\infty$. Since ${∥{\chi }_i(1-{\chi }_i)∥}_{L^p}\le C{\tau }^{1/p}$ and ${∥{\partial }_t{\chi }_i∥}_{L^p}\le C{\tau }^{1/p-1}$, we can infer from (9) and (10) that

$${∥\overline{R^u}∥}_{L_t^p{W}_x^{k,q}}\le C\left({∥R^u∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}\sum _{i=1}^n{∥{\mathcal{R}}^u{v}_i∥}_{L_t^{\infty }(t_i,t_{i+1};W_x^{k,q})}+{\tau }^{1/p}\sum _{i=1}^n{∥v_i∥}_{L_t^{\infty }(t_i,t_{i+1};C_x^k)}^2\right),$$

and similarly, we can infer that

$$\begin{array}{cc}\hfill {∥\overline{R^{\theta }}∥}_{L_t^p{W}_x^{k,q}}& \le C\left({∥R^{\theta }∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}\sum _{i=1}^n{∥{\mathcal{R}}^{\theta }{\phi }_i∥}_{L_t^{\infty }(t_i,t_{i+1};W_x^{k,q})}\right.\hfill \\ \hfill & \left.+{\tau }^{1/p}\sum _{i=1}^n{∥v_i∥}_{L_t^{\infty }(t_i,t_{i+1};C_x^k)}{∥{\phi }_i∥}_{L_t^{\infty }(t_i,t_{i+1};C_x^k)}\right).\hfill \end{array}$$

By the standard energy method, one can obtain the following estimate.

Proposition 1.

Suppose $\nu >0$ and $\mu \ge 0$. Let $1<q<\infty$, $m\ge 3$. Suppose that $\left(u,\theta ,R^u,R^{\theta },p\right)$ are smooth functions that satisfy (4). Under these conditions, for any $\delta >0$, there is a sufficiently small $T^{*}>0$ that the outcome depends only on $m,u,\theta ,R^u$, and $R^{\theta }$, such that the unique smooth solution $(v_i,{\phi }_i)$ of (7) satisfies

$${∥(v_i,{\phi }_i)∥}_{L_t^{\infty }(t_{i-1},t_{i-1}+T^{*};H_x^m)}\le \delta .$$

(12)

In addition, for any $\Delta t\le T^{*}$ and $k\in \mathbb{N}$ with $k\le m-2$,

$${∥{\mathcal{R}}^u{v}_i∥}_{L_t^{\infty }(t_{i-1},t_{i-1}+\Delta t;W_x^{k,q})}\le C_q\left({\int }_{t_i}^{t_i+\Delta t}{∥R^u(s)∥}_{W_x^{k,q}}\mathrm{d}s+\delta \Delta t\left(\delta +{∥u∥}_{L_t^{\infty }{H}_x^m}+1\right)\right),$$

(13)

and

$$\begin{array}{cc}\hfill {∥{\mathcal{R}}^{\theta }{\phi }_i∥}_{L_t^{\infty }\left(t_{i-1},t_{i-1}+\Delta t;W_x^{k,q}\right)}& \le C_q\left({\int }_{t_i}^{t_i+\Delta t}{∥R^{\theta }(s)∥}_{W_x^{k,q}}\mathrm{d}s\right.\hfill \\ \hfill & \left.+\delta \Delta t\left({\parallel u\parallel }_{L_t^{\infty }{H}_x^m}+{\parallel \theta \parallel }_{L_t^{\infty }{H}_x^m}+\delta \right)\right)\hfill \end{array}$$

(14)

where the constant $C_q$ is independent of δ, $T^{*}$, and i.

If we choose $\tau >0$ such that ${\tau }^{\epsilon }\le T^{*}$, i.e., $\Delta t=t_{i+1}-t_i={\tau }^{\epsilon }\le T^{*}$, then, for $k\le m-2$,

$$\begin{array}{cc}\hfill {∥\overline{R^u}∥}_{L_t^p{W}_x^{k,q}}& \le C_q\left({∥R^u∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p}\sum _{i=1}^n{\delta }^2\right.\hfill \\ \hfill & \left.+{\tau }^{1/p-1}\sum _{i=1}^n\left[{\int }_{t_i}^{t_{i+1}}{∥R^u(s)∥}_{W_x^{k,q}}\mathrm{d}s+\delta {\tau }^{\epsilon }\left(\delta +{∥u∥}_{L_t^{\infty }{C}_x^k}+1\right)\right]\right)\hfill \\ \hfill & \le C_q\left({∥R^u∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}{∥R^u∥}_{L_t^1{W}_x^{k,q}}+{\tau }^{1/p-\epsilon }{\delta }^2\right.\hfill \\ \hfill & \left.+\delta {\tau }^{1/p-1}\left(\delta +{∥u∥}_{L_t^{\infty }{C}_x^k}+1\right)\right)\hfill \\ \hfill & \le C_q\left({∥R^u∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}{∥R^u∥}_{L_t^1{W}_x^{k,q}}\right),\hfill \end{array}$$

where we have used ${\tau }^{1/p-1}>{\tau }^{1/p-\epsilon }$ for $\tau ,\epsilon <1$. Similarly,

$${∥\overline{R^{\theta }}∥}_{L_t^p{W}_x^{k,q}}\le C_q\left({∥R^{\theta }∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}{∥R^{\theta }∥}_{L_t^1{W}_x^{k,q}}\right).$$

All the results above can be summarized in the following proposition:

Proposition 2.

For any $\delta >0$, if we suppose that $m\ge 3$, $1\le p\le \infty$, and $1<q<\infty$ and that $\left(u,\theta ,R^u,R^{\theta },p\right)$ are smooth functions that satisfy (4), then there exist other smooth functions $\left(\overline{u},\overline{\theta },\overline{R^u},\overline{R^{\theta }},\overline{p}\right)$ satisfying (4), such that

$${∥\overline{R^u}∥}_{L_t^p{W}_x^{k,q}}\le C_q\left({∥R^u∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}{∥R^u∥}_{L_t^1{W}_x^{k,q}}\right),$$

(15)

$${∥\overline{R^{\theta }}∥}_{L_t^p{W}_x^{k,q}}\le C_q\left({∥R^{\theta }∥}_{L_t^p{W}_x^{k,q}}+{\tau }^{1/p-1}{∥R^{\theta }∥}_{L_t^1{W}_x^{k,q}}\right),$$

(16)

and

$${∥\overline{u}-u∥}_{L_t^{\infty }{W}_x^{k,q}}\le \delta ,{∥\overline{\theta }-\theta ∥}_{L_t^{\infty }{W}_x^{k,q}}\le \delta$$

(17)

hold for $k\le m-2$. In addition, the supports of $R^u$ and $R^{\theta }$ are contained in the union about ${\tau }^{-\epsilon }$ intervals $[t_i-\tau ,t_i+\tau ]$, where the upper bound of τ depends only on m and $\left(u,\theta ,R^u,R^{\theta },p\right)$.

Proof. 

We give only the proof of ${∥\overline{u}-u∥}_{L_t^{\infty }{H}_x^m}$ in (17); the proof of the bounds for ${∥\overline{\theta }-\theta ∥}_{L_t^{\infty }{H}_x^m}$ is similar.

For $t\in [0,T]\setminus [T_1,T_2]$, there are no terms added in (8), such that $(\overline{u},\overline{\theta })\equiv (u,\theta )$, and (17) is obvious.

For $t\in [T_1,T_2]$, there is $i\in \left\{1,2,\cdots ,n\right\}$ such that $t\in [t_{i-1},t_i]$, and from the definition of ${\chi }_i$ and (8),

$$\overline{u}(t)-u(t)={\chi }_i(t)v_i(t).$$

Finally, using (12), we obtain the following uniform bounds:

$${∥\overline{u}(t)-u(t)∥}_{H_x^m}\le {∥v_i(t)∥}_{H^m}\le \delta .$$

□

4. The Perturbation of the Glued Solution

Now we introduce the building blocks of the oscillation part. Building blocks such as Beltrami flows, Mikado flows, and jet flows are used to establish the non-uniqueness of solutions of Navier–Stokes equations; see [5,6,7,8,9,12]. Because the temperature equation in the Boussinesq system is essentially a transport (diffusion) equation, it is difficult to use Beltrami flows to tackle the transport term, as there is no $\nabla p$ to absorb the intermediate part generated from $u·\nabla \theta$, and the situation is similar for jet flows. Mikado flows are introduced to establish the non-uniqueness of solutions to Navier–Stokes equations and transport equations, for example, in [9,12,13,14,15,16]. In this paper, the Mikado flow is a suitable choice of building block.

Suppose $\left(\overline{u},\overline{\theta },\overline{R^u},\overline{R^{\theta }},\overline{p}\right)$ are smooth functions that solve (4). Assume we have chosen $p\ge 1$ and $q\ge 1$; set

$$\delta {∥\overline{R^u}∥}_{L_t^p{L}_x^q}+{∥\overline{R^{\theta }}∥}_{L_t^p{L}_x^q}.$$

(18)

Let $\tilde{\chi }:\mathbb{R}\to \mathbb{R}$ be a smooth function that satisfies $\tilde{\chi }(R)=\delta$ for $0\le \left|R\right|\le \delta$, $\tilde{\chi }(R)=\left|R\right|$ for $\left|R\right|\ge 2\delta$, and $\left|R\right|\le 2\tilde{\chi }(R)\le 4\left|R\right|$ for $\delta <\left|R\right|<2\delta$. Then, ${∥\tilde{\chi }∥}_{C^1}$ is bounded by a universal constant. Define

$${\rho }_{\theta }(t,x)={\displaystyle \frac{2}{{\epsilon }_G}}\tilde{\chi }\left(\left|\overline{R^{\theta }}\right|\right),a_{\xi }^{\theta }(t,x)={\rho }_{\theta }^{1/2}{\gamma }_{\xi }^{\theta }\left(-{\displaystyle \frac{\overline{R^{\theta }}}{{\rho }_{\theta }}}\right),\xi \in {\Lambda }_{\theta },$$

$${\rho }_u(t,x)={\displaystyle \frac{2}{{\epsilon }_G}}\tilde{\chi }\left(\left|\overline{R^u}+\overline{T^{\theta }}\right|\right),a_{\xi }^u(t,x)={\rho }_u^{1/2}{\gamma }_{\xi }^u\left(\mathrm{Id}-{\displaystyle \frac{\overline{R^u}+\overline{T^{\theta }}}{{\rho }_u}}\right),\xi \in {\Lambda }_u,$$

where $\overline{T^{\theta }}(t,x)={\sum }_{\xi \in {\Lambda }_{\theta }}{\left(a_{\xi }^{\theta }(t,x)\right)}^2\xi \otimes \xi$. Then Lemma 1 and Lemma 2 imply

$$\sum _{\xi \in {\Lambda }_{\theta }}{\left(a_{\xi }^{\theta }\right)}^2\xi +\overline{R^{\theta }}=0,\sum _{\xi \in {\Lambda }_u}{\left(a_{\xi }^u\right)}^2\xi \otimes \xi ={\rho }_u\mathrm{Id}-\overline{R^u}-\overline{T^{\theta }}.$$

(19)

In addition, one can establish the following estimates of $a_{\xi }^{\theta }$ and $a_{\xi }^u$:

Lemma 5.

Suppose $p\ge 1$, $q\ge 1$. For $\xi \in {\Lambda }_{\xi }$,

$${∥a_{\xi }^{\theta }∥}_{L_t^{2p}{L}_x^{2q}}\le C_{\Lambda }{\delta }^{1/2},{∥a_{\xi }^{\theta }∥}_{C_{x,t}^k}\le C\left({\delta }^{-1},{∥\overline{R^{\theta }}∥}_{C_{x,t}^k}\right),\forall k\in \mathbb{N},$$

(20)

and for $\xi \in {\Lambda }_u$,

$${∥a_{\xi }^u∥}_{L_t^{2p}{L}_x^{2q}}\le C_{\Lambda }{\delta }^{1/2},{∥a_{\xi }^u∥}_{C_{x,t}^k}\le C\left({\delta }^{-1},{∥(\overline{R^u},\overline{R^{\theta }})∥}_{C_{t,x}^k}\right),\forall k\in \mathbb{N},$$

(21)

where $C_{\Lambda }$ is a constant that depends on Λ.

Choose a smooth function $\psi :{\mathbb{R}}^2\to \mathbb{R}$ with compact support in $B(0,1)$ that satisfies

$${\int }_{{\mathbb{R}}^2}{\psi }^2(x)\mathrm{d}x=1.$$

Let $\kappa \in \mathbb{N}$ be sufficiently large, representing the concentration in space, and set $\tilde{\mathsf{\Psi }}(x)=\kappa \psi (\kappa x)$. Under these conditions, ${∥\tilde{\mathsf{\Psi }}∥}_{L^2}=1$, with the periodized function $\tilde{\mathsf{\Psi }}(x)$ in ${\mathbb{T}}^2$ denoted as $\mathsf{\Psi }(x)$. For any $\xi \in \Lambda$ with orthogonal basis $\left\{\xi ,{\xi }_1,{\xi }_2\right\}$, one can choose points ${\left\{p_{\xi }\right\}}_{\xi \in \Lambda }\subseteq {\mathbb{T}}^3$ such that the lines $l_{\xi }:y=p_{\xi }+\xi ·x$ are disjoint. Then, for $\sigma \in \mathbb{N}$, the concentrated Mikado flow is defined as $W_{\xi }(x)·\xi$, where

$$W_{\xi }(x)=\mathsf{\Psi }(\sigma N_{\Lambda }{\xi }_1·(x-p_{\xi }),\sigma N_{\Lambda }{\xi }_2·(x-p_{\xi })).$$

(22)

Thus, the supports of ${\left\{W_{\xi }(x)\right\}}_{\xi \in \Lambda }$, contained in cylinders whose axes are parallel to $\xi$, are mutually disjoint when a sufficiently large $\sigma$ is chosen. The conditions $N_{\Lambda }\Lambda \subseteq \mathbb{Z}$ and $\sigma \in \mathbb{N}$ ensure that $W_{\xi }(x)$ is also a periodic function. A suitable choice of concentration $\kappa$ is used to control the new Reynolds terms and keep them small; see (3) in Proposition 3. The choice of oscillation $\sigma$ is used to maintain the perturbation w and $\varphi$ in $L_t^{\infty }{L}_x^2$ under control. Then, for any $r\in [1,\infty ]$, we have the estimates

$$\begin{array}{cc}\hfill {∥W_{\xi }(x)∥}_{L^r\left({\mathbb{T}}^3\right)}& \le C_{\xi }{\parallel \tilde{\mathsf{\Psi }}\parallel }_{L^r\left({\mathbb{T}}^2\right)}\hfill \\ \hfill & \le C_{\xi }·\kappa {\parallel \psi (\kappa x)\parallel }_{L^r\left({\mathbb{T}}^2\right)}\hfill \\ \hfill & \le C_{\xi }·{\kappa }^{1-2/r}{\parallel \psi \parallel }_{L^r\left({\mathbb{T}}^2\right)}\hfill \\ \hfill & \le C_{\xi ,\psi }{\kappa }^{1-2/r}\hfill \end{array}$$

(23)

and

$${∥W_{\xi }(x)∥}_{W^{1,r}({\mathbb{T}}^3)}\le C_{\xi }·\sigma {∥\tilde{\mathsf{\Psi }}∥}_{W^{1,r}({\mathbb{T}}^2)}\le C_{\xi ,\psi ,{\psi }^{\prime }}\sigma {\kappa }^{2-2/r}.$$

(24)

From the definition of $W_{\xi }(x)$ in (22), for any smooth function F, we have

$$\mathrm{div}\left(F(W_{\xi }(x))\xi \otimes \xi \right)=\sigma N_{\Lambda }{F}^{\prime }(W_{\xi })\left({\partial }_1\mathsf{\Psi }\sum _{j=1}^3{\xi }_1^j{\xi }^i{\xi }^j+{\partial }_2\mathsf{\Psi }\sum _{j=1}^3{\xi }_2^j{\xi }^i{\xi }^j\right)=0,$$

(25)

and

$$\mathrm{div}\left(F(W_{\xi }(x))\xi \right)=0.$$

(26)

Choose a smooth cutoff function $\overline{\chi }(t):[0,T]\supseteq [T_1,T_2]\to [0,1]$, and set $\overline{\chi }(t)\equiv 1$ for $t\in \left(t_i-\tau ,t_i+\tau \right)$, $i=0,1,\cdots ,n$, with $\mathrm{supp}\overline{\chi }\subseteq {\cup }_{i=0}^n\left(t_i-2\tau ,t_i+2\tau \right)$; one can also require that ${∥{\overline{\chi }}^{\prime }(t)∥}_{L^{\infty }\left([0,T]\right)}<{\displaystyle \frac{2}{\tau }}$. Choose $g\in C_c^{\infty }\left([0,T]\right)$ such that ${∥g∥}_{L^2[0,T]}=1$, and set $g_{\omega }(t)=\sqrt{\omega }g(\omega t)$ with a period of 1, where $\omega$ is sufficiently large. Finally, let $\lambda \gg 1$, and set $\chi (t)=\overline{\chi }(t)g_{\omega }(\lambda t)$. Under these conditions, for $1\le s\le \infty$,

$${∥\chi (t)∥}_{L^s\left([0,T]\right)}\lesssim {\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{s}}},{∥{\chi }^{\prime }(t)\}∥}_{L^s([0,T])}\lesssim \lambda \omega ·{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{s}}}.$$

Let $h_{\omega }(t)={\int }_0^t{g}_{\omega }^2(s)-1\mathrm{d}s$; we define the perturbation of temperature $\varphi ={\varphi }^{(p)}+{\varphi }^{(c)}+{\varphi }^{(t)}$, where

$$\begin{array}{cc}\hfill {\varphi }^{(p)}(t,x)& :=\chi (t)\sum _{\xi \in {\Lambda }_{\theta }}{a}_{\xi }^{\theta }(t,x)W_{\xi }(x),{\varphi }^{(c)}(t):=-{⨏}_{{\mathbb{T}}^3}{\varphi }^{(p)}\mathrm{d}x,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\varphi }^{(t)}(t)& :={\lambda }^{-1}{h}_{\omega }(\lambda t)\mathrm{div}\overline{R^{\theta }},\hfill \end{array}$$

(28)

and the perturbation of velocity $w=w^{(p)}+w^{(c)}+w^{(t)}$, where

$$\begin{array}{cc}\hfill w^{(p)}(t,x)& :=\sum _{\xi \in {\Lambda }_u}\chi (t)a_{\xi }^u{W}_{\xi }(x)·\xi +\sum _{\xi \in {\Lambda }_{\theta }}\chi (t)a_{\xi }^{\theta }{W}_{\xi }(x)·\xi ,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill w^{(c)}& =-\nabla {\Delta }^{-1}\mathrm{div}{w}^{(p)},w^{(t)}:={\lambda }^{-1}{h}_{\omega }(\lambda t)\mathbb{P}\mathrm{div}\overline{R^u},\hfill \end{array}$$

(30)

where $\mathbb{P}$ is the Leray projection.

The New Reynolds Terms

Suppose that $\left(\overline{u},\overline{\theta },\overline{R^u},\overline{R^{\theta }},\overline{p}\right)$ satisfies (4); define

$$\left\{\begin{array}{c}{u}_1=\overline{u}+w,\hfill \\ {\theta }_1=\overline{\theta }+\varphi ,\hfill \end{array}\right.$$

(31)

to be the perturbation of $(\overline{u},\overline{\theta })$ by $(w,\varphi )$. We wish to find relative Reynolds stress $R_1^u$, $R_1^{\theta }$ corresponding to $\overline{u}$, $\overline{\theta }$ and satisfying (4); thus, by substituting (31) into (4), we obtain

$$\begin{array}{cc}\hfill {\partial }_t{u}_1-\nu \Delta u_1+\mathrm{div}(u_1\otimes u_1)& =-\nabla \overline{p}-{\theta }_1{e}_3+\varphi e_3+\mathrm{div}(\overline{R^u}+w\otimes w)\hfill \\ \hfill & +{\partial }_{t}w-\nu \Delta w+\mathrm{div}(\overline{u}\otimes w+w\otimes \overline{u}),\hfill \\ \hfill {\partial }_t{\theta }_1+\mathrm{div}(u_1·{\theta }_1)& -\mu \Delta {\theta }_1=\mathrm{div}(\overline{R^{\theta }}+w·\varphi )+{\partial }_t\varphi -\mu \Delta \varphi +\mathrm{div}(\overline{u}·\varphi +w·\overline{\theta }).\hfill \end{array}$$

Now, in general, we need to split the term $(\overline{R^u}+w\otimes w)$ into the oscillation part $R_{osc}^u$ and the correction part $R_{cor}^u$; by (29),

$$\overline{R^u}+w\otimes w=\left(\overline{R^u}+w^{(p)}\otimes w^{(p)}\right)+R_{cor}^u,$$

where

$$R_{cor}^u:=w^{(p)}\otimes \left(w^{(c)}+w^{(t)}\right)+\left(w^{(c)}+w^{(t)}\right)\otimes w.$$

Be aware that the supports of $W_{\xi }(x)$ are disjoint for different $\xi \in \Lambda ={\Lambda }_u\cup {\Lambda }_{\theta }$; accordingly, by (19),

$$\begin{array}{cc}\hfill w^{(p)}\otimes w^{(p)}+\overline{R^u}=& \overline{R^u}+\sum _{\xi \in {\Lambda }_u}{\chi }^2(t){\left(a_{\xi }^u\right)}^2{W}_{\xi }^2(x)\xi \otimes \xi +\sum _{\xi \in {\Lambda }_{\theta }}{\chi }^2(t){\left(a_{\xi }^{\theta }\right)}^2{W}_{\xi }^2(x)\xi \otimes \xi \hfill \\ \hfill =& \overline{R^u}+\sum _{\xi \in {\Lambda }_u}{\chi }^2(t){\left(a_{\xi }^u\right)}^2\xi \otimes \xi +\sum _{\xi \in {\Lambda }_{\theta }}{\chi }^2(t){\left(a_{\xi }^{\theta }\right)}^2\xi \otimes \xi \hfill \\ \hfill & +{\chi }^2(t)\sum _{\xi \in \Lambda }{a}_{\xi }^2{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi \otimes \xi \hfill \\ \hfill =& \overline{R^u}+{\chi }^2(t)\left({\rho }_{u}Id-\overline{R^u}\right)+{\chi }^2(t)\sum _{\xi \in \Lambda }{a}_{\xi }^2{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi \otimes \xi \hfill \\ \hfill =& \left(1-g_{\omega }^2(\lambda t)\right)\overline{R^u}+{\chi }^2(t)\sum _{\xi \in \Lambda }{a}_{\xi }^2{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi \otimes \xi +{\chi }^2(t){\rho }_{u}Id.\hfill \end{array}$$

As $\overline{\chi }(t)\equiv 1$ on ${\mathrm{supp}}_t\overline{R^u}$, the term $\overline{R^u}$ is canceled for most of the time range. Finally, using (25), we obtain

$$\begin{array}{cc}\hfill {\partial }_t{w}^{(t)}+div\left(w^{(p)}\otimes w^{(p)}+\overline{R^u}\right)=& \left(g_{\omega }^2(\lambda t)-1\right)\mathbb{P}div\overline{R^u}+{\lambda }^{-1}{h}_{\omega }(\lambda t)\mathbb{P}div{\partial }_t\overline{R^u}\hfill \\ \hfill & +\left(1-g_{\omega }^2(\lambda t)\right)div\overline{R^u}+{\chi }^2(t)\nabla {\rho }_u\hfill \\ \hfill & +div\left({\chi }^2(t)\sum _{\xi \in \Lambda }{a}_{\xi }^2{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi \otimes \xi \right)\hfill \\ \hfill =& {\lambda }^{-1}{h}_{\omega }(\lambda t)\mathbb{P}div{\partial }_t\overline{R^u}+{\chi }^2(t)\sum _{\xi \in \Lambda }\xi ·\nabla a_{\xi }^2{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi \hfill \\ \hfill & +{\chi }^2(t)\nabla {\rho }_u-\left(g_{\omega }^2(\lambda t)-1\right)\nabla {\Delta }^{-1}divdiv\overline{R^u}.\hfill \end{array}$$

Similarly, for the advection term $w·\nabla \varphi$ of temperature, we split the oscillation part into

$$\begin{array}{cc}\hfill \mathrm{div}(w·\varphi +\overline{R^{\theta }})& =\mathrm{div}(w^{(p)}·{\varphi }^{(p)}+\overline{R^{\theta }})+\mathrm{div}(w^{(p)}·({\varphi }^{(c)}+{\varphi }^{(t)})+(w^{(c)}+w^{(t)})·\varphi )\hfill \\ \hfill & =\mathrm{div}{R}_{osc}^{\theta }+\mathrm{div}{R}_{cor}^{\theta },\hfill \end{array}$$

where

$$R_{cor}^{\theta }:=w^{(p)}·({\varphi }^{(c)}+{\varphi }^{(t)})+(w^{(c)}+w^{(t)})·\varphi .$$

Finally, by Lemma 2 and (26), the advection part is split as follows:

$${\partial }_t{\varphi }^{(t)}+\mathrm{div}(w^{(p)}·{\varphi }^{(p)}+\overline{R^{\theta }})={\lambda }^{-1}{h}_{\omega }(\lambda t)\mathrm{div}{\partial }_t\overline{R^{\theta }}+{\chi }^2(t)\sum _{\xi \in {\Lambda }_{\theta }}\xi ·\nabla {\left(a_{\xi }^{\theta }\right)}^2·{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x).$$

Then, by the anti-divergence operator, the new Reynolds term can be defined as

$$\begin{array}{cc}\hfill R_1^u& :=R_{osc,x}^u+R_{osc,t}^u+R_{cor}^u+\underset{R_{lin}^u}{\underbrace{{\mathcal{R}}^u\varphi e_3+{\mathcal{R}}^u\left({\partial }_t(w^{(p)}+w^{(c)})-\nu \Delta w\right)+\overline{u}\otimes w+w\otimes \overline{u}}},\hfill \end{array}$$

$$R_1^{\theta }:=R_{osc,x}^{\theta }+R_{osc,t}^{\theta }+R_{cor}^{\theta }+\underset{R_{lin}^{\theta }}{\underbrace{{\mathcal{R}}^{\theta }{\partial }_t({\varphi }^{(p)}+{\varphi }^{(c)})-\mu {\mathcal{R}}^{\theta }\Delta \varphi +w·\overline{\theta }+\overline{u}·\varphi }}$$

and the pressure can be defined as

$$p_1:=\overline{p}-{\chi }^2(t){\rho }_u+\left(g_{\omega }^2(\lambda t)-1\right){\Delta }^{-1}\mathrm{div}\mathrm{div}\overline{R^u},$$

which ensures that the set of functions $(u_1,{\theta }_1,R_1^u,R_1^{\theta },p_1)$ also satisfies Equation (4). Here, $R_{osc,x}^u$ and $R_{osc,x}^{\theta }$ are the oscillation error in space, set as

$$R_{osc,x}^u:={\chi }^2(t)\left[\sum _{\xi \in \Lambda }\xi ·\nabla a_{\xi }^2{\mathcal{R}}^u{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi -{\mathcal{R}}^u\sum _{\xi \in \Lambda }\nabla \left(\xi ·\nabla a_{\xi }^2\right)·{\mathcal{R}}^u\left({\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\xi \right)\right],$$

$$R_{osc,x}^{\theta }:={\chi }^2(t)\left(\sum _{\xi \in {\Lambda }_{\theta }}\xi ·\nabla {\left(a_{\xi }^{\theta }\right)}^2{\mathcal{R}}^{\theta }{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)-{\mathcal{R}}^{\theta }\sum _{\xi \in {\Lambda }_{\theta }}\nabla \left(\xi ·\nabla {(a_{\xi }^{\theta })}^2\right)·{\mathcal{R}}^{\theta }{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)\right).$$

$R_{osc,t}^u$ and $R_{osc,t}^{\theta }$ are the temporal oscillation error:

$$R_{osc,t}^u:={\lambda }^{-1}{h}_{\omega }(\lambda t)\mathbb{P}{\partial }_t\overline{R^u},R_{osc,t}^{\theta }:={\lambda }^{-1}{h}_{\omega }(\lambda t){\partial }_t\overline{R^{\theta }}.$$

From the definitions of $R_1^u$ and $R_1^{\theta }$, we estimate ${∥R_1^u∥}_{L_t^p{L}_x^q}$ and ${∥R_1^{\theta }∥}_{L_t^p{L}_x^q}$ term by term for $1\le p\le \infty$ and $1<q<\infty$:

$$\begin{array}{cc}\hfill {∥R_1^u∥}_{L_t^p{L}_x^q}& \le {∥R_{lin}^u∥}_{L_t^p{L}_x^q}+{∥R_{osc,x}^u∥}_{L_t^p{L}_x^q}+{∥R_{osc,t}^u∥}_{L_t^p{L}_x^q}+{∥R_{cor}^u∥}_{L_t^p{L}_x^q},\hfill \\ \hfill {∥R_1^{\theta }∥}_{L_t^p{L}_x^q}& \le {∥R_{lin}^{\theta }∥}_{L_t^p{L}_x^q}+{∥R_{osc,x}^{\theta }∥}_{L_t^p{L}_x^q}+{∥R_{osc,t}^{\theta }∥}_{L_t^p{L}_x^q}+{∥R_{cor}^{\theta }∥}_{L_t^p{L}_x^q}.\hfill \end{array}$$

The linear parts ${∥R_{lin}^u∥}_{L_t^p{L}_x^q}$ and ${∥R_{lin}^{\theta }∥}_{L_t^p{L}_x^q}$ are estimated as follows.

By the definition of the anti-divergence operator

$${\mathcal{R}}^u\Delta w=\nabla w+\nabla w^{\mathsf{T}},$$

we have

$$\begin{array}{cc}\hfill {∥{\mathcal{R}}^u\Delta w∥}_{L_t^p{L}_x^q({\mathbb{T}}^3)}& \le {∥\mathbb{P}{w}^{(p)}∥}_{L_t^p{W}_x^{1,q}({\mathbb{T}}^3)}+{∥w^{(t)}∥}_{L_t^p{W}_x^{1,q}({\mathbb{T}}^3)}\hfill \\ \hfill & \le C\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}·\sigma ·{\kappa }^{2-{\displaystyle \frac{2}{q}}}+{\lambda }^{-1}\right).\hfill \end{array}$$

where the constant C relies on the glued Reynolds stresses $\overline{R^u}$ and $\overline{R^{\theta }}$. Consider the definition of $\chi (t)$,

$$\begin{array}{cc}\hfill {∥{\mathcal{R}}^u{\partial }_t(w^{(p)}+w^{(c)})∥}_{L_t^p{L}_x^q}& \le C{∥\chi (t)+\left|{\chi }^{\prime }(t)\right|∥}_{L_t^p}·{\sigma }^{-1}{\kappa }^{1-{\displaystyle \frac{2}{q}}}\hfill \\ \hfill & \le C\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}+\lambda \omega ·{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}\right){\sigma }^{-1}{\kappa }^{1-{\displaystyle \frac{2}{q}}},\hfill \end{array}$$

and

$${∥\overline{u}\otimes w+w\otimes \overline{u}∥}_{L_t^p{L}_x^q}\le 2{∥\overline{u}∥}_{L_{t,x}^{\infty }}·{∥w∥}_{L_t^p{L}_x^q}\le C\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}·{\kappa }^{1-{\displaystyle \frac{2}{q}}}+{\lambda }^{-1}\right).$$

As ${\mathcal{R}}^u$ is $L^q$ bounded, it follows that

$${∥{\mathcal{R}}^u\varphi e_3∥}_{L_t^p{L}_x^q}\le C_q{∥\varphi ∥}_{L_t^p{L}_x^q}\le C\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}·{\kappa }^{1-{\displaystyle \frac{2}{q}}}+{\lambda }^{-1}\right).$$

From all the above, we conclude that

$${∥(R_{lin}^u,R_{lin}^{\theta })∥}_{L_t^p{L}_x^q}\le C\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}·\sigma ·{\kappa }^{2-{\displaystyle \frac{2}{q}}}+{\lambda }^{-1}+\lambda \omega ·{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}·{\sigma }^{-1}{\kappa }^{1-{\displaystyle \frac{2}{q}}}\right).$$

The linear parts ${∥R_{lin}^u∥}_{L_t^p{L}_x^q}$ and ${∥R_{lin}^{\theta }∥}_{L_t^p{L}_x^q}$ are estimated as follows.

By the Hölder inequality,

$$\begin{array}{cc}\hfill {∥R_{cor}^u∥}_{L_t^p{L}_x^q}& \le {∥(w^{(c)}+w^{(t)})\otimes w∥}_{L_t^p{L}_x^q}+{∥w^{(p)}\otimes (w^{(c)}+w^{(t)})∥}_{L_t^p{L}_x^q}\hfill \\ \hfill & \le {∥w^{(c)}+w^{(t)}∥}_{L_t^{2p}{L}_x^{2q}}·{∥w∥}_{L_t^{2p}{L}_x^{2q}}+{∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}·{∥w^{(c)}+w^{(t)}∥}_{L_t^{2p}{L}_x^{2q}}\hfill \\ \hfill & \le C\left(\overline{R^u}\right){\lambda }^{-1}{∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}+{∥w^{(c)}∥}_{L_t^{2p}{L}_x^{2q}}·{∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}.\hfill \end{array}$$

For sufficiently large $\sigma$, using an $L^p$ de-correlation argument, we have

$$\begin{array}{cc}\hfill {∥w^{(p)}∥}_{L_x^{2q}}& \le C{∥\chi (t)a_{\xi }·W_{\xi }\xi ∥}_{L_t^{2p}{L}_x^{2q}}\hfill \\ \hfill & \le C\chi (t)·\left({∥a_{\xi }∥}_{L_x^{2q}}·{∥W_{\xi }∥}_{L_x^{2q}}+{\sigma }^{-{\displaystyle \frac{1}{2q}}}·{∥a_{\xi }∥}_{C_{t,x}^1}·{∥W_{\xi }∥}_{L_x^{2q}}\right)\hfill \\ \hfill & \le C\left(\overline{R^u},\overline{R^{\theta }}\right)g_{\omega }\left(\lambda t\right)·{\kappa }^{1-{\displaystyle \frac{1}{q}}}{∥a_{\xi }∥}_{L_x^{2q}}.\hfill \end{array}$$

Hence, the bound of the principal of the perturbation $w^{(p)}$ satisfies

$${∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}\le C{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}·{\kappa }^{1-{\displaystyle \frac{1}{q}}}{\delta }^{1/2}.$$

Finally, as $\nabla {\Delta }^{-1}\mathrm{div}$ is a Calderón-Zygmund operator and $W_{\xi }\xi$ oscillates with period ${\sigma }^{-1}$ on ${\mathbb{T}}^3$, then

$$\begin{array}{cc}\hfill {∥w^{(c)}∥}_{L_t^{2p}{L}_x^{2q}}& ={∥\nabla {\Delta }^{-1}{\mathbb{P}}_{\ne 0}\sum _{\xi }\chi (t)\nabla a_{\xi }·W_{\xi }\xi ∥}_{L_t^{2p}{L}_x^{2q}}\hfill \\ \hfill & \le C{∥{\chi (t)|\nabla |}^{-1}{\mathbb{P}}_{\ne 0}\left(\nabla a_{\xi }·W_{\xi }\right)∥}_{L_t^{2p}{L}_x^{2q}}\hfill \\ \hfill & \le C\left(\overline{R^u},\overline{R^{\theta }}\right)∥\chi (t){\sigma }^{-1}∥W_{\xi }{{\parallel }_{L_x^{2q}}\parallel }_{L_t^{2p}}\hfill \\ \hfill & \le C\left(\overline{R^u},\overline{R^{\theta }}\right){\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}{\sigma }^{-1}{\kappa }^{1-{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

From all the above estimates,

$${∥R_{cor}^u∥}_{L_t^p{L}_x^q}\le C\left(\overline{R^u},\overline{R^{\theta }}\right)\left({\lambda }^{-1}{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}{\kappa }^{1-{\displaystyle \frac{1}{q}}}+{\sigma }^{-1}{\omega }^{1-{\displaystyle \frac{1}{p}}}{\kappa }^{2-{\displaystyle \frac{2}{q}}}\right){\delta }^{1/2}.$$

Furthermore, for $R_{cor}^{\theta }$, we similarly obtain

$$\begin{array}{cc}\hfill {∥R_{cor}^{\theta }∥}_{L_t^p{L}_x^q}& \le {∥{\varphi }^{(c)}+{\varphi }^{(t)}∥}_{L_t^{2p}{L}_x^{2q}}·{∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}+{∥w^{(c)}+w^{(t)}∥}_{L_t^{2p}{L}_x^{2q}}·{\parallel \varphi \parallel }_{L_t^{2p}{L}_x^{2q}}\hfill \\ \hfill & \le C\left(\overline{R^{\theta }}\right){\lambda }^{-1}{∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}+C\left(\overline{R^u}\right){\lambda }^{-1}{\parallel \varphi \parallel }_{L_t^{2p}{L}_x^{2q}}\hfill \\ \hfill & +{∥{\varphi }^{(c)}∥}_{L_t^{2p}{L}_x^{2q}}{∥w^{(p)}∥}_{L_t^{2p}{L}_x^{2q}}+{∥w^{(c)}∥}_{L_t^{2p}{L}_x^{2q}}·{\parallel \varphi \parallel }_{L_t^{2p}{L}_x^{2q}}.\hfill \end{array}$$

The smooth function ${\varphi }^{(c)}$ does not rely on the spatial variable; therefore,

$${∥{\varphi }^{(c)}∥}_{L_t^{2p}{L}_x^{2q}}\le C\left(\overline{R^{\theta }}\right){\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}{\kappa }^{-1},$$

and the same argument as above shows that

$${∥R_{cor}^{\theta }∥}_{L_t^p{L}_x^q}\le C\left(\overline{R^u},\overline{R^{\theta }}\right)\left({\lambda }^{-1}{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}·{\kappa }^{1-{\displaystyle \frac{1}{q}}}+{\omega }^{1-{\displaystyle \frac{1}{p}}}·{\kappa }^{-{\displaystyle \frac{1}{q}}}+{\sigma }^{-1}{\omega }^{1-{\displaystyle \frac{1}{p}}}·{\kappa }^{2-{\displaystyle \frac{2}{q}}}\right){\delta }^{1/2}.$$

The oscillation parts ${∥R_{osc,x}^u∥}_{L_t^p{L}_x^q}$ and ${∥R_{osc,x}^{\theta }∥}_{L_t^p{L}_x^q}$ are estimated as follows.

For $q\ge 1$, the operator ${\mathcal{R}}^u$ is $L^q$ bounded, and

$${∥R_{osc}^u∥}_{L_x^q}\le C{\chi }^2(t){∥a_{\xi }^2∥}_{C_{t,x}^2}{∥{\mathcal{R}}^u{\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)∥}_{L_x^q}.$$

Notice that ${\mathbb{P}}_{\ne 0}{W}_{\xi }^2(x)$ has period ${\sigma }^{-1}$; thus, again,

$${∥R_{osc,x}^u∥}_{L_t^p{L}_x^q}\le C\left(\overline{R^u},\overline{R^{\theta }}\right){\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}{\sigma }^{-1}{\kappa }^{2-{\displaystyle \frac{2}{q}}},$$

and a similar argument gives

$${∥R_{osc,x}^{\theta }∥}_{L_t^p{L}_x^q}\le C\left(\overline{R^{\theta }}\right){\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}{\sigma }^{-1}{\kappa }^{2-{\displaystyle \frac{2}{q}}}.$$

The oscillation parts ${∥R_{osc,t}^u∥}_{L_t^p{L}_x^q}$ and ${∥R_{osc,t}^{\theta }∥}_{L_t^p{L}_x^q}$ are estimated as follows.

As $h_{\omega }$ is defined with a uniform bound, the following is obvious:

$${∥R_{osc,t}^u∥}_{L_t^p{L}_x^q}+{∥R_{osc,t}^{\theta }∥}_{L_t^p{L}_x^q}\le C\left(\overline{R^u},\overline{R^{\theta }}\right){\lambda }^{-1}.$$

By a general estimate in a convex scheme, the estimation of this perturbation satisfies the following proposition:

Proposition 3.

Suppose $1<q<{\displaystyle \frac{5}{4}}$, and suppose that the glued solution $\left(\overline{u},\overline{\theta },\overline{R^u},\overline{R^{\theta }},\overline{p}\right)$ in Proposition 2 satisfies (4). Then, for any $\delta >0$ and $1<s<2$, there exists another solution $\left(u_1,{\theta }_1,R_1^u,R_1^{\theta },p_1\right)$ that satisfies (4), and this solution can be estimated as follows:

$${∥(R_1^u,R_1^{\theta })∥}_{L_t^1{L}_x^q}\le {\displaystyle \frac{1}{2}}\delta ,$$

$${∥u_1(t)-\overline{u}(t)∥}_{L_{t,x}^2}+{∥{\theta }_1-\overline{\theta }∥}_{L_{t,x}^2}\le C_q{∥\left(R^u,R^{\theta }\right)∥}_{L_t^1{L}_x^q}^{1/2}$$

and

$${∥u_1(t)-\overline{u}(t)∥}_{L_t^s{L}_x^{\infty }}+{∥{\theta }_1(t)-\overline{\theta }(t)∥}_{L_t^s{L}_x^{\infty }}\le \delta .$$

(32)

Proof. 

Set the parameter to

$$\omega ={\lambda }^{\alpha },\sigma ={\lambda }^{\beta },\kappa ={\lambda }^{\gamma }\in \mathbb{N},$$

with a sufficiently large $\lambda >0$. For $p=1$ and $1<q<{\displaystyle \frac{5}{4}}$, there exist $\alpha ,\beta ,$ and $\gamma$ that satisfy

$$\begin{array}{cc}\hfill 1& <{\displaystyle \frac{q}{4-3q}}<\gamma <{\displaystyle \frac{q}{q-1}},\beta >\left({\displaystyle \frac{2}{q}}-1\right)\gamma ,\hfill \\ \hfill 2\gamma & <2\beta +4\left(1-{\displaystyle \frac{1}{q}}\right)\gamma <\alpha <2\beta -2+2\left({\displaystyle \frac{2}{q}}-1\right)\gamma .\hfill \end{array}$$

Then, there exists $ϵ>0$ such that

$$\begin{array}{cc}\hfill {\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}\sigma {\kappa }^{2-{\displaystyle \frac{2}{q}}}& ={\lambda }^{(1/2-1/p)\alpha +\beta +(2-2/q)\gamma }<{\lambda }^{-ϵ};\hfill \\ \hfill \lambda {\omega }^{{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{p}}}·{\sigma }^{-1}{\kappa }^{1-{\displaystyle \frac{2}{q}}}& ={\lambda }^{1+(3/2-1/p)\alpha -\beta +(1-2/q)\gamma }<{\lambda }^{-ϵ};\hfill \\ \hfill {\lambda }^{-1}{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}{\kappa }^{1-{\displaystyle \frac{1}{q}}}& ={\lambda }^{-1+(1/2-1/(2p))\alpha +(1-1/q)\gamma }<{\lambda }^{-ϵ};\hfill \\ \hfill {\sigma }^{-1}{\omega }^{1-{\displaystyle \frac{1}{p}}}{\kappa }^{2-{\displaystyle \frac{2}{q}}}& ={\lambda }^{-\beta +(1-1/p)\alpha +(2-2/q)\gamma }<{\lambda }^{-ϵ};\hfill \\ \hfill {\omega }^{1-{\displaystyle \frac{1}{p}}}{\kappa }^{-{\displaystyle \frac{1}{q}}}& ={\lambda }^{(1-1/p)\alpha -\gamma /q}<{\lambda }^{-ϵ}.\hfill \end{array}$$

Then, from the estimates above,

$$\begin{array}{cc}\hfill {∥R_1^u∥}_{L_t^p{L}_x^q}+{∥R_1^{\theta }∥}_{L_t^p{L}_x^q}& \le C\left(\overline{R^u},\overline{R^{\theta }}\right)\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}}\sigma {\kappa }^{2-{\displaystyle \frac{2}{q}}}+\lambda {\omega }^{{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{p}}}·{\kappa }^{1-{\displaystyle \frac{2}{q}}}\right.\hfill \\ \hfill & \left.+\left({\lambda }^{-1}{\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2p}}}{\kappa }^{1-{\displaystyle \frac{1}{q}}}+{\sigma }^{-1}{\omega }^{1-{\displaystyle \frac{1}{p}}}{\kappa }^{2-{\displaystyle \frac{2}{q}}}+{\omega }^{1-{\displaystyle \frac{1}{p}}}{\kappa }^{-{\displaystyle \frac{1}{q}}}\right){\delta }^{1/2}\right)\hfill \\ \hfill & \le C\left(\overline{R^u},\overline{R^{\theta }}\right){\lambda }^{-ϵ}.\hfill \end{array}$$

Accordingly, for sufficiently large $\lambda$, we have

$${∥(R_1^u,R_1^{\theta })∥}_{L_t^1{L}_x^q}\le {\displaystyle \frac{1}{2}}\delta .$$

From $u_1(t)-\overline{u}(t)=w^{(p)}+w^{(c)}+w^{(t)}$ and ${\theta }_1-\overline{\theta }=\varphi$, we can infer that

$${∥u_1-\overline{u}∥}_{L_{t,x}^2}\le C\left({∥(R^u,R^{\theta })∥}_{L_t^1{L}_x^q}^{1/2}+C\left(\overline{R^u}\right){\lambda }^{-1}\right)$$

and

$${∥{\theta }_1-\overline{\theta }∥}_{L_{t,x}^2}={∥\varphi ∥}_{L_{t,x}^2}\le C\left({∥(R^u,R^{\theta })∥}_{L_t^1{L}_x^q}^{1/2}+C\left(\overline{R^{\theta }}\right)({\kappa }^{-1}+{\lambda }^{-1})\right).$$

Notice that $\kappa ={\lambda }^{\gamma }\gg \lambda$; accordingly, choose a sufficiently large $\lambda$ so that

$${∥u_1-\overline{u}∥}_{L_{t,x}^2}+{∥{\theta }_1-\overline{\theta }∥}_{L_{t,x}^2}\le C{∥(R^u,R^{\theta })∥}_{L_t^1{L}_x^q}^{1/2}.$$

Finally, we prove only the first part of (32), and the situation is similar for ${\theta }_1-\overline{\theta }$. For any given $1\le s<2$,

$$\begin{array}{cc}\hfill {∥u_1(t)-\overline{u}(t)∥}_{L_t^s{L}_x^{\infty }}& \le C\left(\overline{R^u},\overline{R^{\theta }}\right)\left({\omega }^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{s}}}\kappa +{\lambda }^{-1}\right)\hfill \\ \hfill & \le C\left(\overline{R^u},\overline{R^{\theta }}\right)\left({\lambda }^{(1/2-1/s)\alpha +\gamma }+{\lambda }^{-1}\right).\hfill \end{array}$$

If $\beta$ additionally satisfies $\beta >\left({\displaystyle \frac{s}{2-s}}-2+{\displaystyle \frac{2}{q}}\right)\gamma$, so that

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{s}}\right)\alpha +\gamma & <\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{s}}\right)\left(2\beta +4\left(1-{\displaystyle \frac{1}{q}}\right)\gamma \right)+\gamma \hfill \\ \hfill & <\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{s}}\right)\left(2\left({\displaystyle \frac{s}{2-s}}-2+{\displaystyle \frac{2}{q}}\right)\gamma +4\left(1-{\displaystyle \frac{1}{q}}\right)\gamma \right)+\gamma =0,\hfill \end{array}$$

then there exists a suitable $ϵ>0$, and for sufficiently large $\lambda$, we have

$${∥u_1-\overline{u}∥}_{L_t^s{L}_x^{\infty }}\le {\lambda }^{-ϵ}\le \delta .$$

□

5. The Proof of Theorem 1

Since on $[0,T]$, both $\left(u^{(1)},{\theta }^{(1)}\right)$ and $\left(u^{(2)},{\theta }^{(2)}\right)$ are smooth solutions of Equation (1), let smooth function $\eta (t)\in C^{\infty }\left([0,T]\right)$ such that $\eta (t)\equiv 1$ for $t\in [0,T_1]$ and $\eta (t)\equiv 0$ for $t\in [T_2,T]$, and define

$$u=\eta (t)u^{(1)}+(1-\eta (t))u^{(2)},\theta =\eta (t){\theta }^{(1)}+(1-\eta (t)){\theta }^{(2)}.$$

Let $\tilde{u}=u^{(1)}-u^{(2)}$, $\tilde{\theta }={\theta }^{(1)}-{\theta }^{(2)}$. Since ${\int }_{{\mathbb{T}}^3}\tilde{\theta }\mathrm{d}x=0$, we can take

$$R^u={\partial }_t\eta ·{\mathcal{R}}^u\tilde{u}-\eta (1-\eta )\left(\tilde{u}\otimes \tilde{u}\right),R^{\theta }={\partial }_t\eta ·{\mathcal{R}}^{\theta }\tilde{\theta }-\eta (1-\eta )\tilde{u}·\tilde{\theta }.$$

Since $\eta$ is 1 on $[0,T_1]$ and 0 on $[T_2,T]$, it follows that $\mathrm{supp}{R}^u\subseteq [T_1,T_2]$ and $\mathrm{supp}{R}^{\theta }\subseteq [T_1,T_2]$. Thus, $(u,\theta ,R^u,R^{\theta },p)$ satisfies Equation (4), where the pressure can be obtained by solving the Poisson Equation (5).

By Proposition 2, we take $p=1$, $1<q<{\displaystyle \frac{5}{4}}$, and we have $W_x^{k,q}\hookrightarrow L_x^{\infty }$ for $k=3$ and $m\ge 5$. If we set $\delta =\parallel (R^u,R^{\theta }){\parallel }_{L_t^1{L}_x^q}$, then there exists a glued solution $\left(\overline{u},\overline{\theta },\overline{R^u},\overline{R^{\theta }},\overline{p}\right)$ that also satisfies (4); then, we have

$${∥(\overline{R^u},\overline{R^{\theta }})∥}_{L_t^1{L}_x^q}\le C_q{∥(R^u,R^{\theta })∥}_{L_t^1{L}_x^q},\parallel \overline{u}{-u\parallel }_{L_{t,x}^{\infty }}+\parallel \overline{\theta }{-\theta \parallel }_{L_{t,x}^{\infty }}\le {\parallel (R^u,R^{\theta })\parallel }_{L_t^1{L}_x^q},$$

and the supports of $\overline{R^u}$ and $\overline{R^{\theta }}$ are contained in the union of no more than $C{\tau }^{-\epsilon }$ (where C is an absolute constant) intervals with a length of no more than $2\tau$, namely, $[t_i-\tau ,t_i+\tau ]$.

By Proposition 3, for any $1<s<2$, there exist smooth functions $(u_1,{\theta }_1,R_1^u,R_1^{\theta },p_1)$ satisfying (4), and

$$\parallel (R_1^u,R_1^{\theta }){\parallel }_{L_t^1{L}_x^q}\le {\displaystyle \frac{1}{2}}{\parallel (R^u,R^{\theta })\parallel }_{L_t^1{L}_x^q}.$$

By combining this with (5), we obtain

$$\parallel u_1{-u\parallel }_{L_{t,x}^2}+{\parallel {\theta }_1-\theta \parallel }_{L_{t,x}^2}\le C_{q}min\left\{\parallel (R^u,R^{\theta }){\parallel }_{L_t^1{L}_x^q},{\parallel (R^u,R^{\theta })\parallel }_{L_t^1{L}_x^q}^{{\displaystyle \frac{1}{2}}}\right\}$$

and

$$\parallel u_1{-u\parallel }_{L_t^s{L}_x^{\infty }}+\parallel {\theta }_1{-\theta \parallel }_{L_t^s{L}_x^{\infty }}\le 2{\parallel (R^u,R^{\theta })\parallel }_{L_t^1{L}_x^q}.$$

The temporal supports of $R_1^u$ and $R_1^{\theta }$ are contained in the union of no more than $C{\tau }^{-\epsilon }$ intervals with a length of no more than $4\tau$, namely, $[t_i-2\tau ,t_i+2\tau ]$. Therefore, outside these intervals, we have $u_1=\overline{u}$ and ${\theta }_1=\overline{\theta }$.

By iterating the above process, we can construct a sequence of smooth solutions $(u_n,{\theta }_n,R_n^u,R_n^{\theta },p_n)$ satisfying Equation (4), such that

$$\parallel (R_n^u,R_n^{\theta }){\parallel }_{L_t^1{L}_x^q}\le {\displaystyle \frac{1}{2^n}}{\parallel (R_0^u,R_0^{\theta })\parallel }_{L_t^1{L}_x^q},$$

$$\parallel u_{n+1}-u_n{\parallel }_{L_{t,x}^2}+{\parallel {\theta }_{n+1}-{\theta }_n\parallel }_{L_{t,x}^2}\le C_{q}min\left\{\parallel (R_n^u,R_n^{\theta }){\parallel }_{L_t^1{L}_x^q},{\parallel (R_n^u,R_n^{\theta })\parallel }_{L_t^1{L}_x^q}^{{\displaystyle \frac{1}{2}}}\right\},$$

and

$$\parallel u_{n+1}-u_n{\parallel }_{L_t^s{L}_x^{\infty }}+\parallel {\theta }_{n+1}-{\theta }_n{\parallel }_{L_t^s{L}_x^{\infty }}\le 2\parallel (R_n^u,R_n^{\theta }){\parallel }_{L_t^1{L}_x^q}\le {\displaystyle \frac{1}{2^{n-1}}}{\parallel (R_0^u,R_0^{\theta })\parallel }_{L_t^1{L}_x^q},$$

so that $R_n^u\to 0$ and $R_n^{\theta }\to 0$ in $L_t^1{L}_x^q$. The sequences $\left\{u_{n+1}-u_n\right\}$ and $\left\{{\theta }_{n+1}-{\theta }_n\right\}$ are absolutely convergent in $L_{t,x}^2\cap L_t^s{L}_x^{\infty }$; accordingly, there exists $u\in L_{t,x}^2\cap L_t^s{L}_x^{\infty }$ such that $u_n\to u$ and ${\theta }_n\to \theta$ in $L_{t,x}^2\cap L_t^s{L}_x^{\infty }$. Therefore, the pair $(u,\theta )$ is a weak solution of (1) under Definition 1.

Finally, since the number of intervals where $(u_n,{\theta }_n)$ differs from $(u,\theta )$ is at most $C{\tau }^{-\epsilon }$ and the length of each interval is at most $4\tau$, it is true outside these intervals that $(u(t),\theta (t))\equiv (u_n(t),{\theta }_n(t))\in C^{\infty }$. Therefore, the Hausdorff dimension of the temporal singular set $S_T$ of the weak solution $(u,\theta )$ of Equation (1) does not exceed its Minkowski dimension:

$${dim}_M(S_T)\le \underset{\tau \to 0}{lim\; sup}{\displaystyle \frac{N(S_T,\tau )}{log(1/\tau )}}\le \underset{\tau \to 0}{lim}{\displaystyle \frac{log(C{\tau }^{-\epsilon })}{-log(4\tau )}}=\epsilon ,$$

where $N(S_T,\tau )$ is the minimum number of intervals with a maximum length of $\tau$ needed to cover the set $S_T$.

6. Conclusions

In this article, we have established the non-uniqueness of very weak (or wild) solutions to the incompressible Boussinesq system on the three-dimensional periodic domain ${\mathbb{T}}^3$ for any positive viscosity $\nu >0$ and non-negative thermal diffusivity $\mu \ge 0$. The results extend the author’s previous work on the non-uniqueness of such solutions in high-dimensional cases.

The proofs are based on a convex integration scheme with a glued-step construction, employing intermittent Mikado flows as building blocks; this allows us to construct non-unique weak solutions in $L_{t,x}^2\cap L_t^s{L}_x^{\infty }$ for $1\le s<2$, which does not satisfy the Serrin criterion. As a result, the Hausdorff dimension of the temporal singular set is less than any prescribed $\epsilon >0$. The results also imply that the energy inequality is not satisfied, as the energy may be increased by connecting the weak solutions to one another using the main theorem, Theorem 1. In summary, this work confirms the non-uniqueness of weak solutions of the 3D Boussinesq system.

Despite these advances, several challenges remain open. For example, it is not known whether such non-unique solutions can be constructed within the Leray–Hopf class, such as in $L_t^{\infty }{L}_x^2\cap L_t^2{H}_x^1$, or particularly in situations where the Serrin regularity criterion does not hold. Furthermore, the physical interpretation of such wild solutions and their potential relevance to turbulent regimes deserve deeper exploration.