Global Regularity for the 2D Boussinesq Equations with Temperature-Dependent Viscosity

Abstract

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.

Appendix A. Besov spaces and some useful facts

This Appendix includes several parts. It recalls the Littlewood–Paley theory, introduces the Besov spaces, provides Bernstein inequalities as well as several facts used in the proof of our main result. We start with the Littlewood–Paley theory. We choose some smooth radial non increasing function $\chi$ with values in [0, 1] such that $\chi \in C_{0}^{\infty }({\mathbb {R}}^{n})$ is supported in the ball ${\mathcal {B}}:=\{\xi \in {\mathbb {R}}^{n}, |\xi |\le \frac{4}{3}\}$ and and with value 1 on $\{\xi \in {\mathbb {R}}^{n}, |\xi |\le \frac{3}{4}\}$, then we set $\varphi (\xi )=\chi \big (\frac{\xi }{2}\big )-\chi (\xi )$. One easily verifies that ${\varphi \in C_{0}^{\infty }({\mathbb {R}}^{n})}$ is supported in the annulus ${\mathcal {C}}:=\{\xi \in {\mathbb {R}}^{n}, \frac{3}{4}\le |\xi |\le \frac{8}{3}\}$ and satisfies

$$\begin{aligned} \chi (\xi )+\sum _{j\ge 0}\varphi (2^{-j}\xi )=1, \quad \forall \xi \in {\mathbb {R}}^{n}. \end{aligned}$$

Let $h={\mathcal {F}}^{-1}(\varphi )$ and ${\widetilde{h}}={\mathcal {F}}^{-1}(\chi )$, then we introduce the dyadic blocks $\Delta _{j}$ of our decomposition by setting

$$\begin{aligned} \Delta _{j}u= & {} 0,\ \ j\le -2; \ \ \ \ \ \Delta _{-1}u=\chi (D)u=\int _{{\mathbb {R}}^{n}}{{\widetilde{h}}(y)u(x-y)\,dy};\\ \Delta _{j}u= & {} \varphi (2^{-j}D)u=2^{jn}\int _{{\mathbb {R}}^{n}}{h(2^{j}y)u(x-y)\,dy},\ \ \forall j\in {\mathbb {N}}. \end{aligned}$$

We shall also denote

$$\begin{aligned} \ S_{j}u:=\sum _{-1\le k\le j-1} \Delta _{k}u,\qquad {\widetilde{\Delta }}_{j} u:=\Delta _{j-1}u+\Delta _{j}u+\Delta _{j+1}u. \end{aligned}$$

The nonhomogeneous Besov spaces are defined through the dyadic decomposition.

Definition A.1

Let $s\in {\mathbb {R}}, (p,r)\in [1,+\infty ]^{2}$. The nonhomogeneous Besov space $B_{p,r}^{s}$ is defined as a space of $f\in S'({\mathbb {R}}^{n})$ such that

$$\begin{aligned} B_{p,r}^{s}=\{f\in S'({\mathbb {R}}^{n}); \Vert f\Vert _{B_{p,r}^{s}}<\infty \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{B_{p,r}^{s}}=\left\{ \begin{aligned}&\Big (\sum _{j\ge -1}2^{jrs}\Vert \Delta _{j}f\Vert _{L^{p}}^{r}\Big )^{\frac{1}{r}}, \quad \ r<\infty ,\\&\sup _{j\ge -1} 2^{js}\Vert \Delta _{j}f\Vert _{L^{p}}, \quad \ r=\infty .\\ \end{aligned}\right. \end{aligned}$$

We shall also need the mixed space–time spaces

$$\begin{aligned} \Vert f\Vert _{L_{T}^{\rho }B_{p,r}^{s}}:= \Big \Vert (2^{js}\Vert \Delta _{j}f\Vert _{L^{p}})_{l_{j}^{r}}\Big \Vert _{L_{T}^{\rho }} \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{{\widetilde{L}}_{T}^{\rho }B_{p,r}^{s}}:= (2^{js}\Vert \Delta _{j}f\Vert _{L_{T}^{\rho }L^{p}})_{l_{j}^{r}}. \end{aligned}$$

The following links are direct consequence of the Minkowski inequality

$$\begin{aligned} L_{T}^{\rho }B_{p,r}^{s}\hookrightarrow {\widetilde{L}}_{T}^{\rho }B_{p,r}^{s},\qquad \text{ if }\,\,r\ge \rho , \quad \text{ and } \quad {\widetilde{L}}_{{T}}^{\rho }B_{p,r}^{s}\hookrightarrow {L}_{T}^{\rho }B_{p,r}^{s},\qquad \text{ if }\,\,\rho \ge r. \end{aligned}$$

In particular,

$$\begin{aligned} {\widetilde{L}}_{{T}}^{r}B_{p,r}^{s}\thickapprox {L}_{T}^{r}B_{p,r}^{s}. \end{aligned}$$

The following lemma provides Bernstein type inequalities for fractional derivatives

Lemma A.1

(see [3]). Assume $1\le a\le b\le \infty$. If the integer $j\ge -1$, then it holds

$$\begin{aligned} \Vert \Lambda ^{k}\Delta _{j}f\Vert _{L^b} \le C_1\, 2^{j k + jn(\frac{1}{a}-\frac{1}{b})} \Vert \Delta _{j}f\Vert _{L^a},\quad k\ge 0. \end{aligned}$$

If the integer $j\ge 0$, then we have

$$\begin{aligned} C_2\, 2^{ j k} \Vert \Delta _{j}f\Vert _{L^b } \le \Vert \Lambda ^{k}\Delta _{j}f\Vert _{L^b } \le C_3\, 2^{ j k + j n(\frac{1}{a}-\frac{1}{b})} \Vert \Delta _{j}f\Vert _{L^a},\quad k\in {\mathbb {R}}, \end{aligned}$$

where $C_1$, $C_2$ and $C_3$ are constants depending on k, a and b only.

Finally, we state the following fundamental commutator estimates which have been used repeatedly.

Lemma A.2

(See [3]). Let $\theta$ be a $C^{1}$ function on ${\mathbb {R}}^{n}$ such that $|x|{\check{\theta }}(x)\in L^{1}$. There exists a constant C such that for any Lipschitz function a with gradient in $L^{p}$ and any function b in $L^{q}$, we have, for any positive $\lambda$,

$$\begin{aligned} \Vert [\theta (\lambda ^{-1}D), a]b\Vert _{L^{r}}\le C\lambda ^{-1}\Vert \nabla a\Vert _{L^{p}}\Vert b\Vert _{L^{q}} \ \ with \ \ \frac{1}{p}+\frac{1}{q}=\frac{1}{r},\,\,(p,q)\in [1,\infty ]^{2}. \end{aligned}$$