On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces

Abstract

We study the existence, uniqueness as well as regularity issues for the two-dimensional incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces. The optimal regularity exponent ranges are considered.

1 Introduction

In the present paper, we consider the two-dimensional incompressible Boussinesq equations

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\theta + u\cdot \nabla _x \theta -{{\,\mathrm{div}\,}}_x(\kappa \nabla _x \theta )=0,\\&\partial _t u+u\cdot \nabla _x u-{{\,\mathrm{div}\,}}_x(\mu S_x u)+\nabla _x\Pi =\beta \theta \mathbf {e_2},\\&{{\,\mathrm{div}\,}}_x u=0, \end{aligned} \right. \end{aligned}$$

(1.1)

where $(t,x)\in [0,\infty )\times \mathbb {R}^2$ denote the time and space variables, respectively. The unknown temperature function $\theta =\theta (t,x):[0,\infty )\times \mathbb {R}^2\rightarrow \mathbb {R}$ satisfies the parabolic-type equation (1.1)$_1$, and the unknown velocity vector field $u=u(t,x): [0,\infty )\times \mathbb {R}^2\rightarrow \mathbb {R}^2$ together with the unknown pressure term $\Pi =\Pi (t,x):[0,\infty )\times \mathbb {R}^2\rightarrow \mathbb {R}$ satisfies the incompressible Navier–Stokes type equations (1.1)$_2$-(1.1)$_3$, respectively. We are going to study the well-posedness and regularity problems for the Boussinesq system (1.1) together with the initial data

$$\begin{aligned} (\theta ,u)\mid _{t=0}=(\theta _0,u_0). \end{aligned}$$

(1.2)

We write $x=\begin{pmatrix}x_1\\ x_2\end{pmatrix}\in \mathbb {R}^2$ with $x_1, x_2$ denoting the horizontal and vertical components, respectively. Let $u=\begin{pmatrix}u^1\\ u^2\end{pmatrix}: [0,\infty )\times \mathbb {R}^2\rightarrow \mathbb {R}^2$, and let

$$\begin{aligned} \frac{1}{2} S_xu:=\frac{1}{2} \big (\nabla _x u+(\nabla _x u)^T\big ),\hbox { with }\nabla _x u=\big (\partial _{x_j}u^i\big )_{1\le i,j\le 2} \end{aligned}$$

denote the symmetric deformation tensor in the second equation (1.1)$_2$ above. The vector field $\mathbf {e_2}$ denotes the unit vector in the vertical direction: $\mathbf {e_2}=\begin{pmatrix}0\\ 1\end{pmatrix}$, and $\beta \theta \mathbf {e_2}$ stands for the buoyancy force, with the constant parameter $\beta >0$ denoting the thermodynamic dilatation coefficient which will be assumed to be 1 in the following context for simplicity.

We consider the cases when the heat diffusion and the viscosity in the fluids are sensitive to the change of temperatures, that is, the thermal diffusivity $\kappa$ and the viscosity coefficient $\mu$ may depend on the temperature function $\theta$ as follows

$$\begin{aligned} \kappa =a(\theta ),\quad \mu =b(\theta ), \quad \hbox { with }\kappa _*\le a \le \kappa ^*,\,\, \mu _*\le b \le \mu ^*, \end{aligned}$$

(1.3)

where $\kappa _*\le \kappa ^*$, $\mu _*\le \mu ^*$ are positive constants. We will not assume any smallness conditions on $\kappa ^*-\kappa _*$ or $\mu ^*-\mu _*$, and large variations in these diffusivity coefficients are permitted.

The Boussinesq system (1.1) arises from the zero order approximation to the corresponding inhomogeneous hydrodynamic systems, which are nonlinear coupling between the Navier–Stokes equations or Euler equations and the thermodynamic equations for the temperature or density functions: The Boussinesq approximation [5] ignores density differences except when they appear in the buoyancy term. They are common geophysical models describing the dynamics from large-scale atmosphere and ocean flows to solar and plasma inner convection, where density stratification is a typical feature [22, 34].

The temperature or density differences in the inhomogeneous fluids may cause density gradients. When the thermodynamical coefficients such as the heat conducting coefficients and the viscosity coefficients are assumed to be constant in the Boussinesq approximation [i.e., $\kappa$, $\mu$ are constants in (1.1)], density gradients influence the motion of the flows only through the buoyancy force, which may lead to finite time singularity in the flows (the formation of the finite time singularity is sensitive to the thermal and viscous dissipation and see Sect. 1.1 below for more references on this topic).

However, the temperature variations do influence the thermal conductivity and the viscosity coefficients effectively, even for simple fluids such as pure water [32, Sect. 6].¹,² In many applications in the engineering, one also aims for effective thermal conductivities in building thermal energy storage materials [21]. Therefore in plenty of physical models, density gradients would influence the motion of the fluids not only through buoyancy force, but also through the variations of the diffusion coefficients. It is then interesting to study the well-posedness and regularity problems of the Boussinesq system (1.1)–(1.3).

1.1 Known results

The well-posedness and regularity problems on the two-dimensional Boussinesq equations have attracted considerable attention from the PDE community. Many interesting mathematical results have been established in the past two decades, mainly in the cases with constant thermal diffusivity coefficient $\kappa$ and viscosity coefficient $\mu$:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\theta + u\cdot \nabla _x \theta -\kappa \Delta _x \theta =0,\\&\partial _t u+u\cdot \nabla _x u-\mu \Delta _x u+\nabla _x\Pi = \theta \mathbf {e_2},\\&{{\,\mathrm{div}\,}}_x u=0,\\&(\theta ,u)\mid _{t=0}=(\theta _0,u_0). \end{aligned} \right. \end{aligned}$$

(1.4)

If $\kappa =\mu =0$, the two-dimensional inviscid Boussinesq equations (1.4) can be compared with the three-dimensional incompressible axisymmetric Euler equations with swirl, where the buoyancy force corresponds to the vortex stretching mechanism [35]. The local-in-time well-posedness as well as some blowup criteria have been well known for decades, see, e.g., [10, 11, 42]. We mention that an (improved) lower bound for the lifespan which tends to infinity as the initial temperature tends to a constant (and correspondingly, as the initial swirl tends to zero for the 3D axisymmetric Euler equations) was given in [11]. The fundamental global regularity problem for the 2D inviscid Boussinesq equations remains still open. Recently, an interesting example of finite-energy strong solutions with a finite weighted Hölder norm in a wedge-shaped domain, which become singular at the origin in finite time, has been given in [19] (see also an interesting example of solutions in Hölder-type spaces with finite-time singularity for 3D axisymmetric Euler equations in [18]).

If $\kappa >0$ and $\mu >0$ are positive constants, on the contrary, the convection terms can be controlled thanks to the strong diffusion effects, and the global-in-time existence and regularity results can be established (see, e.g., [7]). Particular interests then raised if only partial dissipation is present, that is, either $\kappa =0$ whereas $\mu >0$ or $\kappa >0$ whereas $\mu =0$ (see, e.g., H.K. Moffatt’s list of the twenty first century PDE problems [36]). The global-in-time results continue to hold, thanks to a priori estimates in the $L^p$-framework as well as the sharp Sobolev embedding inequality in dimension two with a logarithm correction, which help the partial diffusion terms to control the demanding term $\partial _{x_1}\theta$ successfully (see [9, 27] and see [24] for less regular cases). Further developments were made for horizontal dissipation cases (see, e.g., [13]), for vertical dissipation cases (see, e.g., [8]), and for the fractional dissipation cases (see, e.g., [25, 26]). See the review notes [43] and the references therein for more interesting results and sketchy proofs.

There also have been remarkable progresses in solving the two dimensional Boussinesq equations (1.1)–(1.3) when the thermal and viscosity diffusion coefficients $\kappa , \mu$ are variable and depend smoothly on the unknown temperature function $\theta$. In the variational formulation framework, the global-in-time existence of a solution of (1.1)–(1.3) has been established in [17] [see [20] for a similar formulation of (1.1)–(1.3)] for the motion of the so-called Bingham fluid (as a non-Newtonian fluid), where $\kappa$ is a positive constant, $\beta =0$ and $\mu$ depends not only on $\theta$ but also on Su/|Su|. The Boussinesq–Stefan model has been investigated in [38], where the phase transition was taken into account. The global-in-time existence and the uniqueness of the solutions for (1.1)–(1.3) have been shown in [15, 23, 33] under Dirichlet boundary conditions and in [37] under generalized outflow boundary conditions. We remark that the resolution of the nonhomogeneous Boussinesq system under more physical boundary conditions (e.g., with Dirichlet boundary conditions only on the inflow part of the boundary while with no prescribed assumptions on the outflow part) remains unsolved.

Lorca and Boldrini [33] (see also [15, 23]) studied the initial-boundary value problem of the Boussinesq system (1.1)–(1.3) in dimension two and three under the initial condition (1.2) and Dirichlet boundary conditions, and obtained a global-in-time weak solution

$$\begin{aligned} (\theta ,u)\in \big (L^\infty _\text {loc}\big ([0,\infty ); L^2(\Omega )\big ) \big )^3 \end{aligned}$$

as well as a local-in-time unique strong solution

$$\begin{aligned} (\theta ,u)\in L^\infty _\text {loc}\big ([0,\infty ); H^2(\Omega )\big )\times \big (L^\infty _\text {loc}([0,\infty ); H^{1}(\Omega )\big )^2. \end{aligned}$$

(1.5)

The remarkable global-in-time existence and uniqueness results of the smooth solutions

$$\begin{aligned} (\theta ,u)\in \big (L^\infty _\text {loc}\big ([0,\infty ); H^s(\mathbb {R}^2)\big )\cap L^2_\text {loc}\big ([0,\infty ); H^{s+1}(\mathbb {R}^2)\big )\big )^3,\quad s>2 \end{aligned}$$

(1.6)

have been successfully established by Wang and Zhang [41], which affirms the propagation of high regularities (without finite time singularity) of the two dimensional Boussinesq flow in the presence of viscosity variations (see [39] for the case $s=2$). We remark that the $L^2_x$-norm of the velocity vector field may grow in time due to the buoyancy forcing term, even provided with constant diffusion coefficients and smooth and fast decaying small initial data [6], and hence the norm with respect to the time variable in (1.5) and (1.6) is only locally in time.

It is still not clear whether there will be finite time singularity for the two-dimensional Boussinesq flow (1.1)–(1.3) in the presence of viscosity variations while no heat diffusion (i.e., $\kappa =0$, $\mu =b(\theta )$), and we mention a recent work [2] toward this direction in the case of less heat diffusion (with $\text {div}(\kappa \nabla \theta )$ replaced by $(-\Delta )^{1/2}$) and the small viscosity variation assumption: $|\mu -1|\le \varepsilon$. A closely related question would pertain to the global-in-time well-posedness problem of the two-dimensional inhomogeneous incompressible Navier–Stokes equations with density-dependent viscosity coefficient

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\rho +u\cdot \nabla _x\rho =0,\\&\partial _{t}(\rho u)+{\text {div}}_x(\rho u\otimes u)-{\text {div}}_x( \mu S_xu)+\nabla _x \Pi =0, \\&{\text {div}}_x u=0, \\&(\rho , \, \rho u)\big |_{t=0}=(\rho _{0}, m_{0}). \end{aligned} \right. \end{aligned}$$

(1.7)

The global-in-time existence results of weak solutions of (1.7) (see, e.g., [3, 31]) as well as the local-in-time well-posedness results (see, e.g., [28]) have been well known, while the global-in-time regularities still remain open (see, e.g., [1, 16] for some interesting results under the assumption on the weak inhomogeneity).

To the best of our knowledge, there are no global-in-time regularity propagation results by the two-dimensional Boussinesq flow with temperature-dependent diffusion coefficients (1.1)–(1.2)–(1.3) in the low regularity regime

$$\begin{aligned} H^s,\quad s<2, \end{aligned}$$

or in the general Sobolev setting

$$\begin{aligned} \theta _0\in H^{s_\theta }_x(\mathbb {R}^2),\quad u_0\in (H^{s_u}_x(\mathbb {R}^2))^2 \end{aligned}$$

with different regularity indices $s_\theta$ and $s_u$. In this paper, we are going to investigate the existence, uniqueness as well as the regularity problems in these general Sobolev functional settings.

To conclude this subsection, let us just mention some recent interesting progresses on the stability of the stationary shear flow solutions (together with the corresponding striated temperature function) to the Boussinesq equations (1.4), with full dissipation or partial dissipation, in, e.g., [14, 40, 44] and references therein. It should also be interesting to investigate the stability of the stationary striated solutions of the Boussinesq equations with variable diffusion coefficients (1.1). We mention a recent work in this direction on the incompressible Navier–Stokes equations with constant density function but with variable viscosity coefficient [30].

1.2 Main results

We are going to show the global-in-time existence of weak solutions to the Cauchy problem for the Boussinesq system (1.1)–(1.2)–(1.3) in the whole two-dimensional space $\mathbb {R}^2$ under the low-regularity initial condition $(\theta _0, u_0)\in L^2(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2.$ The uniqueness result holds true if the initial temperature function becomes smoother $(\theta _0, u_0)\in H^1(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2.$ Finally, we will establish the global-in-time regularity of the solutions in the general Sobolev setting $(\theta _0, u_0)\in H^{s_\theta }(\mathbb {R}^2)\times (H^{s_u}(\mathbb {R}^2))^2\subset H^1(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$ with the restriction $s_u-1\le s_\theta \le s_u+2$. These regularity exponent ranges are optimal for the existence, uniqueness and regularity results, respectively, by view of the formulations of the Boussinesq equations (1.1) with temperature-dependent diffusion coefficients (see Remark 1.3 below for more details).

We first define the weak solutions as follows.

Definition 1.1

(Weak solutions) We say that a pair $(\theta ,u)$ is a weak solution of the Boussinesq equations (1.1)–(1.3) with the given initial data $(\theta _0,u_0)\in (L^2(\mathbb {R}^2))^3$ if the following statements hold:

• The temperature function

  $$\begin{aligned} \theta =\theta (t,x)\in C\big ([0,\infty ); L^2_x(\mathbb {R}^2)\big )\cap L^2_\text {loc}\big ([0,\infty ); H^1_x(\mathbb {R}^2)\big ) \end{aligned}$$

  satisfies the initial condition $\theta |_{t=0}=\theta _0$, the energy equality

  $$\begin{aligned} \frac{1}{2}\Vert \theta (T,\cdot )\Vert _{L^2_x(\mathbb {R}^2)}^2+\int \limits ^T_0\int \limits _{\mathbb {R}^2} (\kappa |\nabla \theta |^2)(t, x) \mathrm{d}x{\mathrm {dt}} =\frac{1}{2}\Vert \theta (0,\cdot )\Vert _{L^2_x(\mathbb {R}^2)}^2, \end{aligned}$$

  (1.8)

  for all positive times $T>0$, and the equation

  $$\begin{aligned} \partial _t\theta +u\cdot \nabla \theta -\text {div}_x(\kappa \nabla \theta )=0 \end{aligned}$$

  (1.9)

  in $L^2_\text {loc}([0,\infty ); H^{-1}_x(\mathbb {R}^2))$.

• The velocity vector field

  $$\begin{aligned} u=u(t,x)\in C\big ([0,\infty ); \big (L^2_x(\mathbb {R}^2)\big )^2\big )\cap L^2_\text {loc}\big ([0,\infty ); \big (H^1_x(\mathbb {R}^2)\big )^2\big ) \end{aligned}$$

  satisfies the initial condition $u|_{t=0}=u_0$, the divergence-free condition $\text {div}_x u=0$, the energy equality

  $$\begin{aligned} \begin{aligned}&\frac{1}{2}\Vert u(T,\cdot )\Vert _{L^2_x(\mathbb {R}^2)}^2+\frac{1}{2}\int \limits ^T_0\int \limits _{\mathbb {R}^2} (\mu |S u|^2)(t, x)\mathrm{d}x{\mathrm {dt}} \\&=\frac{1}{2}\Vert u(0,\cdot )\Vert _{L^2_x(\mathbb {R}^2)}^2+\int \limits ^T_0\int \limits _{\mathbb {R}^2} (\theta u_2)(t,x) \mathrm{d}x{\mathrm {dt}} ,\quad \forall T>0, \end{aligned} \end{aligned}$$

  (1.10)

  and the equation

  $$\begin{aligned} \partial _t u+u\cdot \nabla _x u-{{\,\mathrm{div}\,}}_x(\mu S_x u)+\nabla _x\Pi = \theta \mathbf {e_2} \end{aligned}$$

  (1.11)

  in $L^2_\text {loc}([0,\infty ); (H^{-1}_x(\mathbb {R}^2))^2 )$ for some scalar function $\Pi \in L^2_\text {loc}([0,\infty )\times \mathbb {R}^2)$ with $\nabla \Pi \in L^2_\text {loc}([0,\infty ); (H^{-1}_x(\mathbb {R}^2))^2 )$ and $\int \limits _{B_1}\Pi \mathrm{d}x=0$ a.e. t (with $B_1$ denoting the unit disk in $\mathbb {R}^2$).

For any fixed $T>0$, $p\ge 1$, $q\ge 1$, $s\ge 0$ and for any fixed (vector-valued) function $f: [0,T]\times \mathbb {R}^2\mapsto \mathbb {R}^m$, $m\ge 1$, we denote

$$\begin{aligned} \Vert f\Vert _{L^p_T X_x}:=\big \Vert \Vert f(t)\Vert _{X_x(\mathbb {R}^2; \mathbb {R}^m)}\big \Vert _{L^p_t([0,T])} \hbox { with }X= H^s\hbox { or }L^q. \end{aligned}$$

(1.12)

The functional space $L^p([0,T];H^s(\mathbb {R}^2; \mathbb {R}^m))$ consists of all functions $f:[0,\infty )\times \mathbb {R}^2\rightarrow \mathbb {R}^m$ satisfying $\Vert f\Vert _{L^p_TH^s_x}<\infty$. We have the following existence, uniqueness as well as global-in-time regularity results for the solutions of the Cauchy problem for the Boussinesq equations (1.1)-(1.2)-(1.3) on the whole two-dimensional space $\mathbb {R}^2$.

Theorem 1.2

(Existence, uniqueness and global-in-time regularity) For any initial data $\theta _0\in L^2(\mathbb {R}^2)$ and $u_0\in (L^2(\mathbb {R}^2))^2$, there exists a global-in-time weak solution

$$\begin{aligned} (\theta ,u)\in C([0,\infty ); (L^2(\mathbb {R}^2))^3)\cap L^2_\text {loc}([0,\infty ); (H^1(\mathbb {R}^2))^3) \end{aligned}$$

of the initial value problem (1.1)–(1.2)–(1.3).

If $\theta _0\in H^1(\mathbb {R}^2)$, $u_0\in (L^2(\mathbb {R}^2))^2$ and the functions $a\in C^2_b (\mathbb {R}; [\kappa _*, \kappa ^*])$, $b\in C^2_b (\mathbb {R}; [\mu _*, \mu ^*])$ have finite first and second derivatives, then the weak solution is indeed unique, and satisfies

$$\begin{aligned}&\theta \in C\big ([0,\infty );H^1(\mathbb {R}^2)\big )\cap L^2_\text {loc}\big ([0,\infty );H^2(\mathbb {R}^2)\big ), \end{aligned}$$

as well as the following energy estimates for any given $T>0$,

$$\begin{aligned}&\Vert u\Vert ^2_{L^\infty _TL^2_x}+\Vert \nabla u\Vert ^2_{L^2_T L^2_x}\le C\big (T\Vert \theta _0\Vert ^2_{L^2}+\Vert u_0\Vert ^2_{L^2}\big ), \end{aligned}$$

(1.13)

$$\begin{aligned}&\Vert \theta \Vert _{L^\infty _TH^1_x}^2+ \Vert (\partial _t \theta , \nabla ^2\theta )\Vert _{L^2_TL^2_x}^2 \nonumber \\&\le C\Vert \theta _0\Vert _{H^1}^2(1+\Vert \nabla \theta _0\Vert _{L^2}^2)\exp \big (C(T^2\Vert \theta _0\Vert ^4_{L^2}+\Vert u_0\Vert ^4_{L^2})\big ), \end{aligned}$$

(1.14)

where C is a positive constant depending only on $\Vert a\Vert _\mathrm {Lip}, \kappa _*, \kappa ^*, \mu _*$.

Furthermore, the general $H^s$-regularities can be propagated globally in time in the following sense: For any initial data (see the gray closed unbounded quadrangle in Fig. 1 for the admissible regularity exponent range)

$$\begin{aligned} \begin{aligned}&(\theta _0, u_0)\in H^{s_\theta }(\mathbb {R}^2)\times (H^{s_u}(\mathbb {R}^2))^2 \\&\hbox { with }(s_\theta , s_u)\in \{(s_\theta , s_u)\subset [1,\infty )\times [0,\infty )\,|\, s_u-1\le s_\theta \le s_u+2\} \end{aligned}\end{aligned}$$

(1.15)

and the functions $a\in C^2_b\cap C^{[s_\theta ]+1}$, $b\in C^2_b\cap C^{[s_u]+1}$, the unique solution $(\theta , u)$ stays in

$$\begin{aligned} C\big ([0,\infty );H^{s_\theta }(\mathbb {R}^2)\times (H^{s_u}(\mathbb {R}^2))^2\big )\cap L^2_\text {loc}\big ([0,\infty );H^{s_\theta +1}(\mathbb {R}^2)\times (H^{s_u+1}(\mathbb {R}^2))^2\big ). \end{aligned}$$

(1.16)

Fig. 1

Admissible regularity exponents

Theorem 1.2 will be proved in Sect. 2. The proof of the existence of weak solutions is rather standard, and we are going to sketch the proof in Sect. 2.1 for the reason of completeness, as we did not find the proof in the literature. As mentioned before, some well-posedness results have already been established for smooth data in the bounded domain case (see (1.5) above in, e.g., [15, 17, 23, 33]) or in smoother functional frameworks in the whole space case (see (1.6) above in, e.g., [41]). We are going to focus on the proofs of the uniqueness result and the global-in-time regularity result (in the low regularity regimes) in Sects. 2.2 and 2.3, respectively, where different regularity exponents for different unknowns are permitted. The commutator estimates as well as the composition estimates in Lemma 2.1 will play an important role, and the a priori estimates for a general linear parabolic equation in Lemma 2.2 will be of independent interest.

We conclude this introduction part with several remarks on the results in Theorem 1.2.

Remark 1.3

(Optimality of the regularity exponent ranges in Theorem 1.2) We are going to follow the standard procedure to show the existence of weak solutions for $L^2$-initial data by use of the a priori energy (in)equalities (1.8) and (1.10) (see Sect. 2.1 below).

If we take the difference between two different weak solutions $(\theta _1, u_1)$ and $(\theta _2, u_2)$, the difference of the nonlinear viscosity term $\text {div}(\mu Su)$ in the u-equation will become

$$\begin{aligned} \text {div}((\mu _1-\mu _2) Su_1)+\text {div}(\mu _2 S(u_1-u_2)), \end{aligned}$$

which stays in $L^2_\text {loc}([0,\infty ); (H^{-1}(\mathbb {R}^2))^2)$ provided with

$$\begin{aligned}&u_1\hbox { and }(u_1-u_2)\in L^2_\text {loc}([0,\infty ); (H^{1}(\mathbb {R}^2))^2), \\&\hbox {and }\mu _1-\mu _2\in L^\infty _\text {loc}\bigl ([0,\infty ); H^1(\mathbb {R}^2)\bigr )\subset L^\infty _\text {loc}\bigl ([0,\infty ); (L^1(\mathbb {R}^2))'=\hbox {BMO}(\mathbb {R}^2)\bigr ). \end{aligned}$$

Therefore in order to ensure the $L^2_x$-Estimate for the velocity difference $(u_1-u_2)$, we require the $H^1_x$-Estimate for the temperature difference in $(\mu _1-\mu _2)$. And hence the initial condition $\theta _0\in H^1_x$ (i.e., $s_\theta \ge 1$ above) is required for the proof of the uniqueness result (see Sect. 2.2 below).

Under the lower-regularity assumption $\theta _0\in H^s_x$ with $0<s<1$, the coefficients $\kappa , \mu$ are not expected to be continuous uniformly in time, and hence no uniqueness or $H^s$-regularity results for $\theta$ or $H^{s_1}$, $s_1>0$-regularity results for u are expected. Nevertheless with constant diffusion coefficients (e.g., $\kappa =\mu =1$), the uniqueness result for the weak solutions holds true by virtue of the $L^2_x$-energy (in)equalities (similar as the classical global-in-time well-posedness result for the classical two dimensional incompressible Navier–Stokes equations). Furthermore, if $\kappa =1$ is a positive constant, then the $H^s_x$, $s\in (0,1)$-Estimate for $\theta$ holds true, provided with $u\in L^4_\text {loc}(L^4_x(\mathbb {R}^2))^2$ (or with $u_0\in (L^2(\mathbb {R}^2))^2$), simply by an interpolation argument between (1.8) and (1.14). Similarly if $\mu =1$ is a positive constant, then the $H^s_x$, $s>0$-Estimate for u holds true, provided with $\theta \in L^2_\text {loc}(H^{s-1}_x(\mathbb {R}^2))$. Thus with constant diffusion coefficients (e.g., $\kappa =\mu =1$), the Sobolev regularities

$$\begin{aligned} (\theta _0, u_0)\in (H^s(\mathbb {R}^2))\times (L^2(\mathbb {R}^2))^2\hbox { or }(L^2(\mathbb {R}^2))\times (H^s(\mathbb {R}^2))^2,\quad 0<s\le 1 \end{aligned}$$

can be propagated globally in time, and the admissible regularity exponent set (1.15) extends itself indeed to the closed set consisting of all nonnegative admissible regularity exponents:

$$\begin{aligned}\begin{aligned} (s_\theta , s_u)\in \{(s_\theta , s_u)\subset [0,\infty )\times [0,\infty )\,|\, s_u-1\le s_\theta \le s_u+2\}. \end{aligned}\end{aligned}$$

In order to propagate the $H^{s_\theta }$, $s_\theta \ge 2$-regularity of $\theta$, we require the transport term $u\cdot \nabla \theta$ in the $\theta$-equation to be at least in $L^2_\text {loc}([0,\infty ); H^{s_\theta -1}_x)$, which requires $u\in L^2_\text {loc}([0,\infty ); H^{s_\theta -1}_x)$ and hence the initial assumption $u_0\in H^{s_u}$ with the restriction $s_u\ge s_\theta -2$ (as there is a gain of regularity of oder 1 when taking $L^2$-norm in the time variable in general). Similarly, in order the propagate the $H^{s_u}$, $s_u\ge 2$-regularity of u, we require the viscosity term $\text {div}(\mu Su)$ in the u-equation to be at least in $L^2_\text {loc}([0,\infty ); H^{s_u-1}_x)$, which requires $\mu Su\in L^2_\text {loc}([0,\infty ); H^{s_u}_x)$ and hence the initial assumption $\theta _0\in H^{s_\theta }$ with the restriction $s_\theta \ge s_u-1$.

Remark 1.4

(Precise $H^s_x$-Estimates in the high regularity regime) The global-in-time regularity in the high regularity regime (1.15)–(1.16) follows immediately from the following borderline a priori estimates:

• If $\theta _0\in H^{s}(\mathbb {R}^2)$, $u_0\in (L^2(\mathbb {R}^2))^2$ with $s\in (1,2]$ and the function $a\in C^2_b(\mathbb {R})$, then for $s\in (1,2)$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert \theta \Vert ^2_{ L^\infty _T H^{s}_x }+ \Vert \nabla \theta \Vert ^2_{ L^2_T H^{s}_x }\le C(\kappa _*) \Vert \theta _0\Vert _{H^s_x}^2\times \\&\quad \times \exp \Bigl (C(\kappa _*, s, \Vert a\Vert _{C^2}, \Vert \theta \Vert _{L^\infty _TH^1_x})\bigl (\Vert \nabla u\Vert _{L^2_T L^2_x}^2+ \Vert \nabla \theta \Vert _{L^2_T H^1_x}^2\bigr )\Bigr ), \end{aligned} \end{aligned}$$

  (1.17)

  and for $s=2$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert \theta \Vert _{L^\infty _T H^2_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^2_x}^2 \le C(\kappa _*,\Vert a\Vert _{C^2}, \kappa ^*) \Vert \theta _0\Vert ^2_{H^2}(1+\Vert \nabla \theta _0\Vert ^2_{L^2})^2 \\&\quad \times \exp \Bigl (C(\kappa _*,\Vert a\Vert _{\mathrm {Lip}})(\Vert \nabla u\Vert _{L^2_T L^2_x}^2+\Vert u\Vert _{L^4_T L^4_x}^4+\Vert \nabla \theta \Vert _{L^4_T L^4_x}^4)\Bigr ) . \end{aligned} \end{aligned}$$

  (1.18)

• If $\theta _0\in H^1(\mathbb {R}^2)$, $u_0\in (H^{s}(\mathbb {R}^2))^2$ with $s\in (0,2]$ and the function $b\in C^2_b(\mathbb {R})$, then for $s\in (0,2)$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert u\Vert _{L^\infty _T H^s_x}^2+\Vert \nabla u\Vert _{L^2_T H^{s}_x}^2 \le C(\mu _*) ( \Vert u_0\Vert _{H^s_x}^2+T\Vert \theta _0\Vert _{ L^2_x}^2+\Vert \theta \Vert _{L^2_T H^{s-1}_x}^2) \\&\quad \times \exp \Bigl (C(\mu _*,s, \Vert b\Vert _{C^2}, \Vert \theta \Vert _{L^\infty _T H^1_x})\bigl (\Vert \nabla u\Vert _{L^2_T L^2_x}^2+ \Vert \nabla \theta \Vert _{L^2_T H^1_x}^2\bigr )\Bigr ), \end{aligned} \end{aligned}$$

  (1.19)

  and for $s=2$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert u\Vert _{L^\infty _T H^2_x}^2+\Vert \nabla u\Vert _{L^2_T H^2_x}^2 \le (\Vert u\Vert _{L^\infty _T H^1_x}^2+\Vert \nabla u\Vert _{L^2_T H^1_x}^2) +C(\mu _*, \Vert b\Vert _{C^2})\times \\&\quad (1+ \Vert \Delta u_0\Vert _{L^2_x}^2) (1+\Vert u\Vert _{L^\infty _T H^1_x}^2+\Vert \nabla u\Vert _{L^2_T H^1_x}^2)^2 (1+\Vert \theta \Vert _{L^\infty _T H^1_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^1_x}^2) \\&\quad \times \exp \Bigl (C(\mu _*, \Vert b\Vert _{C^2})(\Vert (u, \nabla \theta )\Vert _{L^4_T L^4_x}^4+\Vert \nabla ^2\theta \Vert _{L^2_T L^2_x}^2)\Bigr ). \end{aligned}\end{aligned}$$

  (1.20)

• If $\theta _0\in H^{s}(\mathbb {R}^2)$, $u_0\in (H^{s-2}(\mathbb {R}^2))^2$ with $s>2$ and the function $a\in C^{[s]+1}$, then for $s\in (2,3)$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert \theta \Vert _{L^\infty _T H^s_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^{s}_x}^2 \le C(\kappa _*)\Vert \theta _0\Vert _{H^s_x}^2\times \\&\quad \times \exp \Bigl (C(\kappa _*,s, a, \Vert \theta \Vert _{L^\infty _T L^\infty _x})(\Vert u\Vert _{L^2_T H^{s-1}_x}^2+ \Vert \nabla \theta \Vert _{L^2_T L^\infty _x}^2)\Bigr ), \end{aligned}\end{aligned}$$

  (1.21)

  and for $s\ge 3$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert \theta \Vert _{L^\infty _T H^s_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^{s}_x}^2 \le C(\kappa _*,s) (\Vert \theta _0\Vert _{H^s_x}^2+\Vert \nabla \theta \Vert _{L^\infty _T L^\infty _x}^2\Vert \nabla u\Vert _{L^2_T H^{s-2}_x}^2) \\&\quad \times \exp (C(\kappa _*,s, a, \Vert \theta \Vert _{L^\infty _T L^\infty _x})(\Vert \nabla u\Vert _{L^2_T L^\infty _x}^2+ \Vert \nabla \theta \Vert _{L^2_T L^\infty _x}^2)). \end{aligned}\end{aligned}$$

  (1.22)

• If $\theta _0\in H^{s-1}(\mathbb {R}^2)$, $u_0\in (H^{s}(\mathbb {R}^2))^2$ with $s>2$ and the function $b\in C^{[s]+1}$, then for $s\in (2,3)$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert u\Vert _{L^\infty _T H^s_x}^2+\Vert \nabla u\Vert _{L^2_T H^{s}_x}^2 \le C(\mu _*) ( \Vert u_0\Vert _{H^s_x}^2+T\Vert \theta \Vert _{L^\infty _T H^{s-1}_x}^2)\\&\quad \times \exp (C(\mu _*, s, b, \Vert \theta \Vert _{L^\infty _TL^\infty _x})(\Vert \nabla u\Vert _{L^2_T H^1_x}^2+ \Vert \nabla \theta \Vert _{L^2_T H^{s-1}_x}^2)), \end{aligned}\end{aligned}$$

  (1.23)

  and for $s\ge 3$ it holds

  $$\begin{aligned} \begin{aligned}&\Vert u\Vert _{L^\infty _T H^s_x}^2+\Vert \nabla u\Vert _{L^2_T H^{s}_x}^2 \\&\quad \le C(\mu _*) (\Vert u_0\Vert _{H^s_x}^2+T\Vert \theta \Vert _{L^\infty _T H^{s-1}_x}^2 +\Vert \nabla u\Vert _{L^\infty _T L^\infty _x}^2\Vert \nabla \theta \Vert _{L^2_T H^{s-1}_x}^2) \times \\&\qquad \times \exp ( C(\mu _*, s, b, \Vert \theta \Vert _{L^\infty _TL^\infty _x})(\Vert \nabla u\Vert _{L^2_T L^\infty _x}^2+ \Vert \nabla \theta \Vert _{L^2_T L^\infty _x}^2)). \end{aligned}\end{aligned}$$

  (1.24)

We are going to prove the above borderline estimates one by one in Sect. 2.3 below.

Remark 1.5

($L^2$-in time Estimate V.S. $L^1$-in time Estimate) Instead of the classical $L^\infty _t H^s_x\cap L^1_t H^{s+2}_x$-type estimate in the literature, we derive $L^\infty _t H^s_x\cap L^2_t H^{s+1}_x$-type estimate here, since, e.g., only the $L^2_t \dot{H}^1_x$-a priori estimate for the velocity vector field is available from the energy estimates (roughly speaking, the $L^2_t$-in time norm asks less spatial regularity on the coefficients). See Lemma 2.2 below for the a priori $H^s_x$, $s\in (0,2)$-Estimates for a general linear parabolic equation with divergence-free $L^2_t H^1_x$-velocity vector field, which is of independent interest.

It is in general not true that $\theta \in L^1_t H^{s+2}_x$ (or $u\in L^1_t H^{s+2}_x$) in the low regularity regime, although it holds straightforward in the high regularity regime.

Remark 1.6

(Remarks on the smoothness assumptions on the functions a, b) It is common to assume smooth heat conductivity law and viscosity law [37, I] in fluid models.

The Lipschitz continuity assumption $a,b\in \mathrm {Lip}$ is enough for the $H^1\times L^2$-Estimates (1.13)–(1.14) in Theorem 1.2. As for the uniqueness result, due to the following $\dot{H}^1_x$-Estimate for the difference of the diffusion coefficinets

$$\begin{aligned} \Vert \nabla ( a(\theta _1)-a(\theta _2))\Vert _{L^2_x}&\le \Vert (a'(\theta _1)-a'(\theta _2))\nabla \theta _1\Vert _{L^2_x}+\Vert a(\theta _2)\nabla (\theta _1-\theta _2)\Vert _{L^2_x} \\&\quad \le \Vert a'\Vert _{\mathrm {Lip}} \Vert \nabla \theta _1\Vert _{L^4_x} \Vert \theta _1-\theta _2\Vert _{L^4_x}+\Vert a\Vert _{L^\infty } \Vert \nabla (\theta _1-\theta _2)\Vert _{L^2_x}, \end{aligned}$$

the Lipschitz continuity assumptions $a',b'\in \mathrm {Lip}$ are required.

The dependence on the function a of the constants C in (1.21)–(1.22) reads precisely as [similarly for the constants in (1.23)–(1.24)]

$$\begin{aligned} \sup _{k=0,\cdots , [s]+1}\,\sup _{|y|\le c\Vert \theta \Vert _{L^\infty _T L^\infty _x}} \bigl | \frac{d}{dy^k} a(y)\bigr |, \end{aligned}$$

and hence only $a\in C^{[s]+1}$ (instead of $a\in C^{[s]+1}_b$) is required.

For the integer regularity exponents, we can simply derive the energy estimates by integration by parts (instead of the application of the commutator estimates or the composition estimates in Lemma 2.1 below), such that the requirement for $a\in C^{[s_\theta ]+1}$ and $b\in C^{[s_u]+1}$ can be relaxed, see, e.g., (1.18), (1.20).

2 Proofs

Recall the Cauchy problem for the two-dimensional Boussinesq equations (1.1)–(1.3)

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\theta + u\cdot \nabla \theta -{{\,\mathrm{div}\,}}(\kappa \nabla \theta )=0,\\&\partial _t u+u\cdot \nabla u-{{\,\mathrm{div}\,}}(\mu Su)+\nabla \Pi = \theta \mathbf {e_2},\\&{{\,\mathrm{div}\,}}u=0,\\&(\theta ,u)\mid _{t=0}=(\theta _0,u_0) , \end{aligned} \right. \end{aligned}$$

(2.1)

where $\kappa =a(\theta )\in L^\infty (\mathbb {R}; [\kappa _*, \kappa ^*])$, $\mu =b(\theta )\in L^\infty (\mathbb {R}; [\mu _*, \mu ^*])$ with $\kappa _*, \kappa ^*, \mu _*, \mu ^*$ four positive constants.

We are going to show the existence, uniqueness as well as the global-in-time regularity results in Theorem 1.2 in Sects. 2.1, 2.2 and 2.3, respectively.

Recall the definition of the $\Vert \cdot \Vert _{L^q_T X_x}$-norm in (1.12). The Gagliardo–Nirenberg’s inequality

$$\begin{aligned} \Vert f\Vert _{L^4_TL^4_x(\mathbb {R}^2)}\le C\Vert f\Vert _{L^2_TL^2_x(\mathbb {R}^2)}^{\frac{1}{2}}\Vert \nabla f\Vert _{L^2_TL^2_x(\mathbb {R}^2)}^{\frac{1}{2}} \end{aligned}$$

(2.2)

as well as the equivalence relations between the norms

$$\begin{aligned} \begin{aligned}&\Vert Su\Vert _{L^2_x(\mathbb {R}^2)}^2=2\Vert \nabla u\Vert _{L^2_x(\mathbb {R}^2)}^2\hbox { if }\text {div}u=0, \\&\Vert \Delta \eta \Vert _{L^2_x(\mathbb {R}^2)}\sim \Vert \nabla ^2\eta \Vert _{L^2_x(\mathbb {R}^2)} \end{aligned}\end{aligned}$$

(2.3)

will be used freely in the proof.

2.1 Existence of weak solutions if $(\theta _0,u_0) \in (L^2(\mathbb {R}^2))^3$

We will follow the standard procedure to show the existence of the weak solutions under the initial condition

$$\begin{aligned} (\theta _0,u_0) \in L^2(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2, \end{aligned}$$

namely

Step 1:

We construct a sequence of approximate solutions, which satisfy the energy estimates uniformly.

Step 2:

We show the convergence of this approximate solution sequence to a weak solution and study the property of the weak solution.

We are going to sketch the proof and pay attention to the low-regularity assumptions.

Step 1: Construction of approximate solutions with uniform bounds

We use the Friedrich’s method to construct a sequence of approximate solutions. We consider the following system of $(\theta _n,u_n)$

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\theta _n+ P_n( u_n \cdot \nabla \theta _n)-P_n{{\,\mathrm{div}\,}}(\kappa _n \nabla \theta _n)=0,\\&\partial _t u_n+P_n \mathbb {P}( u_n\cdot \nabla u_n)-P_n \mathbb {P}{{\,\mathrm{div}\,}}(\mu _n Su_n)= \mathbb {P}( \theta _n \mathbf {e_2}),\\&u_n(0,x)=P_nu_0(x),\quad \theta _n(0,x)=P_n\theta _0(x), \end{aligned} \right. \end{aligned}$$

(2.4)

where $\kappa _n=a(\theta _n)$ and $\mu _n=b(\theta _n)$. The operator $P_n$, $n\in \mathbb {N}$, is the low-frequency cutoff operator which is defined as follows

$$\begin{aligned} P_nf(x)={\mathcal {F}}^{-1}(\mathbb {1}_{B_n}(\xi )\mathcal {F} f(\xi ))(x), \end{aligned}$$

where $B_n\subset \mathbb {R}^2$ is the disk with center at 0 and radius n, and ${\mathcal {F}}, {\mathcal {F}}^{-1}$ are the standard Fourier and inverse Fourier transformations. The operator $\mathbb {P}$ in (2.4) denotes the Leray–Helmholtz projector on $\mathbb {R}^2$, which decomposes the tempered distributions $v\in \mathcal {S}'(\mathbb {R}^2; \mathbb {R}^2)$ into div-free and curl-free parts as follows

$$\begin{aligned} v=\nabla ^\perp V_1+\nabla V_2, \end{aligned}$$

(2.5)

where

$$\begin{aligned} \nabla ^\perp V_1=-\nabla ^\perp (-\Delta )^{-1}\nabla ^\perp \cdot v=:\mathbb {P}v, \quad \nabla V_2=-\nabla (-\Delta )^{-1}\nabla \cdot v= (1-\mathbb {P})v \end{aligned}$$

with $\nabla ^\perp =(\partial _{x_2}, \, -\partial _{x_1})^T$. Notice that $\mathbb {P}$ maps $L^p(\mathbb {R}^2; \mathbb {R}^2)$ into itself for any $p\in (1,\infty )$ and it is commutative with the projection operator $P_n$.

We define the Banach spaces $L^2_n$ and $L^{2,\sigma }_n$ as following

$$\begin{aligned} L^2_n(\mathbb {R}^2)&=\{f\in L^2(\mathbb {R}^2)\mid f=P_nf\},\\ L^{2,\sigma }_n(\mathbb {R}^2)&=\{f\in (L^2_n(\mathbb {R}^2))^2\mid \text {div}_x(f)=0\}. \end{aligned}$$

The system (2.4) turns out to be an ordinary differential equation system in $L^2_n(\mathbb {R}^2)\times L^{2,\sigma }_n(\mathbb {R}^2)$. Indeed, the following estimates hold

$$\begin{aligned}&\Vert P_n( u_n \cdot \nabla \theta _n)-P_n{{\,\mathrm{div}\,}}(\kappa _n \nabla \theta _n)\Vert _{L^2_x} \le C n^3 (\Vert u_n\Vert _{L^2_x} +\kappa ^*)\Vert \theta _n\Vert _{L^2_x},\\&\Vert P_n \mathbb {P}( u_n\cdot \nabla u_n)-P_n \mathbb {P}{{\,\mathrm{div}\,}}(\mu _n Su_n)\Vert _{L^2_x} \le C n^3 (\Vert u_n\Vert _{L^2_x} +\mu ^*)\Vert u_n\Vert _{L^2_x}. \end{aligned}$$

Hence, for any $n\in \mathbb {N}$, there exists $T_n>0$ such that the system (2.4) has a solution $(\theta _n,u_n)\in C ([0,T_n];L_n^2(\mathbb {R}^2))\times C ([0,T_n];L_n^{2,\sigma }(\mathbb {R}^2))$.

We take the $L^2(\mathbb {R}^2)$-inner product of Eq. (2.4)$_1$ and $\theta _n$ to derive

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int \limits _{\mathbb {R}^2} \theta _n^2+\int \limits _{\mathbb {R}^2}\kappa _n |\nabla \theta _n|^2=0. \end{aligned}$$

Then, the following uniform estimate for $(\theta _n)$ holds

$$\begin{aligned} \frac{1}{2}\Vert \theta _n\Vert _{L^\infty _T L^2_x}^2+\kappa _*\Vert \nabla \theta _n\Vert _{L^2_TL^2_x}^2\,dt \le \frac{1}{2} \Vert P_n\theta _0\Vert _{L^2_x}^2\le \frac{1}{2} \Vert \theta _0\Vert _{L^2_x}^2,\quad \forall T>0. \end{aligned}$$

(2.6)

Similarly, we take the $L^2(\mathbb {R}^2)$-inner product of Eq. (2.4)$_2$ and $u_n$ to derive

$$\begin{aligned} \frac{1}{2} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert u_n\Vert _{L^2_x}^2+\frac{1}{2}\Vert \mu _n S u_n\Vert ^2_{L^2_x}&\le \Vert \theta _n\Vert _{L^2_x}\Vert u_n\Vert _{L^2_x}\le \frac{1}{2}\left( T\Vert \theta _n\Vert _{L^2_x}^2+\frac{1}{T}\Vert u_n\Vert _{L^2_x}^{2}\right) , \end{aligned}$$

for all positive times $T>0$, and thus by Gronwall’s inequality we arrive at the following uniform estimate for $(u_n)$ (noticing $\Vert Su_n\Vert _{L^2_x}^2=2\Vert \nabla u_n\Vert _{L^2_x}^2$)

$$\begin{aligned} \frac{1}{2} \Vert u_n\Vert _{L^\infty _TL^2_x}^2+\mu _*\Vert \nabla u_n\Vert _{L^2_T L^2_x}^2 \le \frac{e}{2} ( T\Vert \theta _0\Vert _{L^2_x}^2+\Vert u_0\Vert _{L^2_x}^2),\quad \forall T>0. \end{aligned}$$

(2.7)

Thus, the approximate solutions $(\theta _n, u_n)$ exist for all positive times.

Step 2: Passing to the limit

By the above uniform bounds (2.6)–(2.7), there exists a subsequence, still denote by $(\theta _{n},u_{n})$, converging weakly to a limit $(\theta , u)\in L^\infty _\text {loc}([0,\infty ); (L^2_x)^3)\cap L^2_\text {loc}([0,\infty ); ( H^1_x)^3)$:

$$\begin{aligned}&\theta _n\overset{*}{\rightharpoonup } \theta \quad \text {in } L^\infty _\text {loc}([0,\infty ); L^2(\mathbb {R}^2)),\quad \nabla \theta _n\rightharpoonup \nabla \theta \quad \text {in } L^2_\text {loc}([0,\infty ); (L^2(\mathbb {R}^2))^2),\\&u_n\overset{*}{\rightharpoonup } u \quad \text {in } L^\infty _\text {loc}([0,\infty ); (L^2(\mathbb {R}^2))^2), \quad \nabla u_n\rightharpoonup \nabla u \quad \text {in } L^2_\text {loc}([0,\infty ); (L^2(\mathbb {R}^2))^4). \end{aligned}$$

Since by the Gagliardo–Nirenberg’s inequality $(\theta _n, u_n)$ is a bounded sequence in $L^4_TL^4_x$ for any $T>0$, the sequence of the time derivatives $(\partial _t\theta _n, \partial _t u_n)$ is bounded in $L^2_T(H^{-1}_x)$ [by use of the equations in (2.4)], and hence $\{(\theta _n, u_n)\}$ is relatively compact in $L^p_T L^2_x(B_R)$ for any fixed disk $B_R\subset \mathbb {R}^2$ and $p\in [1,\infty )$, which implies the pointwise convergence (up to a subsequence)

$$\begin{aligned} \theta _n\rightarrow \theta ,\quad u_n\rightarrow u \hbox { for almost every }t\in \mathbb {R}^+,\quad x\in \mathbb {R}^2, \end{aligned}$$

as well as the convergence of the nonlinear terms (noticing, e.g., $u_n\varphi \rightarrow u\varphi$ in $L^4_TL^4_x$ for fixed $\varphi \in C_c^\infty ((0,T)\times \mathbb {R}^2)$)

$$\begin{aligned}&u_n\theta _n\rightarrow u\theta ,\quad u_n\otimes u_n\rightarrow u\otimes u\hbox { in }\mathcal {D}'((0,T)\times \mathbb {R}^2) \hbox { and hence weakly in }L^2_TL^2_x. \end{aligned}$$

Consequently, $\kappa _n=a(\theta _n)\rightarrow \kappa =a(\theta )$ and $\mu _n=b(\theta _n)\rightarrow \mu =b(\theta )$ almost everywhere and

$$\begin{aligned}&\kappa _n\nabla \theta _n\rightharpoonup \kappa \nabla \theta ,\quad \mu _n S u_n\rightharpoonup \mu S u \hbox { in }L^2_TL^2_x. \end{aligned}$$

Thus, the equation (noticing $P_n\rightarrow \hbox {Id}$ as an operator from $H^s(\mathbb {R}^2)$ to itself)

$$\begin{aligned} \partial _t\theta +\text {div}(u\theta )-\text {div}(\kappa \nabla \theta )=0\hbox { holds in }L^2_\text {loc}((0,\infty ); H^{-1}_x(\mathbb {R}^2)), \end{aligned}$$

and we can test it by $\theta \in L^2_\text {loc}([0,\infty ); H^1_x)$ to arrive at the energy equality (1.8) for $\theta$, such that $\theta |_{t=0}=\theta _0$ and $\theta \in C([0,\infty ); L^2_x)$ hold true.

Similarly, the equation

$$\begin{aligned} \partial _t u+\mathbb {P}\text {div}(u\otimes u-\mu Su)=\mathbb {P}(\theta \mathbf {e}_2) \hbox { holds in }L^2_\text {loc}((0,\infty ); (H^{-1}_x(\mathbb {R}^2))^2), \end{aligned}$$

(2.8)

and we can test it by the divergence-free velocity field $u\in L^2_\text {loc}((0,\infty ); (H^{1}_x(\mathbb {R}^2))^2)$ to arrive at the energy equality (1.10), which implies $u\in C([0,\infty ); (L^2_x(\mathbb {R}^2))^2)$ and $u|_{t=0}=u_0$. We take the solution $\Pi$ of the Poisson equation

$$\begin{aligned} \Delta \Pi =\text {div}(1-\mathbb {P})(\theta \mathbf {e}_2-\text {div}(u\otimes u-\mu Su)) \end{aligned}$$

(2.9)

under the renormalization condition $\int \limits _{B_1}\Pi \mathrm{d}x=0$, such that

$$\begin{aligned} \nabla \Pi = (1-\mathbb {P})(\theta \mathbf {e}_2-\text {div}(u\otimes u-\mu Su)) \in L^2_\text {loc}((0,\infty ); (H^{-1}_x(\mathbb {R}^2))^2), \end{aligned}$$

and the equation (1.11) holds in $L^2_\text {loc}((0,\infty ); (H^{-1}_x(\mathbb {R}^2))^2)$.

2.2 Energy estimates and uniqueness of the weak solutions if $(\theta _0,u_0) \in H^1(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$

We first introduce a scalar function $\eta$, which is given in terms of the temperature function as follows (recalling now $\kappa =a(\theta )\in C^2_b(\mathbb {R}; [\kappa _*, \kappa ^*])$)

$$\begin{aligned} \eta =A(\theta ),\hbox { with } A(z):=\int \limits _0^z a(\alpha )\,d\alpha \hbox { the primitive function of }a. \end{aligned}$$

(2.10)

As $A'(\theta )=a(\theta )\ge \kappa _*>0$, the function A is invertible and we can write

$$\begin{aligned} \theta =A^{-1}(\eta ), \end{aligned}$$

(2.11)

where $(A^{-1})'(\eta )=\frac{1}{a(A^{-1}(\eta ))}\le \frac{1}{\kappa ^*}$. We have the following equivalence relations³

$$\begin{aligned} \begin{aligned}&\kappa _*\Vert \theta \Vert _{L^2_x}\le \Vert \eta \Vert _{L^2_x}\le \kappa ^*\Vert \theta \Vert _{L^2_x},\\&\kappa _*\Vert \nabla \theta \Vert _{L^2_x}\le \Vert \nabla \eta \Vert _{L^2_x}=\Vert a(\theta )\nabla \theta \Vert _{L^2_x}\le \kappa ^*\Vert \nabla \theta \Vert _{L^2_x}, \\&\kappa _*\Vert \partial _t\theta \Vert _{L^2_x}\le \Vert \partial _t \eta \Vert _{L^2_x}=\Vert a(\theta )\partial _t \theta \Vert _{L^2_x}\le \kappa ^*\Vert \partial _t\theta \Vert _{L^2_x}, \\&\Vert \nabla ^2\eta \Vert _{L^2_x}\le \Vert a\Vert _{\mathrm {Lip}}\Vert \nabla \theta \Vert _{L^4_x}^2+\kappa ^*\Vert \nabla ^2 \theta \Vert _{L^2_x} \le \big (C\Vert a\Vert _{\mathrm {Lip}}\Vert \nabla \theta \Vert _{L^2_x}+\kappa ^*\big )\Vert \nabla ^2 \theta \Vert _{L^2_x}, \\&\Vert \nabla ^2 \theta \Vert _{L^2_x}\le \frac{\Vert a\Vert _{\mathrm {Lip}}}{\kappa _*^3}\Vert \nabla \eta \Vert _{L^4_x}^2+\frac{1}{\kappa _*}\Vert \nabla ^2 \eta \Vert _{L^2_x}\le \left( C\frac{\Vert a\Vert _{\mathrm {Lip}}}{\kappa _*^3}\Vert \nabla \eta \Vert _{L^2_x}+\frac{1}{\kappa _*}\right) \Vert \nabla ^2 \eta \Vert _{L^2_x}. \end{aligned}\end{aligned}$$

(2.12)

That is,

$$\begin{aligned} \theta (t,\cdot )\in H^k_x(\mathbb {R}^2)\Leftrightarrow \eta (t,\cdot )\in H^k_x(\mathbb {R}^2),\quad k=0,1,2. \end{aligned}$$

(2.13)

Let $(\theta , u)\in C([0,\infty ); (L^2(\mathbb {R}^2))^3)\cap L^2_\text {loc}([0,\infty ); (H^1(\mathbb {R}^2))^3)$ be a weak solution of the Cauchy problem (2.1) in the sense of Definition 1.1 with

$$\begin{aligned} \partial _t \theta +u\cdot \nabla \theta -\text {div}(\kappa \nabla \theta )=0 \hbox { holding in } L^2_\text {loc}([0,\infty ); H^{-1}_x(\mathbb {R}^2)). \end{aligned}$$

(2.14)

Since $Y:=L^\infty _{t,x}([0,\infty )\times \mathbb {R}^2)\cap L^2_\text {loc}([0,\infty ); H^1_x(\mathbb {R}^2))$ is an algebra (in the sense that the product of any two elements in Y still belongs to Y), we can multiply the above $\theta$-equation by $\kappa =a(\theta )$ (with $a(\theta )-a(0)\in Y$), to arrive at the parabolic equation for $\eta =A(\theta )\in C([0,\infty ); L^2(\mathbb {R}^2))\cap L^2_\text {loc}([0,\infty ); H^1(\mathbb {R}^2))$:

$$\begin{aligned} \partial _t \eta +u\cdot \nabla \eta -\kappa \Delta \eta =0\hbox { holding in the dual space }Y'. \end{aligned}$$

(2.15)

We are going to derive the $H^1$-Estimate for $\eta$ (and hence for $\theta$⁴) as well as the $L^2$-Estimate for u first. Then, we will show the uniqueness result of the weak solutions by considering the difference of two possible weak solutions. The procedure is standard (see, e.g., Sect. 2 [29]) and we are going to sketch the proof.

$H^1\times L^2$-Estimate for $(\theta , u)$

By virtue of the energy equalities (1.8) and (1.10) and the derivation of the uniform estimates (2.6) and (2.7), we have the $L^2$-Estimate

$$\begin{aligned} \Vert \theta \Vert ^2_{L^\infty _TL^2_x}+\Vert \nabla \theta \Vert ^2_{L^2_T L^2_x}\le C(\kappa _*) \Vert \theta _0\Vert ^2_{L^2}, \end{aligned}$$

(2.16)

and the $L^2$-Estimate (1.13) for u. By Gagliardo–Nirenberg’s inequality (2.2) it holds

$$\begin{aligned} \Vert u\Vert _{L^4_TL^4_x} \le C(\mu _*)\big (\sqrt{T}\Vert \theta _0\Vert _{L^2}+\Vert u_0\Vert _{L^2}\big ). \end{aligned}$$

(2.17)

We assume a priori that the function $\eta$ is smooth and decay sufficiently fast at infinity. We test the $\eta$-equation (2.15) by $\Delta \eta$ to derive by integration by parts that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2} |\nabla \eta |^2\mathrm{d}x+\int \limits _{\mathbb {R}^2}\kappa |\Delta \eta |^2\mathrm{d}x&=\int \limits _{\mathbb {R}^2}u\cdot \nabla \eta \Delta \eta \mathrm{d}x\le \Vert u\Vert _{L^4_x}\Vert \nabla \eta \Vert _{L^4_x}\Vert \Delta \eta \Vert _{L^2_x}. \end{aligned}$$

By Gagliardo–Nirenberg’s inequality (2.2), the equivalence $\Vert \Delta \eta \Vert _{L^2_x(\mathbb {R}^2)}\sim \Vert \nabla ^2\eta \Vert _{L^2_x(\mathbb {R}^2)}$ and Young’s inequality we arrive at

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2} |\nabla \eta |^2\mathrm{d}x+\frac{\kappa _*}{2}\int \limits _{\mathbb {R}^2}|\Delta \eta |^2\mathrm{d}x\le C(\kappa _*)\Vert u\Vert _{ L^4_x}^4\Vert \nabla \eta \Vert _{ L^2_x}^2. \end{aligned}$$

Gronwall’s inequality gives

$$\begin{aligned} \Vert \nabla \eta (T)\Vert _{L^2_x}^2+\Vert \nabla ^2\eta \Vert _{L^2_T L^2_x}^2&\le C(\kappa _*)\Vert \nabla \eta _0\Vert _{L^2_x}^2\exp \left( C(\kappa _*)\Vert u\Vert _{L^4_T L^4_x}^4\right) \end{aligned}$$

for any positive time $T>0$. Thus by the $\eta$-equation

$$\begin{aligned} \Vert \partial _t\eta \Vert _{L^2_T L^2_x}&= \Vert u\cdot \nabla \eta - \kappa \Delta \eta \Vert _{L^2_T L^2_x} \le \Vert u\Vert _{L^4_T L^4_x} \Vert \nabla \eta \Vert _{L^4_T L^4_x}+\kappa ^*\Vert \Delta \eta \Vert _{L^2_T L^2_x} \\&\le C(\kappa _*, \kappa ^*)\Vert \nabla \eta _0\Vert _{L^2_x}\exp \left( C(\kappa _*) \Vert u\Vert _{L^4_T L^4_x}^4\right) . \end{aligned}$$

By virtue of the equivalence relation (2.12):

$$\begin{aligned} \Vert \nabla \theta \Vert ^2_{L^\infty _TL^2_x}+\Vert \nabla ^2 \theta \Vert ^2_{L^2_T L^2_x} \le C(\kappa _*,\Vert a\Vert _\mathrm {Lip}) \left( \Vert \nabla \eta \Vert _{L^\infty _T L^2_x}^2+ \left( 1+ \Vert \nabla \eta \Vert _{L^\infty _T L^2_x}^2\right) \Vert \nabla ^2\eta \Vert _{L^2_T L^2_x}^2\right) \end{aligned}$$

and (2.16)-(2.17), we have the a priori $H^1$-Estimate (1.14) for $\theta$:

$$\begin{aligned} \begin{aligned}&\Vert \theta \Vert ^2_{L^\infty _TH^1_x}+\Vert \nabla \theta \Vert ^2_{L^2_T H^1_x} +\Vert \partial _t\theta \Vert ^2_{L^2_TL^2_x} \\&\le C(\kappa _*,\Vert a\Vert _\mathrm {Lip}, \kappa ^*) \Vert \theta _0\Vert ^2_{H^1}\left( 1+\Vert \nabla \theta _0\Vert ^2_{L^2}\right) \exp \left( C(\kappa _*) \Vert u\Vert _{L^4_T L^4_x}^4\right) . \end{aligned}\end{aligned}$$

(2.18)

Therefore, both the parabolic equations (2.14) and (2.15) for $\theta$ and $\eta$ hold in $L^2_\text {loc}([0,\infty ); L^2(\mathbb {R}^2))$. A standard density argument ensures the $H^1$-Estimate (1.14) for $\theta$, and hence $\theta \in C([0,\infty ); H^1_x(\mathbb {R}^2))$.

Proof of the uniqueness

Let $(\theta _1,u_1,\Pi _1)$ and $(\theta _2,u_2,\Pi _2)$ be two weak solutions of the Cauchy problem (2.1) with the same initial data $(\theta _0,u_0) \in H^1(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$, which satisfy the energy estimates (1.13)–(1.14). Recall (2.10) for the definition of the function A, and we set

$$\begin{aligned} \eta _1=A(\theta _1),\quad \eta _2=A(\theta _2). \end{aligned}$$

We consider the difference

$$\begin{aligned} ({\dot{\eta }},\dot{u},\nabla {\dot{\Pi }})=(\eta _1-\eta _2,u_1-u_2,\nabla \Pi _1-\nabla \Pi _2), \end{aligned}$$

which lies in

$$\begin{aligned}&\bigl ( C([0,\infty );H^1(\mathbb {R}^2))\cap L^2_\text {loc}([0,\infty );H^2(\mathbb {R}^2))\bigr )\\&\quad \times \bigl (C([0,\infty ); (L^2(\mathbb {R}^2))^2)\cap L^2_\text {loc}([0,\infty ); (H^1(\mathbb {R}^2))^2)\bigr )\times L^2_\text {loc}([0,\infty ); (H^{-1}(\mathbb {R}^2))^2)\bigr ). \end{aligned}$$

It satisfies the following Cauchy problem

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t{\dot{\eta }}+ u_1\cdot \nabla {\dot{\eta }}-\kappa _1\Delta {\dot{\eta }} ={\dot{\kappa }}\Delta \eta _2 -\dot{u}\cdot \nabla \eta _2,\\&\partial _t \dot{u}+u_1\cdot \nabla \dot{u}-{{\,\mathrm{div}\,}}(\mu _1 S\dot{u})+\nabla {\dot{\Pi }}= {\dot{\theta }} \mathbf {e_2}-\dot{u}\cdot \nabla u_2+{{\,\mathrm{div}\,}}({\dot{\mu }} Su_2),\\&{{\,\mathrm{div}\,}}\dot{u}=0, \\&({\dot{\eta }}_0,\dot{u}_0)=(0,0), \end{aligned} \right. \end{aligned}$$

(2.19)

where

$$\begin{aligned} \kappa _1=a(\theta _1),\,\mu _1=b(\theta _1),\,{\dot{\theta }}=\theta _1-\theta _2,\,{\dot{\kappa }}=a(\theta _1)-a(\theta _2),\,{\dot{\mu }}=b(\theta _1)-b(\theta _2). \end{aligned}$$

Similarly as in (2.12) we have the following equivalence relationships

$$\begin{aligned} \begin{aligned}&\kappa _*\Vert {\dot{\theta }}\Vert _{L^2_x}\le \Vert {\dot{\eta }}\Vert _{L^2_x}\le \kappa ^*\Vert {\dot{\theta }}\Vert _{L^2_x},\\&\Vert \nabla {\dot{\eta }}\Vert _{L^2_x}\le \Vert a\Vert _\mathrm {Lip}\Vert \nabla \theta _1\Vert _{L^4_x}\Vert {\dot{\theta }}\Vert _{L^4_x} +\kappa ^*\Vert \nabla {\dot{\theta }}\Vert _{L^2_x}, \\&\Vert \nabla {\dot{\theta }}\Vert _{L^2_x}\le \frac{\Vert a\Vert _\mathrm {Lip}}{\kappa _*^3}\Vert \nabla \eta _1\Vert _{L^4_x}\Vert {\dot{\eta }}\Vert _{L^4_x} +\frac{1}{\kappa _*}\Vert \nabla {\dot{\eta }}\Vert _{L^2_x}, \end{aligned}\end{aligned}$$

(2.20)

and correspondingly we have

$$\begin{aligned} \begin{aligned} \Vert ({\dot{\kappa }}, {\dot{\mu }})\Vert _{H^1_x}&\le C(\Vert (a,b)\Vert _{\mathrm {Lip}}, \Vert (a', b')\Vert _\mathrm {Lip},\kappa _*)(\Vert \nabla \eta _1\Vert _{L^4_x}\Vert {\dot{\eta }}\Vert _{L^4_x}+\Vert {\dot{\eta }}\Vert _{H^1_x}) \\&\le C(\Vert (a,b)\Vert _{\mathrm {Lip}}, \Vert (a', b')\Vert _\mathrm {Lip},\kappa _*)(1+\Vert \nabla \eta _1\Vert _{L^4_x}) \Vert {\dot{\eta }}\Vert _{H^1_x}. \end{aligned}\end{aligned}$$

(2.21)

We are going to sketch the derivation of the $H^1\times L^2$-Estimate for $({\dot{\eta }}, \dot{u})$.

• (i) $L^2$ estimate of ${\dot{\eta }}$. We take the $L^2(\mathbb {R}^2)$-inner product between (2.19)$_1 and {\dot{\eta }}$ to derive

  $$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2}|{\dot{\eta }}|^2+\int \limits _{\mathbb {R}^2}\kappa _1 |\nabla \dot{\eta }|^2 \le \int \limits _{\mathbb {R}^2}|{\dot{\eta }} \nabla \kappa _1\cdot \nabla {\dot{\eta }}|+ |{\dot{\eta }}\dot{u}\cdot \nabla \eta _2|+ |{\dot{\kappa }}\Delta \eta _2{\dot{\eta }}| . \end{aligned} \end{aligned}$$

  (2.22)

  The right hand side can be bounded by

  $$\begin{aligned}&\Vert {\dot{\eta }}\Vert _{L^4_x} \Vert \nabla \kappa _1\Vert _{L^4_x}\Vert \nabla {\dot{\eta }}\Vert _{L^2_x} + \Vert \nabla \eta _2\Vert _{L^2_x}\Vert \dot{u}\Vert _{L^4_x}\Vert {\dot{\eta }}\Vert _{L^4_x} +\Vert \Delta \eta _2\Vert _{L^2_x}\Vert {\dot{\kappa }}\Vert _{L^4_x}\Vert {\dot{\eta }}\Vert _{L^4_x} \\&\quad \le C(\Vert a\Vert _\mathrm {Lip}) \Bigl ( \Vert \nabla \theta _1\Vert _{L^4_x}\Vert {\dot{\eta }}\Vert _{L^2_x}^{\frac{1}{2}} \Vert \nabla {\dot{\eta }}\Vert _{L^2_x}^{\frac{3}{2}} + \Vert \nabla \eta _2\Vert _{L^2_x}\Vert \dot{u}\Vert _{L^2_x}^{\frac{1}{2}}\Vert \nabla \dot{u}\Vert _{L^2_x}^{\frac{1}{2}}\Vert {\dot{\eta }}\Vert _{L^2_x}^{\frac{1}{2}}\Vert \nabla {\dot{\eta }}\Vert _{L^2_x}^{\frac{1}{2}} \\&\qquad + \Vert \Delta \eta _2\Vert _{L^2_x}\Vert {\dot{\theta }}\Vert _{L^2_x}^{\frac{1}{2}}\Vert \nabla {\dot{\theta }}\Vert _{L^2_x}^{\frac{1}{2}}\Vert {\dot{\eta }}\Vert _{L^2_x}^{\frac{1}{2}}\Vert \nabla {\dot{\eta }}\Vert _{L^2_x}^{\frac{1}{2}}\Bigr ) \\&\quad \le \frac{\kappa _*}{2}\Vert \nabla {\dot{\eta }}\Vert _{L^2_x}^2+\frac{\mu _*}{4}\Vert \nabla \dot{u}\Vert _{L^2_x}^2 \\&\qquad + C(\Vert a\Vert _\mathrm {Lip}, \kappa _*, \mu _*) \Bigl (\Vert \nabla \theta _1\Vert _{L^4_x}^4 + \Vert \nabla \eta _2\Vert _{L^2_x}^2+ \Vert \Delta \eta _2\Vert _{L^2_x}^2\Bigr ) \times \bigl (\Vert {\dot{\eta }}\Vert _{L^2_x}^{2}+\Vert \dot{u}\Vert _{L^2_x}^{2} \bigr ). \end{aligned}$$

• (ii) $L^2$ estimate of $\dot{u}$. We take the $L^2$ inner product of the equation (2.19)$_2$ and $\dot{u}$ to derive

  $$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2}|\dot{u}|^2+\frac{1}{2}\int \limits _{\mathbb {R}^2}\mu _1|S \dot{u}|^2 \le \int \limits _{\mathbb {R}^2} |{\dot{\theta }} \dot{u}|+ |\dot{u}|^2|\nabla u_2|+ | {\dot{\mu }} S u_2:\nabla \dot{u}|. \end{aligned} \end{aligned}$$

  (2.23)

  The right hand side can be bounded by

  $$\begin{aligned} \Vert \dot{u}\Vert _{L^2_x}\Vert {\dot{\theta }}\Vert _{L^2_x} +\Vert \dot{u}\Vert _{L^4_x}^2\Vert \nabla u_2\Vert _{L^2_x} +\Vert {\dot{\mu }}\Vert _{H^1_x}\Vert S u_2:\nabla \dot{u}\Vert _{H^{-1}_x} \end{aligned}$$

  which, by use of the Sobolev embedding $L^1(\mathbb {R}^2)\hookrightarrow H^{-1}(\mathbb {R}^2)$, is bounded by

  $$\begin{aligned}&\frac{\mu _*}{4}\Vert \nabla \dot{u}\Vert _{L^2_x}^2 +C(\Vert a\Vert _\mathrm {Lip},\kappa _*, \mu _*)(1+\Vert \nabla u_2\Vert _{L^2_x}^2) (\Vert \dot{u}\Vert _{L^2_x}^2+\Vert {\dot{\eta }}\Vert _{L^2_x}^2)\\&\quad +C(\mu _*)\Vert \nabla u_2\Vert _{L^2_x}^2\Vert {\dot{\mu }}\Vert _{H^1_x}^2. \end{aligned}$$

• (iii) $L^2$ estimate of $\nabla {\dot{\eta }}$. We take the $L^2$ inner product of Eq. (2.19)$_1 and \Delta {\dot{\eta }}$ to derive

  $$\begin{aligned}&\frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2}|\nabla {\dot{\eta }}|^2+\int \limits _{\mathbb {R}^2}\kappa _1 |\Delta {\dot{\eta }}|^2 \le \int \limits _{\mathbb {R}^2}| u_1\cdot \nabla {\dot{\eta }}\Delta {\dot{\eta }}| + |\dot{u}\cdot \nabla \eta _2\Delta {\dot{\eta }}|+ |{\dot{\kappa }}\Delta \eta _2\Delta {\dot{\eta }}| \end{aligned}$$

  By $L^1(\mathbb {R}^2)\hookrightarrow H^{-1}(\mathbb {R}^2)$ again, the right hand side is bounded similarly by

  $$\begin{aligned}&\frac{\kappa _*}{2}\Vert \Delta {\dot{\eta }}\Vert _{L^2_x}^2+\frac{\mu _*}{4}\Vert \nabla \dot{u}\Vert _{L^2_x}^2 + C(\kappa _*, \mu _*) \left( \Vert u_1\Vert _{L^4_x}^4+\Vert \nabla \eta _2\Vert _{L^4_x}^4\right) \left( \Vert \dot{u}\Vert _{L^2_x}^2+\Vert \nabla {\dot{\eta }}\Vert _{L^2_x}^2\right) \\&\quad +C(\kappa _*)\Vert \Delta \eta _2\Vert _{L^2_x}^2\Vert {\dot{\kappa }}\Vert _{H^{1}_x}^2. \end{aligned}$$

To conclude, by virtue of the above estimates and (2.21), we have the following inequality

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{{d}}}{\mathrm{{d}}t}\left( \Vert {\dot{\eta }} \Vert _{H^1_x}^2+\Vert \dot{u}\Vert _{L^2_x}^2\right) +\Vert \nabla \dot{u}\Vert _{L^2_x}^2+\Vert \nabla {\dot{\eta }}\Vert _{H^1_x}^2 \\&\quad \le C\left( \Vert (a,b)\Vert _{\mathrm {Lip}}, \Vert (a', b')\Vert _\mathrm {Lip}, \kappa _*, \mu _*\right) B(t) \left( \Vert \dot{u}\Vert _{L^2_x}^2+\Vert {\dot{\eta }}\Vert _{H^1_x}^2\right) , \end{aligned} \end{aligned}$$

where

$$\begin{aligned} B(t)&=\Bigl (\Vert \nabla \theta _1\Vert _{L^4_x}^4 + \Vert \nabla \eta _2\Vert _{L^2_x}^2+ \Vert \Delta \eta _2\Vert _{L^2_x}^2+1+\Vert \nabla u_2\Vert _{L^2_x}^2 +\Vert u_1\Vert _{L^4_x}^4+\Vert \nabla \eta _2\Vert _{L^4_x}^4\Bigr ) \\&\quad \times \left( 1+\Vert \nabla \eta _1\Vert _{L^4_x}\right) \in L^1_\text {loc}([0,\infty )). \end{aligned}$$

Gronwall’s inequality implies then ${\dot{\eta }}=0$ and $\dot{u}=0$. The uniqueness of the weak solutions follows.

2.3 Propagation of the general $H^s$-regularities

After the derivation of the a priori $H^s_x$, $s\in (0,2)$-Estimate for a general linear parabolic equation in Sect. 2.3.1, we are going to derive the precise $H^s_x$-Estimates (1.17)–(1.24) in Remark 1.4 in the subsequent subsections:

• In Sect. 2.3.2 the global-in-time $H^s_x(\mathbb {R}^2)\times (L^2_x(\mathbb {R}^2))^2$, $s\in (1,2)$-regularities [i.e., (1.17)] will be established, where the endpoint case $s=2$ [i.e., (1.18)] will be treated separately.

• In Sect. 2.3.3, the global-in-time $H^1_x(\mathbb {R}^2)\times (H^s_x(\mathbb {R}^2))^2$, $s\in (0,2)$-regularities [i.e., (1.19)] will be established, where the endpoint case $s=2$ [i.e., (1.20)] will be treated separately.

• In Sect. 2.3.4 the global-in-time $H^{s}_x(\mathbb {R}^2)\times (H^{s-2}_x(\mathbb {R}^2))^2$ [i.e., (1.21)–(1.22)] and $H^{s-1}_x\times (H^s_x(\mathbb {R}^2))^2$, $s>2$-regularities [i.e., (1.23)–(1.24)] will be established, respectively.

As far as the borderline estimates (1.17)–(1.24) are established, the global-in-time regularity (1.16) follows immediately.

For readers’ convenience, we recall here briefly the Littlewood–Paley dyadic decomposition and the definition of the $H^s(\mathbb {R}^n)$-norms (see, e.g., Chapter 2 in the book [4] for more details). We fix a nonincreasing radial function $\chi \in C^\infty _c(B_{\frac{4}{3}})$ with $\chi (x)=1$ for $x\in B_1$, where $B_r\subset \mathbb {R}^n$ denotes the ball centered at 0 with radius r. We define the function $\varphi (\xi )=\chi (\frac{\xi }{2})-\chi (\xi )$ and $\varphi _j(\xi )=\varphi (2^{-j}\xi )$ with $j\ge 0$. We do the Littlewood–Paley decomposition in the following way

$$\begin{aligned} g=\Delta _{-1}g+\sum _{j\ge 0} \Delta _j g, \end{aligned}$$

(2.24)

where

$$\begin{aligned} \mathcal {F}(\Delta _{-1}g)(\xi )=\chi (\xi )\mathcal {F}(g)(\xi ),\quad \mathcal {F}(\Delta _{j}g)(\xi )=\varphi _j(\xi )\mathcal {F}(g)(\xi ), \quad j\ge 0, \end{aligned}$$

and $\mathcal {F}$ denotes the Fourier transform. We have the following Bernstein’s inequalities for some universal constant C (depending only on n)

$$\begin{aligned}&\Vert \Delta _{-1}g\Vert _{L^2(\mathbb {R}^n)}\le C\Vert g\Vert _{L^2(\mathbb {R}^n)}, \nonumber \\&C^{-1}2^{j}\Vert \Delta _j g\Vert _{L^2(\mathbb {R}^n)}\le \Vert \nabla (\Delta _j g)\Vert _{L^2(\mathbb {R}^n)}\le C2^{j}\Vert \Delta _j g\Vert _{L^2(\mathbb {R}^n)},\quad \forall j\ge 0. \end{aligned}$$

(2.25)

Let $s\ge 0$ and $p,\,r\ge 1$. We define the nonhomogeneous Besov spaces $B^s_{p,r}(\mathbb {R}^n)$ as the spaces consisting of all tempered distributions $g\in \mathcal {S}'(\mathbb {R}^n)$ satisfying

$$\begin{aligned} \Vert g\Vert _{B^s_{p,r}(\mathbb {R}^n)}=\big \Vert \Big (2 ^{js}\Vert \Delta _j g\Vert _{L^p(\mathbb {R}^n)}\Big )_{j\ge -1}\big \Vert _{l^r}<\infty . \end{aligned}$$

The inhomogeneous Sobolev spaces $H^s(\mathbb {R}^n)= B^s_{2,2}(\mathbb {R}^n)$ can be defined by

$$\begin{aligned} H^s(\mathbb {R}^n)=\left\{ g\in \mathcal {S}'(\mathbb {R}^n)\mid \Vert g\Vert _{H^s(\mathbb {R}^n)}=\left( \int \limits _{\mathbb {R}^n}(1+|\xi |^2)^{\frac{s}{2}}|\mathcal {F}(g)(\xi )|^2\,d\xi \right) ^{1/2}<\infty \right\} , \end{aligned}$$

where the $H^s(\mathbb {R}^n)$-norm reads in terms of Littlewood–Paley decomposition as follows

$$\begin{aligned} \Vert g\Vert _{H^s(\mathbb {R}^n)}\sim \Vert g\Vert _{L^2(\mathbb {R}^n)}+\left( \sum _{j\ge 0}2^{2js}\Vert \Delta _j g\Vert _{L^2(\mathbb {R}^n)}^2\right) ^{\frac{1}{2}}. \end{aligned}$$

(2.26)

It is straightforward to derive the following interpolation inequality

$$\begin{aligned} \Vert u\Vert _{H^{t_\sigma }}\le C\Vert u\Vert _{H^{t_0}}^{1-\sigma }\Vert u\Vert _{H^{t_1}}^\sigma , \hbox { where } t_\sigma =(1-\sigma )t_0+\sigma t_1 , \sigma \in [0,1]. \end{aligned}$$

(2.27)

We are going to use the following known estimates to control the nonlinear terms in the Boussinesq system (1.1).

Lemma 2.1

We have the following commutator, product, and composition estimates.

• (1) [12, Proposition 2.4] In the low regularity regime where $(s,\nu )\in \mathbb {R}^2$ satisfy

  $$\begin{aligned} -1<s<\nu +1,\quad \text {and } -1<\nu <1, \end{aligned}$$

  the following commutator estimate holds true (in $\mathbb {R}^2$):

  $$\begin{aligned} \Vert \big (2^{js}\Vert [\phi ,\Delta _j]\nabla \psi \Vert _{L^{2}(\mathbb {R}^2)}\big )_{j\ge 1}\Vert _{l^1}\le C\Vert \nabla \phi \Vert _{H^{\nu }(\mathbb {R}^2)}\Vert \nabla \psi \Vert _{H^{s-\nu }(\mathbb {R}^2)}, \end{aligned}$$

  (2.28)

  where C is a constant depending only on $s,\nu$.

• (2) [4, Lemma 2.100] For any $s>0$, the following commutator estimate holds true

  $$\begin{aligned} \begin{aligned}&\Vert \big (2^{js}\Vert [\phi ,\Delta _j]\nabla \psi \Vert _{L^{2}(\mathbb {R}^n)}\big )_{j\ge 1}\Vert _{l^2} \\&\le C\bigl ( \Vert \nabla \phi \Vert _{L^\infty (\mathbb {R}^n)}\Vert \nabla \psi \Vert _{H^{s-1}(\mathbb {R}^n)}+\Vert \nabla \phi \Vert _{H^{s-1}(\mathbb {R}^n)}\Vert \nabla \psi \Vert _{L^\infty (\mathbb {R}^n)}). \end{aligned}\end{aligned}$$

  (2.29)

• (3) [4, Corollary 2.86] For any $s>0$, the following product estimate holds true

  $$\begin{aligned} \Vert \phi \psi \Vert _{H^{s}(\mathbb {R}^n)}\le C\bigl ( \Vert \phi \Vert _{L^\infty (\mathbb {R}^n)}\Vert \psi \Vert _{H^{s}(\mathbb {R}^n)}+\Vert \phi \Vert _{H^{s}(\mathbb {R}^n)}\Vert \psi \Vert _{L^\infty (\mathbb {R}^n)}). \end{aligned}$$

  (2.30)

• (4) [4, Theorems 2.87 and Theorem 2.89] For any $s>0$ and $g\in C^{k+1}$ with $k=[s]\in \mathbb {N}$, the following composition estimate holds true

  $$\begin{aligned} \Vert \nabla (g\circ \theta )\Vert _{H^{s-1}(\mathbb {R}^n)}\le C(g, \Vert \theta \Vert _{L^\infty (\mathbb {R}^n)})\Vert \nabla \theta \Vert _{H^{s-1}(\mathbb {R}^n)}. \end{aligned}$$

  (2.31)

  If $g\in C^{k+1}_b$ with $k=[s]\in \mathbb {N}$, then the above estimate can be improved in the spatial dimension two as follows

  $$\begin{aligned} \Vert \nabla (g\circ \theta )\Vert _{H^{s-1}(\mathbb {R}^2)}\le C(\Vert g\Vert _{C^{k+1}}, \Vert \theta \Vert _{H^1(\mathbb {R}^2)})\Vert \nabla \theta \Vert _{H^{s-1}(\mathbb {R}^2)}. \end{aligned}$$

  (2.32)

The commutator estimate (2.28) will present its power in the low regularity regime (see Sects. 2.3.1–2.3.3 below), and the classical commutator estimate (2.29) will help in the high regularity regime (see Sect. 2.3.4 below).

The composition estimate (2.32) will help to bound the diffusion coefficients $\kappa , \mu$ in terms of $\theta$ in the low regularity regime, where only $H^1(\mathbb {R}^2)$-norm (instead of $L^\infty _x$-norm) of $\theta$ is available.

2.3.1 Estimates for the general parabolic equations

We derive in this paragraph a priori $H^s$, $s\in (0,2)$-Estimates for a general linear parabolic equation, which should be of independent interest.

Lemma 2.2

Let $\psi =\psi (t,x): [0,\infty )\times \mathbb {R}^2\mapsto \mathbb {R}^m$, $m\ge 1$ be a smooth solution with sufficiently decay of the following linear parabolic equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\psi + u\cdot \nabla _x \psi -{{\,\mathrm{div}\,}}_x(\kappa \nabla _x \psi )=f,\\&\psi |_{t=0}=\psi _0, \end{aligned} \right. \end{aligned}$$

(2.33)

where

• $u=u(t,x): \mathbb {R}^+\times \mathbb {R}^2\mapsto \mathbb {R}^2$ is a given divergence-free vector field: $\text {div}_x u=0$;

• $\kappa =\kappa (t,x):\mathbb {R}^+\times \mathbb {R}^2\rightarrow [\kappa _*,\kappa ^*]$ with $\kappa _*,\kappa ^*\in (0,\infty )$;

• $f=f(t,x): \mathbb {R}^+\times \mathbb {R}^2\mapsto \mathbb {R}^m$ denotes the given external force.

Then, the following a priori $H^s_x$-Estimates for (2.33) holds true:

$$\begin{aligned} \begin{aligned}&\Vert \psi \Vert _{L^\infty _T H^s_x}^2+\Vert \nabla \psi \Vert _{L^2_T H^{s}_x}^2 \le C(\kappa _*)\Bigl ( \Vert \psi _0\Vert _{H^s_x}^2+\Vert f\Vert _{L^2_T H^{s-1}_x}^2 \Bigr )\\&\quad \times \exp \Bigl (C(\kappa _*,s,\nu )(\Vert \nabla u\Vert _{L^2_T L^2_x}^2+\Vert \nabla \kappa \Vert _{L^{\frac{2}{\nu }}_T H^\nu _x}^{2/\nu }+ \Vert f\Vert _{L^1_T H^{-s}_x})\Bigr ) \\&\qquad \hbox { for any }s\in (0,2) \hbox { and }\nu \in (s-1,1)\subset (-1,1). \end{aligned}\end{aligned}$$

(2.34)

Proof

It is straightforward to derive the following $L^2_x$-Estimate by simply taking the $L^2(\mathbb {R}^2)$ inner product of the equation (2.33) and $\psi$ itself

$$\begin{aligned} \Vert \psi \Vert _{L^\infty _TL^2_x}^2+\Vert \nabla \psi \Vert ^2_{L^2_TL^2_x}\le C(\kappa _*) \Bigl ( \Vert \psi _0\Vert _{L^2_x}^2+\int ^T_0 \langle \psi , f\rangle _{H^{s}_x, H^{-s}_x}\,\mathrm{{d}}t\Bigr ), \quad \forall s\in \mathbb {R}. \end{aligned}$$

(2.35)

We next consider the a priori estimates for the $H^s(\mathbb {R}^2)$-norm. By virtue of the description (2.26) of the $H^s(\mathbb {R}^2)$-norm, we consider the dyadic piece of $\psi$:

$$\begin{aligned} \psi _j:=\Delta _j\psi , \quad j\ge 0. \end{aligned}$$

(2.36)

where the operator $\Delta _j$ is defined in (2.24). We apply $\Delta _j$ to the linear $\psi$-equation to derive the equation for $\psi _j$:

$$\begin{aligned} \partial _t\psi _j+u\cdot \nabla \psi _j-{{\,\mathrm{div}\,}}(\kappa \nabla \psi _j)=[u,\Delta _j]\cdot \nabla \psi -\text {div}([\kappa ,\Delta _j]\nabla \psi )+f_j, \quad j\ge 0. \end{aligned}$$

(2.37)

We take the $L^2$ inner product of the equation (2.37) and $\psi _j$ and make use of $\text {div}u=0$ and $\kappa \ge \kappa _*$ to derive

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert \psi _j\Vert _{L^2_x}^2+\kappa _*\Vert \nabla \psi _j\Vert _{L^2_x}^2 \le \Vert \psi _j\Vert _{L^2_x}\Vert [u,\Delta _j]\cdot \nabla \psi \Vert _{L^2_x} \\&\quad +\Vert \nabla \psi _j\Vert _{L^2_x}\Vert [\kappa ,\Delta _j]\nabla \psi \Vert _{L^2_x}+\Vert f_j\Vert _{L^2_x}\Vert \psi _j\Vert _{L^2_x},\, j\ge 0. \end{aligned} \end{aligned}$$

By use of Bernstein’s inequality (2.25), we have

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert \psi _j\Vert _{L^2_x}^2+2^{2j}\Vert \psi _j\Vert _{L^2_x}^2\\&\quad \le C(\kappa _*)\Vert \psi _j\Vert _{L^2_x}\left( \Vert [u,\Delta _j]\cdot \nabla \psi \Vert _{L^2_x}+2^j\Vert [\kappa ,\Delta _j]\nabla \psi \Vert _{L^2_x}+\Vert f_j\Vert _{L^2_x}\right) , \end{aligned} \end{aligned}$$

that is,

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert \psi _j\Vert _{L^2_x}+2^{2j}\Vert \psi _j\Vert _{L^2_x}\\&\quad \le C(\kappa _*) \left( \Vert [u,\Delta _j]\cdot \nabla \psi \Vert _{L^2_x}+2^j\Vert [\kappa ,\Delta _j]\nabla \psi \Vert _{L^2_x}+\Vert f_j\Vert _{L^2_x}\right) , \quad j\ge 0. \end{aligned} \end{aligned}$$

(2.38)

We make use of the commutator estimate (2.28) in Lemma 2.1 to estimate the commutators $\Vert [u,\Delta _j]\cdot \nabla \psi \Vert _{L^2_x}$ and $2^j\Vert [\kappa ,\Delta _j]\nabla \psi _j\Vert _{L^2_x}$ in the above inequality in the following way. Let $(l_j)_{j\ge 0}$ be a normalized sequence in $\ell ^1(\mathbb {N})$ such that $l_j\ge 0$ and $\sum _{j\ge 0}l_j=1$. Then, we have

$$\begin{aligned} \begin{aligned}&\Vert [u,\Delta _j]\nabla \psi \Vert _{L^{2}} \le C(s) l_j 2^{j(1-s)}\Vert \nabla u\Vert _{L^2_x}\Vert \nabla \psi \Vert _{H^{s-1}_x}, \hbox { for } s\in (0,2),\\&\quad 2^j\Vert [\kappa ,\Delta _j]\nabla \psi \Vert _{L^2_x} \le C(s,\nu ) l_j 2^{j(1-s)}\Vert \nabla \kappa \Vert _{H^{\nu }_x}\Vert \nabla \psi \Vert _{H^{s-\nu }_x} \\&\qquad \hbox { for } \nu \in (-1,1),\, s\in (-1,\nu +1). \end{aligned} \end{aligned}$$

(2.39)

Therefore, we have

$$\begin{aligned}&\frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert \psi _j\Vert _{L^2_x}+2^{2j}\Vert \psi _j\Vert _{L^2_x} \\&\quad \le C(\kappa _*,s,\nu ) l_j 2^{j(1-s)}\left( \Vert \nabla u\Vert _{L^2_x}\Vert \nabla \psi \Vert _{H^{s-1}_x}+\Vert \nabla \kappa \Vert _{H^{\nu }_x}\Vert \nabla \psi \Vert _{H^{s-\nu }_x}\right) +C(\kappa _*)\Vert f_j\Vert _{L^2_x}\\&\qquad \hbox { for }\nu \in (-1,1),\,s\in (0,\nu +1),\,j\ge 0. \end{aligned}$$

We use Duhamel’s Principle to derive

$$\begin{aligned} \begin{aligned}&\Vert \psi _j\Vert _{L^2_x} \le e^{- t2^{2j}}\Vert (\psi _0)_j\Vert _{L^2_x}+C(\kappa _*) \int \limits _0^t e^{- (t-\tau )2^{2j}}\Vert f_j(\tau )\Vert _{L^2_x}\,d\tau \\&\quad +C(\kappa _*,s,\nu ) 2^{j(1-s)}l_j\int \limits _0^t e^{- (t-\tau )2^{2j}} \big (\Vert \nabla u(\tau )\Vert _{L^2_x}\Vert \nabla \psi (\tau )\Vert _{H^{s-1}_x}\\&\quad +\Vert \nabla \kappa (\tau )\Vert _{H^{\nu }_x}\Vert \nabla \psi (\tau )\Vert _{H^{s-\nu }_x}\big )\,d\tau ,\quad j\ge 0. \end{aligned} \end{aligned}$$

(2.40)

We multiply the inequality (2.40) by $2^{js}$ to derive

$$\begin{aligned} \begin{aligned}&2^{js}\Vert \psi _j\Vert _{L^2_x} \le 2^{js}e^{- t2^{2j}}\Vert (\psi _0)_j\Vert _{L^2_x}+C(\kappa _*) 2^{js}\int \limits _0^t e^{- (t-\tau )2^{2j}} \Vert f_j\Vert _{L^2_x}\,d\tau \\&\quad +C(\kappa _*,s,\nu ) 2^{j}l_j\int \limits _0^t e^{-(t-\tau )2^{2j}} \big (\Vert \nabla u(\tau )\Vert _{L^2_x}\Vert \nabla \psi (\tau )\Vert _{H^{s-1}_x}\\&\quad +\Vert \nabla \kappa (\tau )\Vert _{H^{\nu }_x}\Vert \nabla \psi (\tau )\Vert _{H^{s-\nu }_x}\big )\,d\tau ,\quad j\ge 0. \end{aligned} \end{aligned}$$

(2.41)

We take $L^\infty ([0,T])$-norm in t of (2.41) and the $L^2([0,T])$-norm in t of $2^j\cdot$(2.41), to derive by use of Young’s inequality that

$$\begin{aligned} \begin{aligned}&2^{js}\Vert \psi _j\Vert _{L^\infty _T L^2_x}+2^{j(s+1)}\Vert \psi _j\Vert _{L^2_T L^2_x} \le 2^{js}\Vert (\psi _0)_j\Vert _{L^2_x}+C(\kappa _*) 2^{j(s-1)}\Vert f_j\Vert _{L^2_T L^2_x}\\&\quad +C(\kappa _*,s,\nu ) l_j \Bigl \Vert \Vert \nabla u\Vert _{L^2_x}\Vert \nabla \psi \Vert _{H^{s-1}_x}+\Vert \nabla \kappa \Vert _{H^{\nu }_x}\Vert \nabla \psi \Vert _{H^{s-\nu }_x}\Bigr \Vert _{L^2_T}. \end{aligned} \end{aligned}$$

(2.42)

We take square of (2.42) and sum them up for $j\in \mathbb {N}$ to derive

$$\begin{aligned}&\sum _{j\ge 0}\Bigl ( 2^{2js}\Vert \psi _j\Vert _{L^\infty _T L^2_x}^2+2^{2j(s+1)}\Vert \psi _j\Vert _{L^2_T L^2_x}^2\Bigr )\\&\quad \lesssim _{\kappa _*,s,\nu } \sum _{j\ge 0}\Bigl (2^{2js}\Vert (\psi _0)_j\Vert _{L^2_x}^2+ 2^{2j(s-1)}\Vert f_j\Vert _{L^2_T L^2_x}^2\Bigr )\\&\qquad +\int ^T_0 \Vert \nabla u\Vert _{L^2_x}^2\Vert \nabla \psi \Vert _{H^{s-1}_x}^2+\Vert \nabla \kappa \Vert _{H^\nu _x}^2\Vert \nabla \psi \Vert _{H^{s-\nu }_x}^2 \mathrm{{d}}t,\quad j\ge 0, \end{aligned}$$

that is, by virtue of the $L^2$-Estimate (2.35),

$$\begin{aligned}&\Vert \psi \Vert _{L^\infty _T H^s_x}^2+\Vert \nabla \psi \Vert _{L^2_T H^{s}_x}^2 \lesssim _{\kappa _*,s,\nu } \Bigl ( \Vert \psi _0\Vert _{H^s_x}^2+\Vert f\Vert _{L^2_T H^{s-1}_x}^2+\int ^T_0 \Vert \psi \Vert _{H^s_x}\Vert f\Vert _{H^{-s}_x} \mathrm{{d}}t\\&\qquad +\int ^T_0 \Vert \nabla u\Vert _{L^2_x}^2\Vert \nabla \psi \Vert _{H^{s-1}_x}^2+\Vert \nabla \kappa \Vert _{H^\nu _x}^2\Vert \nabla \psi \Vert _{H^{s-\nu }_x}^2 \mathrm{{d}}t\Bigr ). \end{aligned}$$

We next consider the norm $\Vert \nabla \psi \Vert _{H^{s-\nu }_x}$. By the interpolation inequality (2.27), we have

$$\begin{aligned} \Vert \nabla \psi \Vert _{H^{s-\nu }_x} \le C\Vert \nabla \psi \Vert _{H^{s-1}_x}^{\nu }\Vert \nabla \psi \Vert _{H^s_x}^{1-\nu }, \quad \nu \in (0,1), \end{aligned}$$

which implies by Young’s inequality that

$$\begin{aligned} \int ^T_0 \Vert \nabla \kappa \Vert _{H^\nu _x}^2\Vert \nabla \psi \Vert _{H^{s-\nu }_x}^2 \mathrm{{d}}t&\le \int ^T_0 \Vert \nabla \kappa \Vert _{H^\nu _x}^2\Vert \nabla \psi \Vert _{H^{s-1}_x}^{2\nu }\Vert \nabla \psi \Vert _{H^s_x}^{2(1-\nu )} \mathrm{{d}}t \\&\le \varepsilon \Vert \nabla \psi \Vert _{L^2_T H^s_x}^2+C_{\varepsilon }\int ^T_0 \Vert \nabla \kappa \Vert _{H^\nu _x}^{2/\nu }\Vert \nabla \psi \Vert _{H^{s-1}_x}^{2}\,\mathrm{{d}}t. \end{aligned}$$

To conclude, by taking $\varepsilon$ small enough and Gronwall’s inequality, we derive the $H^s$-Estimate (2.34). $\square$

2.3.2 Case $(\theta _0,u_0)\in H^{s}(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$, $s\in (1,2]$

In this subsection, we are going to prove the $H^s$-Estimates (1.17) for the unique solution $(\theta , u)$ of the Boussinesq equations (1.1) with the initial data $(\theta _0, u_0)\in H^{s}(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$, $s\in (1,2)$, following exactly the procedure in Sect. 2.3.1. We will pay more attention on the “nonlinearities” in the equations such as $\kappa =a(\theta )$, $u\cdot \nabla u$ when using the commutator estimates and will sketch the proof.

The endpoint estimate (1.18) for $(\theta _0, u_0)\in H^{2}(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$ follows similarly as in the proof for the $H^1$-Estimate for $\theta$ in Sect. 2.2 and we will sketch the proof.

Case $(\theta _0,u_0)\in H^{s}(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$, $s\in (1,2)$

Similarly as (2.38), we have the following preliminary estimate for $\theta _j=\Delta _j \theta$:

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert \theta _j\Vert _{L^2_x}+2^{2j}\Vert \theta _j\Vert _{L^2_x}\le C(\kappa _*) \left( \Vert [u,\Delta _j]\cdot \nabla \theta \Vert _{L^2_x}+2^j\Vert [\kappa ,\Delta _j]\nabla \theta \Vert _{L^2_x}\right) ,\quad j\ge 0. \end{aligned}$$

(2.43)

By use of the commutator estimates (2.39) and the action estimate (2.32):

$$\begin{aligned} \Vert \nabla \kappa \Vert _{H^\nu }\le C(\Vert a\Vert _{C^2}, \Vert \theta \Vert _{H^1})\Vert \nabla \theta \Vert _{H^\nu }\hbox { for }\nu \in (0,1), \end{aligned}$$

we derive similar as (2.42)

$$\begin{aligned} \begin{aligned}&2^{2js}\Vert \theta _j\Vert _{L^\infty _T L^2_x}^2+2^{2j(s+1)}\Vert \theta _j\Vert _{L^2_T L^2_x}^2 \le 2^{2js}\Vert (\theta _0)_j\Vert _{L^2_x}^2 \\&\quad +C(\kappa _*, s, \nu , \Vert a\Vert _{C^2},\Vert \theta \Vert _{L^\infty _T H^1_x}) (l_j)^2 \int ^T_0 \Bigl ( \Vert \nabla u\Vert _{L^2_x}^2\Vert \nabla \theta \Vert _{H^{s-1}_x}^2\\&\quad +\Vert \nabla \theta \Vert _{H^{\nu }_x}^2\Vert \nabla \theta \Vert _{H^{s-\nu }_x}^2\Bigr ) \mathrm{{d}}t, \quad 1<s<\nu +1<2. \end{aligned} \end{aligned}$$

(2.44)

By using the interpolation inequality (2.27), we have

$$\begin{aligned} \Vert \nabla \theta \Vert _{H^{\nu }_x}\Vert \nabla \theta \Vert _{H^{s-\nu }_x} \le C\Vert \nabla \theta \Vert _{L^2_x}^{1-\nu }\Vert \nabla \theta \Vert _{H^1_x}^\nu \Vert \nabla \theta \Vert _{H^{s-1}_x}^{\nu }\Vert \nabla \theta \Vert _{H^{s}_x}^{1-\nu }, \quad 0< \nu <1. \end{aligned}$$

Recall the $L^2$-Estimate (2.16) for $\theta$:

$$\begin{aligned} \Vert \theta \Vert _{L^\infty _T L^2_x}^2+\Vert \nabla \theta \Vert _{L^2_T L^2_x}^2\le C(\kappa _*) \Vert \theta _0\Vert _{L^2_x}^2. \end{aligned}$$

(2.45)

Therefore by Young’s inequality, we arrive at

$$\begin{aligned}&\Vert \theta \Vert _{L^\infty _T H^s_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^{s}_x}^2 \le C(\kappa _*)\Vert \theta _0\Vert _{H^s_x}^2\\&+C(\kappa _*, s, \nu , \Vert a\Vert _{C^2},\Vert \theta \Vert _{L^\infty _T H^1_x}) \int ^T_0 \Bigl (\Vert \nabla u\Vert _{L^2_x}^2+ \Vert \nabla \theta \Vert _{H^1_x}^2\Bigr )\Vert \nabla \theta \Vert _{H^{s-1}_x}^2 \mathrm{{d}}t, \end{aligned}$$

which, together with Gronwall’s inequality, implies (1.17).

Endpoint case $(\theta _0, u_0)\in H^2(\mathbb {R}^2)\times (L^2(\mathbb {R}^2))^2$

We recall the function $\eta =A^{-1}(\theta )$ defined in (2.11), and the parabolic $\eta$-equation (2.15):

$$\begin{aligned} \partial _t\eta +u\cdot \nabla \eta -\kappa \Delta \eta =0. \end{aligned}$$

(2.46)

We are going to derive the a priori $H^2$-Estimate for $\eta$ under the conditions

$$\begin{aligned} \text {div}u=0,\quad \nabla u\in L^2_\text {loc}([0,\infty ); (L^2(\mathbb {R}^2))^4)\hbox { and }\nabla \kappa \in L^4_\text {loc}([0,\infty ); (L^4(\mathbb {R}^2))^2). \end{aligned}$$

We test the above $\eta$-equation (2.46) by $\Delta ^2\eta$, to arrive at

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2}|\Delta \eta |^2\mathrm{d}x+\int \limits _{\mathbb {R}^2}\kappa |\nabla \Delta \eta |^2\mathrm{d}x=-\int \limits _{\mathbb {R}^2} \Bigl ( u\cdot \nabla \eta \Delta ^2\eta +\nabla \kappa \cdot \nabla \Delta \eta \Delta \eta \Bigr )\mathrm{d}x. \end{aligned}$$

By integration by parts, $\text {div}u=0$ and the embedding $L^1(\mathbb {R}^2)\hookrightarrow H^{-1}(\mathbb {R}^2)$, we derive

$$\begin{aligned} -\int \limits _{\mathbb {R}^2} u\cdot \nabla \eta \Delta ^2\eta \mathrm{d}x&=\int \limits _{\mathbb {R}^2}\nabla \Delta \eta \cdot \nabla u\cdot \nabla \eta - \nabla u:\nabla ^2\eta \Delta \eta \mathrm{d}x\\&\le 4\Vert \nabla u\Vert _{L^2_x}\Big ( \Vert \nabla \Delta \eta \Vert _{L^2_x}\Vert \nabla \eta \Vert _{H^1_x}+\Vert \nabla ^2\eta \Vert _{L^4_x}^2\Big ) \\&\le \frac{\kappa _*}{4}\Vert \nabla \Delta \eta \Vert _{L^2_x}^2 +C(\kappa _*)\Vert \nabla u\Vert _{L^2_x}^2\Vert \nabla \eta \Vert _{H^1_x}^2. \end{aligned}$$

Similarly, we have

$$\begin{aligned} -\int \limits _{\mathbb {R}^2} \nabla \kappa \cdot \nabla \Delta \eta \Delta \eta \mathrm{d}x&\le \Vert \nabla \kappa \Vert _{L^4_x}\Vert \nabla \Delta \eta \Vert _{L^2_x}\Vert \Delta \eta \Vert _{L^4_x} \\&\le \frac{\kappa _*}{4}\Vert \nabla \Delta \eta \Vert _{L^2_x}^2 +C(\kappa _*)\Vert \nabla \kappa \Vert _{L^4_x}^4\Vert \Delta \eta \Vert _{L^2_x}^2. \end{aligned}$$

To conclude, we have the following a priori $\dot{H}^2_x$-Estimate for $\eta$ and any positive time $T>0$ by Gronwall’s inequality

$$\begin{aligned} \Vert \Delta \eta (T)\Vert _{L^2_x}^2+\Vert \nabla \Delta \eta \Vert _{L^2_T L^2_x}^2&\le C(\kappa _*)\Big (\Vert \Delta \eta _0\Vert _{L^2_x}^2+\Vert \nabla u\Vert _{L^2_T L^2_x}^2\Vert \nabla \eta \Vert _{L^\infty _TL^2_x}^2\Big ) \\&\qquad \times \exp \Bigl (C\Big (\kappa _*)(\Vert \nabla u\Vert _{L^2_T L^2_x}^2+\Vert \nabla \kappa \Vert _{L^4_T L^4_x}^4\Big )\Bigr ). \end{aligned}$$

By view of the equivalence relation (2.12) as well as⁵

$$\begin{aligned} \Vert \nabla ^3\theta \Vert _{L^2_TL^2_x}\le C(\kappa _*, \Vert a\Vert _{C^2})\Bigl ( (\Vert \nabla \eta \Vert _{L^4_TL^4_x}^2+\Vert \nabla ^2\eta \Vert _{L^2_TL^2_x})\Vert \nabla \eta \Vert _{L^\infty _TL^2_x}+\Vert \nabla ^3\eta \Vert _{L^2_TL^2_x}\Bigr ), \end{aligned}$$

we derive the $H^2$-Estimate (1.18) for $\theta$ by virtue of the $H^1$-Estimate (2.18):

$$\begin{aligned}&\Vert \theta \Vert _{L^\infty _T H^2_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^2_x}^2 \le C\Big (\kappa _*,\Vert a\Vert _{C^2}, \kappa ^*\Big ) \Vert \theta _0\Vert ^2_{H^2}\Big (1+\Vert \nabla \theta _0\Vert ^2_{L^2}\Big )^2 \\&\quad \times \exp \Bigl (C(\kappa _*, \Vert a\Vert _{C^1})\Big (\Vert \nabla u\Vert _{L^2_T L^2_x}^2+\Vert u\Vert _{L^4_T L^4_x}^4+\Vert \nabla \theta \Vert _{L^4_T L^4_x}^4\Big )\Bigr ) . \end{aligned}$$

2.3.3 Case $(\theta _0,u_0)\in H^{1}(\mathbb {R}^2)\times (H^s(\mathbb {R}^2))^2$, $s\in (0,2]$

In this subsection we are going to sketch the proof of the $H^s$, $s\in (0,2)$-Estimate (1.19) for the divergence-free vector field u of the unique solution $(\theta , u)$ to the Boussinesq equations (1.1), under the assumption that $\theta _0\in H^1(\mathbb {R}^2)$, following the procedure in Sect. 2.3.1.

We deal with the endpoint case $(\theta _0, u_0)\in H^1(\mathbb {R}^2)\times (H^2(\mathbb {R}^2))^2$ similarly as for the endpoint case above.

Case $(\theta _0,u_0)\in H^{1}(\mathbb {R}^2)\times (H^s(\mathbb {R}^2))^2$, $s\in (0,2)$

Recall (2.5) for the definition of the Leray–Helmholtz projector $\mathbb {P}$ such that

$$\begin{aligned} \mathbb {P}u=u,\quad \mathbb {P}\nabla \Pi =0. \end{aligned}$$

We apply $\mathbb {P}$ to the velocity equation (1.1)$_2$ to arrive at

$$\begin{aligned} \partial _t u+\mathbb {P}(u\cdot \nabla u)-\mathbb {P}{{\,\mathrm{div}\,}}(\mu Su)=\mathbb {P}(\theta \mathbf {e_2}). \end{aligned}$$

(2.47)

We apply $\Delta _j$ to the above equation (2.47) to arrive at the equation for $u_j:=\Delta _j u$

$$\begin{aligned} \partial _tu_j+\mathbb {P}u\cdot \nabla u_j-\mathbb {P}{{\,\mathrm{div}\,}}(\mu Su_j) =\mathbb {P}[u,\Delta _j]\cdot \nabla u-\mathbb {P}\text {div}([\mu ,\Delta _j] Su) +\mathbb {P}(\theta _j \mathbf {e_2}). \end{aligned}$$

(2.48)

We take the $L^2(\mathbb {R}^2)$-inner product between (2.48) and the divergence-free dyadic piece $u_j=\mathbb {P}u_j$ and follow the similar argument as to arrive at (2.38), to deduce

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert u_j\Vert _{L^2_x}+2^{2j}\Vert u_j\Vert _{L^2_x}\le C(\mu _*) \left( \Vert [u,\Delta _j]\cdot \nabla u\Vert _{L^2_x}+2^j\Vert [\mu ,\Delta _j]\nabla u\Vert _{L^2_x}+\Vert \theta _j\Vert _{L^2_x}\right) . \end{aligned}$$

(2.49)

By use of the commutator estimate (2.28) in Lemma 2.1 again, we have the following commutator estimates as in (2.39):

$$\begin{aligned}\begin{aligned}&\Vert [u,\Delta _j]\nabla u\Vert _{L^{2}_x} \le C l_j 2^{j(1-s)}\Vert \nabla u\Vert _{L^2_x}\Vert \nabla u\Vert _{H^{s-1}_x}, \hbox { for } s\in (0,2),\\&2^j\Vert [\mu ,\Delta _j]\nabla u\Vert _{L^2_x} \le C l_j 2^{j(1-s)}\Vert \nabla \mu \Vert _{H^{\nu }_x}\Vert \nabla u\Vert _{H^{s-\nu }_x}, \hbox { for } \nu \in (-1,1),\, s\in (-1,\nu +1). \end{aligned}\end{aligned}$$

By virtue of the composition estimate (2.32) in Lemma 2.1:

$$\begin{aligned} \Vert \nabla \mu \Vert _{H^\nu _x}\le C(\Vert b\Vert _{C^{[\nu ]+2}}, \Vert \theta \Vert _{H^1})\Vert \nabla \theta \Vert _{H^\nu _x}, \end{aligned}$$

we derive similar as (2.42) that, for $0<s<\nu +1<2$,

$$\begin{aligned} \begin{aligned}&2^{2js}\Vert u_j\Vert _{L^\infty _T L^2_x}^2+2^{2j(s+1)}\Vert u_j\Vert _{L^2_T L^2_x}^2 \le 2^{2js}\Vert (u_0)_j\Vert _{L^2_x}^2 \\&\quad +C(\mu _*)\int ^T_0 2^{2j(s-1)}\Vert \theta _j\Vert _{L^2_x}^2 \mathrm{{d}}t +C(\mu _*,s,\nu ,\Vert b\Vert _{C^{[\nu ]+2}}, \Vert \theta \Vert _{L^\infty _T H^1_x}) (l_j)^2\\&\quad \times \int ^T_0 \Bigl ( \Vert \nabla u\Vert _{L^2_x}^2\Vert \nabla u\Vert _{H^{s-1}_x}^2+\Vert \nabla \theta \Vert _{H^{\nu }_x}^2\Vert \nabla u\Vert _{H^{s-\nu }_x}^2 \Bigr ){\mathrm {d}}t. \end{aligned} \end{aligned}$$

(2.50)

By the interpolation inequality (2.27):

$$\begin{aligned} \Vert \nabla \theta \Vert _{H^{\nu }_x}\Vert \nabla u\Vert _{H^{s-\nu }_x} \le C\Vert \nabla \theta \Vert _{L^2_x}^{1-\nu }\Vert \nabla \theta \Vert _{H^1_x}^\nu \Vert \nabla u\Vert _{H^{s-1}_x}^{\nu }\Vert \nabla u\Vert _{H^{s}_x}^{1-\nu }, \quad \hbox { for }\nu \in (0,1), \end{aligned}$$

and the $L^2$-Estimate (1.13):

$$\begin{aligned} \Vert u\Vert _{L^\infty _T L^2_x}^2+\Vert \nabla u\Vert _{L^2_T L^2_x}^2\le C(\mu _*) \left( \Vert u_0\Vert _{L^2_x}^2+T\Vert \theta _0\Vert _{L^2_x}^2\right) , \end{aligned}$$

(2.51)

we arrive at the following by Young’s inequality

$$\begin{aligned}&\Vert u\Vert _{L^\infty _T H^s_x}^2+\Vert \nabla u\Vert _{L^2_T H^{s}_x}^2 \le C(\mu _*)\Bigl ( \Vert u_0\Vert _{H^s_x}^2+T\Vert \theta _0\Vert _{ L^2_x}^2+\Vert \theta \Vert _{L^2_T H^{s-1}_x}^2\Bigr ) \\&\quad +C(\mu _*,s,\nu ,\Vert b\Vert _{C^2}, \Vert \theta \Vert _{L^\infty _T H^1_x}) \int ^T_0 (\Vert \nabla u\Vert _{L^2_x}^2 + \Vert \nabla \theta \Vert _{H^1_x}^2)\Vert \nabla u\Vert _{H^{s-1}_x}^2 \mathrm{{d}}t, \end{aligned}$$

which, together with Gronwall’s inequality, implies (1.19).

Endpoint case $(\theta _0, u_0)\in H^1(\mathbb {R}^2)\times (H^2(\mathbb {R}^2))^2$

We recall the u-equation (2.47) where

$$\begin{aligned} \text {div}(\mu Su)=\mu \Delta u+\nabla \mu \cdot Su. \end{aligned}$$

We test (2.47) by the divergence-free vector field $\Delta ^2 u$, to arrive at

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \limits _{\mathbb {R}^2}|\Delta u|^2\mathrm{d}x+\int \limits _{\mathbb {R}^2}\mu |\nabla \Delta u|^2\mathrm{d}x\\&\quad = \int \limits _{\mathbb {R}^2} \Bigl ( -u\cdot \nabla u \Delta ^2 u+\nabla \mu \cdot Su\cdot \Delta ^2 u -\nabla \mu \cdot \nabla \Delta u\cdot \Delta u +\Delta \theta \Delta u_2 \Bigr )\mathrm{d}x. \end{aligned}$$

By use of the embedding $L^1(\mathbb {R}^2)\hookrightarrow H^{-1}(\mathbb {R}^2)$ again, the right hand side can be bounded by

$$\begin{aligned}&C\left( \Vert \nabla u\Vert _{L^4_x}^2+\Vert u\Vert _{L^4_x}\Vert \nabla ^2u\Vert _{L^4_x}+\Vert \nabla ^2\mu \Vert _{L^2_x} \Vert \nabla u\Vert _{H^1_x}+\Vert \nabla \mu \Vert _{L^4_x} \Vert \nabla ^2u\Vert _{L^4_x}\right) \Vert \nabla \Delta u\Vert _{L^2_x} \\&\quad +\Vert \Delta \theta \Vert _{L^2_x}\Vert \Delta u\Vert _{L^2_x}. \end{aligned}$$

Thus, we have the following a priori $\dot{H}^2_x$-Estimate for u and any positive time $T>0$ by Young’s inequality and Gronwall’s inequality

$$\begin{aligned}&\Vert \Delta u(T)\Vert _{L^2_x}^2+\Vert \nabla \Delta u\Vert _{L^2_T L^2_x}^2 \\&\quad \le C(\mu _*)\left( \Vert \Delta u_0\Vert _{L^2_x}^2+\Vert \nabla u\Vert _{L^4_T L^4_x}^4+\Vert \nabla ^2\mu \Vert _{L^2_T L^2_x}^2\Vert \nabla u\Vert _{L^\infty _T L^2_x}^2+\Vert \Delta \theta \Vert _{L^2_T L^2_x}\Vert \Delta u\Vert _{L^2_T L^2_x} \right) \\&\quad \times \exp \Bigl (C\Bigl (\mu _*)(\Vert (u, \nabla \mu )\Vert _{L^4_T L^4_x}^4+\Vert \nabla ^2\mu \Vert _{L^2_T L^2_x}^2\Bigr )\Bigr ), \end{aligned}$$

which gives (1.20).

2.3.4 Case $(\theta _0, u_0)\in H^{s}(\mathbb {R}^2)\times (H^{s-2}(\mathbb {R}^2))^2$ or $H^{s-1}\times (H^s(\mathbb {R}^2))^2$, $s>2$

We are going to use the estimates in the high regularity regime in Lemma 2.1 to derive the $H^s$-Estimates (1.21)–(1.22)–(1.23)–(1.24) in Remark 1.4. Let $(l'_j)_{j\ge 0}$ be a normalized sequence in $\ell ^2(\mathbb {N})$ such that $l'_j\ge 0$ and $\sum _{j\ge 0}(l'_j)^2=1$.

Case $(\theta _0,u_0)\in H^{s}(\mathbb {R}^2)\times (H^{s-2}(\mathbb {R}^2))^2$, $s>2$

We can view the transport term $u\cdot \nabla \theta$ simply as a source term of the $\theta$-equation:

$$\begin{aligned} \partial _t\theta -\text {div}(\kappa \nabla \theta )=-u\cdot \nabla \theta \end{aligned}$$

Then, the preliminary estimate for $\theta _j=\Delta _j\theta$ in (2.43) can be replaced by

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\Vert \theta _j\Vert _{L^2_x}+2^{2j}\Vert \theta _j\Vert _{L^2_x} \le C(\kappa _*) \left( \Vert (u\cdot \nabla \theta )_j\Vert _{L^2_x}+2^j\Vert [\kappa ,\Delta _j]\nabla \theta \Vert _{L^2_x}\right) ,\quad j\ge 0. \end{aligned}$$

We apply Lemma 2.1 to derive the following estimates for $s>1$:

$$\begin{aligned}&\Vert (u\cdot \nabla \theta )_j\Vert _{L^2_x} \le C l'_j 2^{j(1-s)} \left( \Vert u\Vert _{L^\infty _x}\Vert \nabla \theta \Vert _{H^{s-1}_x}+\Vert u\Vert _{H^{s-1}_x}\Vert \nabla \theta \Vert _{L^\infty _x}\right) , \\&2^j\Vert [\kappa ,\Delta _j]\nabla \theta \Vert _{L^2_x} \le C l'_j 2^{ j(1-s)} \left( \Vert \nabla \kappa \Vert _{L^\infty _x}\Vert \nabla \theta \Vert _{H^{s-1}_x}+\Vert \nabla \kappa \Vert _{H^{s-1}_x}\Vert \nabla \theta \Vert _{L^\infty _x}\right) . \end{aligned}$$

Therefore, we have the following estimate similarly as in (2.44) for $s\in (2,3)$ by virtue of $H^{s-1}(\mathbb {R}^2)\hookrightarrow L^\infty (\mathbb {R}^2)$:

$$\begin{aligned} \begin{aligned}&2^{2js}\Vert \theta _j\Vert _{L^\infty _T L^2_x}^2+2^{2j(s+1)}\Vert \theta _j\Vert _{L^2_T L^2_x}^2 \le 2^{2js}\Vert (\theta _0)_j\Vert _{L^2_x}^2 \\&\quad + C(\kappa _*, s) (l'_j)^2\int ^T_0 \Vert u\Vert _{H^{s-1}_x}^2\Vert \nabla \theta \Vert _{H^{s-1}_x}^2{\mathrm {d}}t \\&\quad + C(\kappa _*, a, \Vert \theta \Vert _{L^\infty _T L^\infty _x})(l_j')^2 \int ^T_0 \Vert \nabla \theta \Vert _{L^\infty _x}^2\Vert \nabla \theta \Vert _{H^{s-1}_x}^2 \mathrm{{d}}t,\quad j\ge 0, \end{aligned} \end{aligned}$$

which, together with the $L^2$-Estimate (2.45) and the Gronwall’s inequality, implies (1.21).

For $s\ge 3$, we make use of the following commutator estimate

$$\begin{aligned} \Vert [u,\Delta _j]\nabla \theta \Vert _{L^{2}_x}&\le C l'_j 2^{j(1-s)}( \Vert \nabla u\Vert _{L^\infty _x}\Vert \nabla \theta \Vert _{H^{s-2}_x}+\Vert \nabla u\Vert _{H^{s-2}_x}\Vert \nabla \theta \Vert _{L^\infty _x}), \end{aligned}$$

(2.52)

such that the estimate (1.22) follows.

Case $(\theta _0,u_0)\in H^{s-1}(\mathbb {R}^2)\times (H^{s}(\mathbb {R}^2))^2$, $s>2$

We recall the preliminary estimate for $u_j$ in (2.49). We apply Lemma 2.1 to derive the following commutator estimates for $s\in (2,3)$ and $\nu \in (s-2,1)\subset (0,1)$

$$\begin{aligned}\begin{aligned}&\Vert [u,\Delta _j]\nabla u\Vert _{L^{2}_x} \le C l_j 2^{j(1-s)}\Vert \nabla u\Vert _{H^{\nu }_x}\Vert \nabla u\Vert _{H^{s-1-\nu }_x}, \\&\quad 2^j\Vert [\mu ,\Delta _j]\nabla u\Vert _{L^2_x} \le C l'_j 2^{ j(1-s)} ( \Vert \nabla \mu \Vert _{L^\infty _x}\Vert \nabla u\Vert _{H^{s-1}_x}+\Vert \nabla \mu \Vert _{H^{s-1}_x}\Vert \nabla u\Vert _{L^\infty _x}), \end{aligned}\end{aligned}$$

which implies then

$$\begin{aligned}&2^{2js}\Vert u_j\Vert _{L^\infty _T L^2_x}^2+2^{2j(s+1)}\Vert u_j\Vert _{L^2_T L^2_x}^2 \le 2^{2js}\Vert (u_0)_j\Vert _{L^2_x}^2 +C(\mu _*)\int ^T_0 2^{2j(s-1)}\Vert \theta _j\Vert _{L^2_x}^2 \mathrm{{d}}t \\&\quad + C(\mu _*,s,\nu ) (l_j)^2 \int ^T_0\Vert \nabla u\Vert _{H^{\nu }_x}^2\Vert \nabla u\Vert _{H^{s-1-\nu }_x}^2{\mathrm {d}}t \\&\quad +C(\mu _*,s,\Vert b\Vert _{C^{[s]+1}}, \Vert \theta \Vert _{L^\infty _TH^1_x}) (l'_j)^2\int ^T_0 \Vert \nabla \theta \Vert _{L^\infty _x}^2\Vert \nabla u\Vert _{H^{s-1}_x}^2 +\Vert \nabla \theta \Vert _{H^{s-1}_x}^2\Vert \nabla u\Vert _{L^\infty _x}^2 \mathrm{{d}}t, \quad j\ge 0. \end{aligned}$$

This, together with the $L^2$-Estimate (2.51) and Sobolev’s embedding $H^{s-1}(\mathbb {R}^2)\hookrightarrow L^\infty (\mathbb {R}^2)$, implies (1.23) where $\nu \in (0,1)$ is taken to be a small constant bigger than $s-2$.

For $s\ge 3$, we use the commutator estimate (2.52) with $\theta$ replaced by u, to arrive at (1.24).