Global well-posedness for a rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number

Abstract

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

Introduction

In geophysics, thermal convection is often influenced by planetary rotation. Coherence structures in convection under moderate rotation are exclusively cyclonic. However, for rapid rotation, experiments have revealed a transition to equal populations of cyclonic and anticyclonic structures. For instance, the flow visualization experiments of Vorobieff and Ecke [17] identified a striking topological change in the dynamics of the vortices. In the strongly nonlinear and turbulent regimes, plume generation in the thermal boundary layer results in a new population of anticyclonic plumes, in addition to the cyclonic population. Also, the distribution of cyclonic and anticyclonic coherent structures approaches a balance as the Rossby number approaches zero. In order to study such interesting phenomenon numerically, Sprague et al. [14] derived a reduced system of equations for rotationally constrained convection valid in the asymptotic limit of thin columnar structures and rapid rotation. Performing a numerical simulation of Rayleigh–Bénard convection in an infinite layer rotating uniformly about the vertical axis, visualization indicates the existence of cyclonic and anticyclonic vortical population in [14], which is consistent with the experimental results described in [17]. Also see [9, 10].

The reduced three-dimensional rapidly rotating convection model of tall columnar structure introduced in [14] is given by the system:

(1.1)
(1.2)
(1.3)
(1.4)
(1.5)

The above system is considered subject to periodic boundary conditions in \({\mathbb {R}}^3\) with fundamental periodic domain \(\Omega = [0,2\pi L]^2 \times [0,2\pi ]\). The unknowns are functions of (xyzt), where \((x,y,z)\in \Omega \) and \(t\ge 0\). In (1.1)–(1.5), \(\nabla _h=(\frac{\partial }{\partial x}, \frac{\partial }{\partial y})\) denotes the horizontal gradient and \(\Delta _h=\frac{\partial ^2}{\partial x^2}+ \frac{\partial ^2}{\partial y^2}\) denotes the horizontal Laplacian. In the model, \({\mathbf {u}}=(u,v)\) is the horizontal component of the three-dimensional velocity vector field (uvw), and \(\omega = \nabla _h \times {\mathbf {u}}=\partial _x v - \partial _y u\) denotes the vertical component of the vorticity. Moreover, the stream function for the horizontal flow is denoted by \(\psi =\Delta _h ^{-1}\omega \) such that its horizontal average \(\overline{\psi } = \frac{1}{4 \pi ^2 L^2} \int _{[0,2\pi L]^2} \psi (x,y,z,t) \mathrm{d}x\,\mathrm{d}y = 0\). In addition, \(\theta '=\theta - \overline{\theta }\) represents the horizontal fluctuation of the temperature \(\theta \), where \(\overline{\theta }(z,t)=\frac{1}{4\pi ^2 L^2} \int _{[0,2\pi L]^2} \theta (x,y,z,t) \mathrm{d}x\,\mathrm{d}y\) is the horizontal-mean temperature. In the above system, Ra is the Rayleigh number and Pr is the Prandtl number. We comment that the assumption \(\nabla _h \cdot {\mathbf {u}}=0\) means that the horizontal flow is divergence-free.

The global regularity for system (1.1)–(1.5) is unknown. The main difficulty of analyzing (1.1)–(1.5) lies in the fact that the physical domain is three-dimensional, whereas the regularizing viscosity acts only on the horizontal variables, and the equations contain troublesome terms \(\frac{\partial \phi }{\partial z}\) and \(\frac{\partial w}{\partial z}\) involving the derivative in the vertical direction. In our recent paper [4], system (1.1)–(1.5) was regularized by a weak dissipation term and the global well-posedness of strong solutions was established for the regularized system.

System (1.1)–(1.5) is a reduced model derived from the three-dimensional Boussinesq equations by using the asymptotic theory. Generally speaking, the Boussinesq approximation for buoyancy-driven flow is applied to problems where the fluid varies in temperature from one place to another, driving a flow of fluid and heat transfer. In particular, the Boussinesq approximation to the Rayleigh–Bénard convection is a system of equations coupling the three-dimensional Navier–Stokes equations to a heat advection-diffusion equation. For small Rossby number (i.e., rapid rotation) and large ratio of the depth of the fluid layer to the horizontal scale (i.e., tall columnar structures), the 3D Boussinesq equations under the influence of a Coriolis force term can be reduced to system (1.1)–(1.5) asymptotically. The derivation of model (1.1)–(1.5) was motivated by the Taylor–Proudman constraint [13, 15] which suggests that rapidly rotating convection takes place in tall columnar structures.

According to the derivation of model (1.1)–(1.5) in [14] from the 3D Boussinesq equations, the state variables, i.e., the velocity, pressure and temperature, are expanded in terms of the small parameter Ro, which stands for the Rossby number. For rapidly rotating flow, i.e., Ro \(\ll 1\), the leading-order flow is horizontally divergence-free (see [14]). Roughly speaking, if the Rossby number is small in the Boussinesq equations, then the Coriolis force and the pressure gradient force are relatively large, which results in an equation in which the leading order terms are in geostrophic balance: the pressure gradient force is balanced by the Coriolis effect. Then taking the curl of the geostrophic balance equation implies that the horizontal flow is divergence-free, namely \(\nabla _h \cdot {\mathbf {u}}=u_x + v_y=0\). In addition, the term \(\frac{\partial \psi }{\partial z}\) in Eq. (1.1) also originates from the geostrophic balance.

Since the original Boussinesq equations are considered in a tall column \((x,y,{\tilde{z}}) \in [0,2\pi L]^2 \times [0,2\pi \ell ]\), where the aspect ratio \(\ell /L \gg 1\), it is natural to introduce the scaled vertical variable \(z={\tilde{z}}/\ell \in [0,2\pi ]\) which appears in system (1.1)–(1.5). Set \(u(x,y,z)=\tilde{u}(x,y,{\tilde{z}})\), \(v(x,y,z)={\tilde{v}}(x,y,{\tilde{z}})\) and \(w(x,y,z)={\tilde{w}}(x,y,{\tilde{z}})\), where \(({\tilde{u}}, {\tilde{v}}, {\tilde{w}})\) represents the velocity vector field for the original Boussinesq equations. Note, the divergence-free condition of \(({\tilde{u}}, {\tilde{v}}, {\tilde{w}})\) reads \(\frac{\partial \tilde{u}}{\partial x} + \frac{\partial {\tilde{v}}}{\partial y} + \frac{\partial {\tilde{w}}}{\partial {\tilde{z}}} =0\), which implies that \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{1}{\ell } \frac{\partial w}{\partial z}=0\). Then, for \(\ell \gg 1\), one can ignore the term \(\frac{1}{\ell } \frac{\partial w}{\partial z}\) and obtain that \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}=0\). In sum, the fast rotation and the tall columnar structure both imply that to leading order terms the horizontal flow is divergence-free. Moreover, the absence of vertical diffusion in system (1.1)–(1.5) is also a consequence of the large aspect ratio of the fluid region. Furthermore, it is remarked in [14] that in the classical small-aspect-ratio (flat) regime, the strong stable stratification permits weak vertical motions only, while in the present large-aspect-ratio (tall) case, the unstable stratification permits substantial vertical motions.

There are two dimensionless numbers in system (1.1)–(1.5). They are the Prandtl number Pr and the Rayleigh number Ra. The Prandtl number Pr represents the ratio of molecular diffusion of momentum to molecular diffusion of heat. More precisely, one defines

$$\begin{aligned} \text {Pr}=\frac{\text {momentum diffusivity}}{\text {thermal diffusivity}} = \frac{\nu }{\alpha }= \frac{\mu /\rho }{k/(c_p \rho )} =\frac{c_p \mu }{k}. \end{aligned}$$
(1.6)

Here, \(\nu \) represents momentum diffusivity, i.e., kinematic viscosity. Notice that \(\nu =\mu /\rho \), where \(\mu \) is dynamics viscosity and \(\rho \) is the constant density. Also, \(\alpha \) stands for the thermal diffusivity, which is equal to \(k/(c_p \rho )\), where k is thermal conductivity and \(c_p\) represents the specific heat capacity of the fluid. Fluids with small Prandtl numbers are free-flowing liquids with high thermal conductivity and are therefore a good choice for heat transfer liquids. Liquid metals such as mercury have small Prandtl numbers. On the other hand, with increasing viscosity, the Prandtl number also increases, leads to the phenomenon that the momentum transport dominates over the heat transport and acts on a faster time scale. For instance, concerning the engine oil, convection is very effective in transferring energy in comparison with pure conduction, so momentum diffusivity is dominant. Another example is Earth’s mantle, which has extremely large Prandtl number.

The Rayleigh number Ra is another dimensionless number appeared in model (1.1)–(1.5). It represents the strength of the buoyancy in the fluid driven by the heat gradient.

In this manuscript, we consider rapidly rotating convection in the limit of infinite Prandtl number Pr. Under such scenario, the dynamics of the momentum acts on a much faster time scale than the heat dynamics. For this problem, the appropriate time scale is the horizontal thermal diffusion time. Using the substitution (cf. [14])

$$\begin{aligned} t \rightarrow \text {Pr}\; t, \;\;\; {\mathbf {u}} \rightarrow \frac{1}{\text {Pr}} {\mathbf {u}}, \;\;\; w \rightarrow \frac{1}{\text {Pr}} w, \end{aligned}$$

system (1.1)–(1.5) becomes

$$\begin{aligned}&\frac{1}{\text {Pr}}\left( \frac{\partial w}{\partial t} + {\mathbf {u}} \cdot \nabla _h w\right) + \frac{\partial \psi }{\partial z} = \text {Ra}\, \theta ' + \Delta _h w, \end{aligned}$$
(1.7)
$$\begin{aligned}&\frac{1}{\text {Pr}}\left( \frac{\partial \omega }{\partial t} + {\mathbf {u}} \cdot \nabla _h \omega \right) - \frac{\partial w}{\partial z} = \Delta _h \omega , \end{aligned}$$
(1.8)
$$\begin{aligned}&\frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} = \Delta _h \theta ', \end{aligned}$$
(1.9)
$$\begin{aligned}&\frac{\partial (\overline{\theta ' w})}{\partial z} = \frac{\partial ^2 {\overline{\theta }}}{\partial z^2}, \end{aligned}$$
(1.10)
$$\begin{aligned}&\nabla _h \cdot {\mathbf {u}}=0. \end{aligned}$$
(1.11)

Then in the limit of infinite Prandtl number, i.e., letting \(\text {Pr} \rightarrow \infty \) in system (1.7)–(1.11), one formally obtains the following system of equations

$$\begin{aligned}&\frac{\partial \psi }{\partial z} = \text {Ra}\,\theta ' + \Delta _h w, \end{aligned}$$
(1.12)
$$\begin{aligned}&- \frac{\partial w}{\partial z} = \Delta _h \omega , \end{aligned}$$
(1.13)
$$\begin{aligned}&\frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} = \Delta _h \theta ', \end{aligned}$$
(1.14)
$$\begin{aligned}&\frac{\partial (\overline{\theta ' w})}{\partial z} = \frac{\partial ^2 {\overline{\theta }}}{\partial z^2}, \end{aligned}$$
(1.15)
$$\begin{aligned}&\nabla _h \cdot {\mathbf {u}}=0. \end{aligned}$$
(1.16)

The system is considered subject to periodic boundary conditions in \({\mathbb {R}}^3\) with fundamental periodic domain \(\Omega = [0,2\pi L]^2 \times [0,2\pi ]\). Here, \(\omega = \nabla _h \times {\mathbf {u}}\), \(\psi = \Delta _h^{-1} \omega \) such that its horizontal average \(\overline{\psi }=0\). Recall that the horizontal thermal fluctuation \(\theta '\) of the temperature \(\theta \) is defined as \(\theta '=\theta - \overline{\theta }\), where \(\overline{\theta }(z,t)=\frac{1}{4\pi ^2 L^2} \int _{[0,2\pi L]^2} \theta (x,y,z,t) \mathrm{d}x\,\mathrm{d}y\) is the horizontal-mean temperature. In system (1.12)–(1.16), the velocity field acting on a very fast time scale adjusts instantaneously to the dynamics of the thermal fluctuations, demonstrated by the linear equations (1.12) and (1.13). Therefore, the initial condition is imposed on \(\theta '\) only: \(\theta '(0)=\theta '_0\). The purpose of this work is to prove the global well-posedness of weak and strong solutions for system (1.12)–(1.16) defined on a fundamental periodic space domain \(\Omega =[0,2\pi L]^2 \times [0,2\pi ]\). Also, in order to obtain the uniqueness of the temperature \(\theta \), we assume that the average temperature is zero, i.e., \(\int _{\Omega } \theta (x,y,z,t) \mathrm{d}x\,\mathrm{d}y \mathrm{d}z =0\), for all \(t\ge 0\).

For the sake of simplicity and clarity, we adopt the periodic boundary conditions for the model. In particular, under periodic boundary conditions, it is rather convenient to express the explicit solutions of linear equations (1.12) and (1.13) in terms of Fourier series, which enable us to find the precise relationship between the regularities of the temperature and velocity field.

We remark that the original model derived in [14] involves multiple time scales t and T. In fact, there was an additional term \(\partial _T \overline{\theta }\) in Eq. (1.15), where \(T:= A_T^{-1} t\), \(A_T \gg 1\), represents a slow time. Since we are mainly interested in the evolution of the horizontal temperature fluctuation \(\theta '\) with respect to time t, and since the horizontal-mean temperature \(\overline{\theta }\) varies slowly in t, we drop the term \(\partial _T \overline{\theta }\) from the original model.

In the literature, there were some analytical studies for the three-dimensional Boussinesq equations in the limit of infinite Prandtl number. Wang [18] rigorously justified the infinite Prandtl number convection model as the limit of the Boussinesq equations when the Prandtl number approaches infinity (see also [19, 20]). Also, for infinite Prandtl number convection, there have been several rigorous derivation of upper bounds of the upwards heat flux, as given by the Nusselt number Nu, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime Ra \(\gg 1\). For example, the work [5] by Constantin and Doering was one of the early papers in the literature for this topic. More recently, by combining the background field method and the maximal regularity in \(L^{\infty }\), Otto and Seis [12] showed that \(\text {Nu}\lessapprox \text {Ra}^{1/3} (\log \log \text {Ra})^{1/3}\)—an estimate that is only a double logarithm away from the supposedly optimal scaling \(\text {Nu} \sim \text {Ra}^{1/3}\). See also [6,7,8, 11, 21, 22] and references therein.

It is worth mentioning that in [2], Cao, Farhat and Titi established the global regularity for an inviscid three-dimensional slow limiting ocean dynamics model, which was derived as a strong rotation limit of the rotating and stratified Boussinesq equations.

The paper is organized as follows. In Sect. 2, we state main results of the paper, i.e., the global well-posedness of weak solutions and strong solutions for the infinite Prandtl number convection (1.12)–(1.16). In Sect. 3, we provide some auxiliary inequalities and some well-known identities, which will be used repeatedly in our energy estimates. In Sect. 4, we give a detailed proof for the global well-posedness of weak solutions. Finally, Sect. 5 is devoted to the proof for the global well-posedness of strong solutions.

Main results

In this section, we give definitions of weak solutions as well as strong solutions for system (1.12)–(1.16). Then, we state the main results of the manuscript, namely the global well-posedness of weak solutions and strong solutions for system (1.12)–(1.16), subject to periodic boundary conditions on a three-dimensional fundamental periodic domain \(\Omega =[0,2\pi L]^2 \times [0,2\pi ]\).

Weak solutions

For a periodic function f defined on the periodic domain \(\Omega =[0,2\pi L]^2 \times [0,2\pi ]\), the horizontal mean of f is defined as

$$\begin{aligned} {\overline{f}}(z)= \frac{1}{4 \pi ^2 L^2}\int _{[0, 2\pi L]^2} f(x,y,z) \mathrm{d}x \mathrm{d}y. \end{aligned}$$
(2.1)

We define the space \(H^1_h(\Omega )\) of periodic functions on \(\Omega \) with horizontal average zero by

$$\begin{aligned} H^1_h(\Omega )=\{f\in L^2(\Omega ): \nabla _h f \in L^2(\Omega ) \; \text {and} \; \overline{f}=0 \}, \end{aligned}$$

with the norm \(\Vert f\Vert _{H^1_h(\Omega )}= \left( \int _{\Omega } |\nabla _h f|^2 \mathrm{dx}\,\mathrm{d}y\,\mathrm{d}z\right) ^{1/2}\). We denote by \((H^1_h(\Omega ))'\) the dual space of \(H^1_h(\Omega )\).

For \(s>0\), we denote the space of \(H^s\) periodic functions on \([0,2\pi ]\) with average value zero by:

$$\begin{aligned} \dot{H}^s(0,2\pi )= \{\phi \in H^s(0,2\pi ): \int _0^{2\pi } \phi (z) \mathrm{d}z =0\}. \end{aligned}$$
(2.2)

Also, we denote the dual space of \(\dot{H}^1(0,2\pi )\) by \(H^{-1}(0,2\pi )= (\dot{H}^1(0,2\pi ))'\).

Recall \(\theta '=\theta - {\overline{\theta }}\) represents the fluctuation of the temperature \(\theta \), about the horizontal average. Also, \(({\mathbf {u}}, w)=(u,v,w)\) is the three-dimensional velocity vector field on the periodic domain \(\Omega \).

Let us define a weak solution for system (1.12)–(1.16).

Definition 2.1

We call \((\theta ', \overline{\theta }, {\mathbf {u}}, w)\) a weak solution on [0, T] for system (1.12)–(1.16) if

$$\begin{aligned}&\theta ' \in L^2(0,T;H^1_h(\Omega )) \cap C([0,T];L^2(\Omega )); \;\;\; \theta '_t \in L^2(0,T;(H^1_h(\Omega ))'); \nonumber \\&\quad \overline{\theta } \in L^2(0,T; \dot{H}^1(0,2\pi )); \nonumber \\&\quad \Delta _h {\mathbf {u}}, \; \Delta _h w \in C ([0,T];L^2(\Omega )); \;\; {\mathbf {u}}_z, w_z, \Delta _h \omega , \nabla _h^3 w \in L^2(\Omega \times (0,T)), \end{aligned}$$

and the equations hold in the function spaces specified below:

$$\begin{aligned}&\frac{\partial \psi }{\partial z} = \text {Ra} \, \theta ' + \Delta _h w, \;\; \text {in} \;\; L^2(0,T;H^1_h(\Omega )) \cap C([0,T];L^2(\Omega )), \end{aligned}$$
(2.3)
$$\begin{aligned}&- \frac{\partial w}{\partial z} = \Delta _h \omega , \;\; \text {in} \;\; L^2(\Omega \times (0,T)), \end{aligned}$$
(2.4)
$$\begin{aligned}&\frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} = \Delta _h \theta ', \;\; \text {in} \;\; L^2(0,T;(H^1_h(\Omega ))'), \end{aligned}$$
(2.5)
$$\begin{aligned}&\frac{\partial (\overline{\theta ' w})}{\partial z} = \frac{\partial ^2 {\overline{\theta }}}{\partial z^2}, \;\; \text {in} \;\; L^2(0,T;H^{-1}(0,2\pi )), \end{aligned}$$
(2.6)

with \(\nabla _h \cdot {\mathbf {u}}=0\), \(\omega = \nabla _h \times {\mathbf {u}}\), \(\psi = \Delta _h^{-1} \omega \), and \(\overline{\theta '}=\overline{w}=\overline{\omega }=\overline{\psi }=0\), \(\overline{\mathbf{u}}=0\), such that the initial condition \(\theta '(0)=\theta '_0 \in L^2(\Omega )\) is satisfied.

According to the derivation of model (1.1)–(1.5) in [14], the quantities \(\theta '\), \({\mathbf {u}}\) and w are “fluctuating” quantities about the horizontal mean, i.e., the original quantities subtracted by their horizontal means. Therefore, in the above definition of weak solutions, all quantities are demanded to have horizontal average zero.

In Definition 2.1, the horizontal-mean temperature \(\overline{\theta }(z,t) = \frac{1}{4\pi ^2 L^2}\int _{[0,2\pi L]^2} \theta (x,y,z,t) \mathrm{d}x\,\mathrm{d}y\) belongs to the space \(\dot{H}^1(0,2\pi )\), which demands that \(\int _0^{2\pi } \overline{\theta }(z,t) \mathrm{d}z =0\) due to (2.2). Therefore, \(\int _{\Omega } \theta (x,y,z,t) \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z =0\), namely the average temperature is zero for all \(t \in [0,T]\).

In the next theorem, we state the existence and uniqueness of global weak solutions to system (1.12)–(1.16) as well as the continuous dependence on initial data.

Theorem 2.2

(Global well-posedness of weak solutions) Assume \(\theta '_0 \in L^2(\Omega )\) with \(\overline{\theta _0'}= 0\). Then, system (1.12)–(1.16) has a unique weak solution \((\theta ', \overline{\theta }, {\mathbf {u}}, w)\) for all \(t\ge 0\), in the sense of Definition 2.1. Moreover, the solution satisfies the following energy equality:

$$\begin{aligned} \frac{1}{2} \Vert \theta '(t)\Vert _2^2 + \int _0^t \Vert \nabla _h \theta ' (s)\Vert _2^2 \mathrm{d}s + 4\pi ^2 L^2 \int _0^t \int _0^{2\pi } |\partial _z \overline{\theta }(z,s)|^2 \mathrm{d}z \mathrm{d}s = \frac{1}{2} \Vert \theta '_0\Vert _2^2, \end{aligned}$$
(2.7)

for all \(t\ge 0\). Also, the following decay estimates are valid:

$$\begin{aligned}&\Vert \theta '(t)\Vert _2^2 + \Vert \Delta _h {\mathbf {u}}(t)\Vert _2^2 + \Vert \Delta _h w(t)\Vert _2^2 \le C e^{-\frac{2}{L^2} t} \Vert \theta '_0\Vert _2^2, \;\; \text {for all} \;\; t\ge 0; \\&\quad \int _0^{2\pi } |\overline{\theta }_z(z,t)|^2 \mathrm{d}z \le C e^{-\frac{4}{L^2} t} \Vert \theta '_0\Vert _2^4, \;\; \text {for all} \;\; t\ge 0. \end{aligned}$$

In addition, if \(\{{\theta '_{0,n}}\}\) is a sequence of initial data in \(L^2(\Omega )\) such that \({\theta '_{0,n}} \rightarrow \theta '_0\) in \(L^2(\Omega )\), then the corresponding weak solutions \(\{(\theta '_n, \overline{\theta }_n, {\mathbf {u}}_n, w_n)\}\) and \((\theta ',\overline{\theta },{\mathbf {u}}, w)\) with \(\theta '_n(0)=\theta '_{0,n}\) and \(\theta '(0)=\theta '_0\) satisfy \(\theta '_n \rightarrow \theta '\) in \(C([0,T];L^2(\Omega )) \cap L^2(0,T;H^1_h(\Omega ))\), \(\overline{\theta }_n \rightarrow \overline{\theta }\) in \(L^2(0,T; \dot{H}^1(0,2\pi ))\), and \(({\mathbf {u}}_n,w_n)\rightarrow ({\mathbf {u}}, w)\) in \(L^2(0,T;H^1(\Omega ))\).

Strong solutions

We define a strong solution of system (1.12)–(1.16).

Definition 2.3

We call \((\theta ', \overline{\theta }, {\mathbf {u}}, w)\) a strong solution on [0, T] for system (1.12)–(1.16) if

$$\begin{aligned}&\theta ' \in L^{\infty } (0,T;H^1(\Omega ))\cap C([0,T];L^2(\Omega )) \, \text {such that}\; \Delta _h \theta ',\; \nabla _h \theta '_z \in L^2(\Omega \times (0,T)) ; \end{aligned}$$
(2.8)
$$\begin{aligned}&\theta '_t \in L^2(\Omega \times (0,T)) ; \end{aligned}$$
(2.9)
$$\begin{aligned}&\overline{\theta } \in L^{\infty }(0,T; \dot{H}^2(0,2\pi )); \end{aligned}$$
(2.10)
$$\begin{aligned}&\Delta _h {\mathbf {u}}, \; \Delta _h w \in L^{\infty } (0,T;H^1(\Omega )); \;\; {\mathbf {u}}_z, w_z \in L^{\infty }(0,T; L^2(\Omega )), \end{aligned}$$
(2.11)

and the equations hold in the function spaces specified below:

$$\begin{aligned}&\frac{\partial \psi }{\partial z} = \text {Ra} \, \theta ' + \Delta _h w, \;\; \text {in} \;\; L^{\infty }(0,T;H^1(\Omega )), \end{aligned}$$
(2.12)
$$\begin{aligned}&- \frac{\partial w}{\partial z} = \Delta _h \omega , \;\; \text {in} \;\; L^{\infty }(0,T;L^2(\Omega )), \end{aligned}$$
(2.13)
$$\begin{aligned}&\frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} = \Delta _h \theta ', \;\; \text {in} \;\; L^2(\Omega \times (0,T)), \end{aligned}$$
(2.14)
$$\begin{aligned}&\frac{\partial (\overline{\theta ' w})}{\partial z} = \frac{\partial ^2 {\overline{\theta }}}{\partial z^2}, \;\; \text {in} \;\; L^{\infty }(0,T;L^2(0,2\pi )), \end{aligned}$$
(2.15)

with \(\nabla _h \cdot {\mathbf {u}}=0\), \(\omega = \nabla _h \times \mathbf{u}\), \(\psi = \Delta _h^{-1} \omega \), and \(\overline{\theta '}=\overline{w}=\overline{\omega }=\overline{\psi }=0\), \(\overline{\mathbf{u}}=0\), such that the initial condition \(\theta '(0) =\theta '_0 \in H^1_0(\Omega )\) is satisfied.

The following theorem states the existence and uniqueness of global strong solutions to system (1.12)–(1.16).

Theorem 2.4

(Global well-posedness of strong solutions) Assume \(\theta '_0 \in H^1(\Omega )\) with \(\overline{\theta _0'}= 0\). Then, system (1.12)–(1.16) has a unique strong solution \((\theta ', \overline{\theta }, {\mathbf {u}}, w)\) for all \(t\ge 0\), in the sense of Definition 2.3. Also, energy equality (2.7) is valid.

Preliminaries

We state some inequalities which will be useful in our estimates. Let \(\Omega =[0,2\pi L]^2 \times [0,2\pi ]\) be a three-dimensional fundamental periodic domain.

The following is an anisotropic Ladyzhenskaya-type inequality which has been proved in [3].

Lemma 3.1

Let \(f\in H^1(\Omega )\), \(g\in L^2(\Omega )\) with \(\nabla _h g\in L^2(\Omega )\), and \(h\in L^2(\Omega )\). Then

$$\begin{aligned}&\int _{\Omega } |fgh| \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \nonumber \\&\quad \le C (\Vert f\Vert _2+\Vert \nabla _h f\Vert _2)^{\frac{1}{2}} \left( \Vert f\Vert _2 + \Vert f_z\Vert _2 \right) ^{\frac{1}{2}} \Vert g\Vert _2^{\frac{1}{2}} (\Vert g\Vert _2+\Vert \nabla _h g\Vert _2)^{\frac{1}{2}} \Vert h\Vert _2. \end{aligned}$$
(3.1)

Definition 3.2

Let \(s\ge 0\). We say \(\partial _z^{s} f \in L^2(\Omega )\) if \(f\in L^2(\Omega )\) satisfying

$$\begin{aligned} \Vert \partial _z^{s} f \Vert _2^2 = \sum _{{\mathbf {k}} = (k_1,k_2,k_3) \in {\mathbb {Z}}^3} |k_3|^{2s} |{\hat{f}} ({\mathbf {k}})|^2 < \infty . \end{aligned}$$

Lemma 3.3

Let \(s>1/2\). Assume f, \(\partial _z^s f \in L^2(\Omega )\), then

$$\begin{aligned} \sup _{z\in [0,2\pi ]} \int _{[0,2\pi L]^2} |f(x,y,z)|^2 \mathrm{d}x\,\mathrm{d}y \le C(\Vert f\Vert _2^2+\Vert \partial _z^s f\Vert _2^2). \end{aligned}$$
(3.2)

Proof

Thanks to the one-dimensional imbedding \(\Vert \phi \Vert _{L^{\infty }(0,2\pi )} \le C\Vert \phi \Vert _{H^{s}(0,2\pi )}\), when \(s>1/2\), then for a.e. \(z\in [0,2\pi ]\),

$$\begin{aligned}&\int _{[0,2\pi L]^2} |f(x,y,z)|^2 \mathrm{d}x\,\mathrm{d}y \\&\quad \le C\int _{[0,2\pi L]^2} \left( \int _0^{2\pi } (|f|^2 + |\partial _z^s f|^2) \mathrm{d}z \right) \mathrm{d}x\,\mathrm{d}y = C (\Vert f\Vert _2^2 + \Vert \partial _z^s f\Vert _2^2). \end{aligned}$$

\(\square \)

Recall the periodic domain \(\Omega =[0,2\pi L]^2 \times [0,2\pi ]\). For any periodic function \(f\in H^1_h(\Omega )\), i.e., \(f\in L^2(\Omega )\) with \(\nabla _h f \in L^2(\Omega )\) and \(\overline{f}=0\), the Poincaré inequality is valid:

$$\begin{aligned} \Vert f\Vert _2^2 \le L^2 \Vert \nabla _h f\Vert _2^2. \end{aligned}$$
(3.3)

Next, we state some identities which will be employed in the energy estimate. For sufficiently smooth periodic functions \({\mathbf {u}}\), f and g on \(\Omega \), such that \(\nabla _h \cdot \mathbf {u}=0\), an integration by parts shows

$$\begin{aligned} \int _{\Omega }(\mathbf {u} \cdot \nabla _h f) g \, \mathrm{d}x\,\mathrm{d}y \mathrm{d}z=- \int _{\Omega } (\mathbf {u} \cdot \nabla _h g) f \, \mathrm{d}x\,\mathrm{d}y \mathrm{d}z. \end{aligned}$$
(3.4)

This implies

$$\begin{aligned} \int _{\Omega } (\mathbf {u} \cdot \nabla _h f) f \, \mathrm{d}x\,\mathrm{d}y \mathrm{d}z=0, \end{aligned}$$
(3.5)

if \(\nabla _h \cdot \mathbf {u}=0\).

Note that the horizontal velocity \({\mathbf {u}}\), the vertical component \(\omega \) of the vorticity and the horizontal stream function \(\psi \) such that \(\overline{\psi }=0\) have the following relations:

$$\begin{aligned} \omega =\nabla _h \times {\mathbf {u}}=v_x-u_y, \;\;\; \omega =\Delta _h \psi ,\;\;\; {\mathbf {u}} = (u,v) =(-\psi _y,\psi _x). \end{aligned}$$
(3.6)

Weak solutions

In this section, we prove the global well-posedness of weak solutions to system (1.12)–(1.16) by using the Galerkin method.

Existence of weak solutions

Galerkin approximation

Let \(P_m\) be an orthogonal projection onto lower Fourier modes, namely \(P_m \phi = \sum _{\begin{array}{c} \mathbf {k} \in {\mathbb {Z}}^3 \\ |{\mathbf {k}}| \le m \end{array} } {\hat{\phi }}({\mathbf {k}}) e_{\mathbf{k}}\). Here, \(e_{{\mathbf {k}}} = \frac{1}{(2\pi )^3 L^2} \exp [{i\left( \frac{k_1x+k_2 y}{L} + k_3 z \right) }]\), \(\mathbf {k} \in \mathbb Z^3\), form an orthonormal basis for \(L^2(\Omega )\), where \(\Omega =[0,2\pi L]^2 \times [0,2\pi ]\) is a three-dimensional periodic domain.

We consider the Galerkin approximation for system (1.12)–(1.16):

$$\begin{aligned}&\frac{\partial \psi _m}{\partial z} = \text {Ra} \, \theta '_m + \Delta _h w_m, \end{aligned}$$
(4.1)
$$\begin{aligned}&- \frac{\partial w_m}{\partial z} = \Delta _h \omega _m, \end{aligned}$$
(4.2)
$$\begin{aligned}&\frac{\partial \theta '_m}{\partial t} + P_m({\mathbf {u}}_m \cdot \nabla _h \theta '_m) + P_m(w_m \frac{\partial \overline{\theta _m}}{\partial z}) = \Delta _h \theta '_m, \end{aligned}$$
(4.3)
$$\begin{aligned}&\frac{\partial (\overline{\theta '_m w_m})}{\partial z} = \frac{\partial ^2 \overline{\theta _m}}{\partial z^2}, \end{aligned}$$
(4.4)

such that \(\nabla _h \cdot {\mathbf {u}}_m=0\), with the initial condition \(\theta '_m(0)=P_m \theta '_0\) where \(\overline{\theta _0'}=0\). Also, \({\mathbf {u}}_m=(u_m,v_m)=(-\partial _y \psi _m, \partial _x \psi _m)\), and \(\omega _m=\Delta _h \psi _m\) such that \(\overline{\psi _m}=0\). Moreover, the temperature \(\theta _m = \theta _m' + \overline{\theta _m}\), where the horizontal-mean temperature \(\overline{\theta _m} (z,t)= \frac{1}{4\pi ^2 L^2} \int _{[0,2\pi L]^2} \theta _m(x,y,z,t) \mathrm{d}x\,\mathrm{d}y\). We assume the horizontal average zero condition: \(\overline{\theta _m'}=\overline{w_m}=\overline{\omega _m}=\overline{\psi _m}=0\) and \(\overline{{\mathbf {u}}_m}=0\). In addition, we demand the average value of \(\theta _m\) is zero:

$$\begin{aligned} \int _{\Omega } \theta _m(x,y,z,t) \mathrm{d}x\,\mathrm{d}y \mathrm{d}z =0, \;\;\text {i.e.}, \;\;\int _0^{2\pi } \overline{\theta _m}(z,t) \mathrm{d}z =0, \;\; \text {for all}\,\, t\ge 0. \end{aligned}$$
(4.5)

Thus, the unknown functions in the Galerkin system can be expressed as finite sums of Fourier modes:

$$\begin{aligned}&\theta '_m= \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } {\widehat{\theta '_m}(\mathbf {k},t)} e_{{\mathbf {k}}}, \;\;\;\; \theta _m= \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ 0<|{\mathbf {k}}| \le m \end{array} } {\widehat{\theta _m}(\mathbf {k},t)} e_{{\mathbf {k}}}, \;\;\;\; \end{aligned}$$
(4.6)
$$\begin{aligned}&{\mathbf {u}}_m = \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3)\in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } {\widehat{\mathbf{u}_m}(\mathbf {k},t)} e_{{\mathbf {k}}}, \;\;\;\; w_m = \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3)\in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } {\widehat{w_m}(\mathbf {k},t)} e_{{\mathbf {k}}}. \end{aligned}$$
(4.7)
$$\begin{aligned}&\psi _m = \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } {\widehat{\psi _m}(\mathbf {k},t)} e_{{\mathbf {k}}}, \;\;\;\; \omega _m = \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } {\widehat{\omega _m}(\mathbf {k},t)} e_{{\mathbf {k}}}. \end{aligned}$$
(4.8)

Since Eqs. (4.1) and (4.2) are linear, they can be solved explicitly if \(\theta _m'\) is given. Indeed, since \(\omega _m=\Delta _h \psi _m\), we have

$$\begin{aligned}&ik_3 {\widehat{\psi _m}(\mathbf {k})} = \text {Ra} \, {\widehat{\theta '_m}(\mathbf {k})} - \left( \frac{k_1^2+k_2^2}{L^2}\right) {\widehat{w_m}(\mathbf {k})} \\&-ik_3 {\widehat{w_m}(\mathbf {k})} = \left( \frac{k_1^2+k_2^2}{L^2}\right) ^2 {\widehat{\psi _m}(\mathbf {k})}. \end{aligned}$$

The above linear system can be written as

$$\begin{aligned} \begin{pmatrix} ik_3 &{}\quad \frac{k_1^2+k_2^2}{L^2} \\ -\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2 &{}\quad -i k_3 \end{pmatrix} \begin{pmatrix} {\widehat{\psi _m}(\mathbf {k})} \\ {\widehat{w_m}(\mathbf {k})} \end{pmatrix} = \text {Ra} \begin{pmatrix} {\widehat{\theta '_m}(\mathbf {k})} \\ 0 \end{pmatrix}. \end{aligned}$$
(4.9)

For the \(2\times 2\) matrix in (4.9), its determinant \(k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3>0\) because \(k_1^2+k_2^2 \not =0\). Thus we can take the inverse of this matrix. It follows that

$$\begin{aligned}&\begin{pmatrix} {\widehat{\psi _m}(\mathbf {k})} \\ {\widehat{w_m}(\mathbf {k})} \end{pmatrix} = \text {Ra} \begin{pmatrix} ik_3 &{}\quad \frac{k_1^2+k_2^2}{L^2} \\ -\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2 &{}\quad -i k_3 \end{pmatrix}^{-1} \begin{pmatrix} {\widehat{\theta '_m}(\mathbf {k})} \\ 0 \end{pmatrix} \nonumber \\&= \frac{ \text {Ra} }{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3 } \begin{pmatrix} -ik_3 &{}\quad - \frac{k_1^2+k_2^2}{L^2}\\ \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2 &{} \quad i k_3 \end{pmatrix} \begin{pmatrix} {\widehat{\theta '_m}(\mathbf {k})} \\ 0 \end{pmatrix} \\&\quad =\frac{ \text {Ra} }{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3} \begin{pmatrix} -ik_3 \\ \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2 \end{pmatrix} {\widehat{\theta '_m}(\mathbf {k})}. \end{aligned}$$

Consequently,

$$\begin{aligned}&{\widehat{\psi _m}(\mathbf {k},t)}= \text {Ra} \left( \frac{-i k_3}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3}\right) {\widehat{\theta '_m}(\mathbf {k},t)}, \end{aligned}$$
(4.10)
$$\begin{aligned}&{\widehat{w_m}(\mathbf {k},t)} = \text {Ra} \left( \frac{\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3} \right) {\widehat{\theta '_m}(\mathbf {k},t)}. \end{aligned}$$
(4.11)

Furthermore, since \({\mathbf {u}}_m=(u_m,v_m)=(-\partial _y \psi _m, \partial _x \psi _m)\), we obtain from (4.10) that

$$\begin{aligned}&{\widehat{u_m}(\mathbf {k},t)} = \text {Ra} \left( \frac{-\frac{k_2}{L} k_3}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3}\right) {\widehat{\theta '_m}(\mathbf {k},t)}, \;\;\;\;\nonumber \\&{\widehat{v_m}(\mathbf {k},t)} = \text {Ra} \left( \frac{\frac{k_1}{L} k_3}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3}\right) {\widehat{\theta '_m}(\mathbf {k},t)}. \end{aligned}$$
(4.12)

Due to (4.4), we have \(\frac{\partial \overline{\theta _m}}{\partial z} = \overline{\theta '_m w_m} + c(t)\), for some constant c(t) depending only on t. Then, since \(\overline{\theta _m}\) is periodic on \([0,2\pi ]\), we obtain \(0=\overline{\theta _m}(2\pi ) - \overline{\theta _m}(0) = \int _0^{2\pi } \overline{\theta '_m w_m} \mathrm{d}z+ c(t) 2\pi \), which implies that \(c(t) = - \frac{1}{2\pi } \int _0^{2\pi } \overline{\theta '_m w_m} \mathrm{d}z \). Therefore,

$$\begin{aligned} \frac{\partial \overline{\theta _m}}{\partial z} = \overline{\theta '_m w_m} - \frac{1}{2\pi } \int _0^{2\pi } \overline{\theta '_m w_m} \mathrm{d}z. \end{aligned}$$
(4.13)

By substituting (4.11) and (4.12) and (4.13) into Eq. (4.3), we obtain a system of first order nonlinear ordinary differential equations with unknowns \(\{\widehat{\theta '_m}({\mathbf {k}},t): \mathbf {k}\in {\mathbb {Z}}^3, k_1^2+k_2^2\not =0, |{\mathbf {k}}| \le m\}.\) By the classical theory of ordinary differential equations, for each \(m\in {\mathbb {N}}\), there exists a solution \(\{\widehat{\theta '_m}({\mathbf {k}},t): |\mathbf{k}|\le m\}\) defined on \([0,T_m^{\text {max}})\) for the system of ODEs. Then, thanks to (4.10)–(4.12), we obtain \(\widehat{\psi _m}(\mathbf {k},t)\), \(\widehat{u_m}(\mathbf {k},t)\), \(\widehat{v_m}(\mathbf {k},t)\) and \(\widehat{w_m}(\mathbf {k},t)\), for \(|{\mathbf {k}}| \le m\). Next, we substitute \(\theta _m'\) and \(w_m\) into (4.13) to get \(\frac{\partial \overline{\theta _m}}{\partial z}\), and along with the assumption \(\int _0^{2\pi } \overline{\theta _m}(z,t) \mathrm{d}z =0\) from (4.5), we obtain \(\overline{\theta _m}\). Finally, \(\theta _m = \theta _m' + \overline{\theta _m}\) which satisfies \(\int _{\Omega } \theta _m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z =0\).

Assume the ODE system has finite time of existence, i.e., \(T_m^{\text {max}}<\infty \). By estimate (4.16) below, we know that \(\Vert \theta '_m(t)\Vert _2^2 \le \Vert \theta '_m(0)\Vert _2^2 \le \Vert \theta '_0\Vert _2^2\) for all \(t\in [0,T_m^{\text {max}})\). Therefore, \(\theta '_m(t)\) can be extended beyond the finite time \(T_m^{\text {max}}\), which is a contradiction. It follows that \(T_m^{\text {max}} = \infty \). As a result, for every \(m\in \mathbb N\), the Galerkin system (4.1)–(4.4) has a global solution on \([0,\infty )\).

Energy estimate

Taking the inner product of (4.3) with \(\theta '_m\) and using (3.5), one has

$$\begin{aligned} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \theta '_m\Vert _2^2 + \Vert \nabla _h \theta '_m\Vert _2^2 + \int _{\Omega } [P_m(w_m \frac{\partial \overline{\theta _m}}{\partial z})] \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z =0, \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(4.14)

Recall that the horizontal mean of a function f is defined as \({\overline{f}}(z)= \frac{1}{4 \pi ^2 L^2}\int _{[0, 2\pi L]^2} f(x,y,z) \mathrm{d}x \mathrm{d}y\). Using \(\frac{\partial (\overline{\theta '_m w_m})}{\partial z} = \frac{\partial ^2 \overline{\theta _m}}{\partial z^2}\) from Eq. (4.4), we find that

$$\begin{aligned}&\int _{\Omega } [P_m(w_m \frac{\partial \overline{\theta _m}}{\partial z})] \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z=\int _{\Omega } w_m \frac{\partial \overline{\theta _m}}{\partial z} \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z = 4\pi ^2 L^2 \int _0^{2\pi } (\overline{w_m \theta '_m}) \frac{\partial \overline{\theta _m}}{\partial z} \mathrm{d}z \nonumber \\&\quad = - 4\pi ^2 L^2 \int _0^{2\pi } \frac{\partial (\overline{w_m \theta '_m})}{\partial z} \overline{\theta _m} \mathrm{d}z =- 4\pi ^2 L^2 \int _0^{2\pi } \left( \frac{\partial ^2 \overline{\theta _m}}{\partial z^2}\right) \overline{\theta _m} \mathrm{d}z \\&\quad = 4\pi ^2 L^2 \int _0^{2\pi } \left| \frac{\partial \overline{\theta _m}}{\partial z}\right| ^2 \mathrm{d}z, \end{aligned}$$

for all \(t\ge 0\). Therefore,

$$\begin{aligned} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \theta '_m\Vert _2^2 + \Vert \nabla _h \theta '_m\Vert _2^2 +4\pi ^2 L^2 \int _0^{2\pi } \left| \frac{\partial \overline{\theta _m}}{\partial z}\right| ^2 \mathrm{d}z =0, \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(4.15)

Integrating over [0, t] yields

$$\begin{aligned}&\frac{1}{2} \Vert \theta '_m(t)\Vert _2^2 + \int _0^t \Vert \nabla _h \theta '_m (s)\Vert _2^2 \mathrm{d}s + 4\pi ^2 L^2 \int _0^t \int _0^{2\pi } \left| \frac{\partial \overline{\theta _m}(z,s)}{\partial z}\right| ^2 \mathrm{d}z \mathrm{d}s \nonumber \\&\quad = \frac{1}{2} \Vert \theta '_m(0)\Vert _2^2 \le \frac{1}{2} \Vert \theta '_0\Vert _2^2, \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(4.16)

Next, we estimate \({\mathbf {u}}_m\) and \(w_m\). By (4.12), we calculate

$$\begin{aligned} \Vert \Delta _h {\mathbf {u}}_m\Vert _2^2&= \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \frac{ \left| \frac{k_1^2+k_2^2}{L^2}\right| ^3 k_3^2 }{ \left| k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3\right| ^2 } |{\widehat{\theta '_m}(\mathbf {k})}|^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2. \end{aligned}$$
(4.17)

Furthermore,

$$\begin{aligned} \Vert \partial _z {\mathbf {u}}_m\Vert _2^2 = \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{k_1^2+k_2^2}{L^2}\right| \left| \frac{ k_3^2}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3}\right| ^2 |{\widehat{\theta '_m}(\mathbf {k})}|^2 \le \text {Ra}^2 \Vert \nabla _h \theta '_m\Vert _2^2. \end{aligned}$$
(4.18)

In addition, by using Young’s inequality, one has

$$\begin{aligned} \Vert \partial _z^{\frac{2}{3}} {\mathbf {u}}_m\Vert _2^2 = \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{ \Big (\frac{k_1^2 + k_2^2}{L^2}\Big )^{1/2} k_3^{5/3}}{k_3^2 + \Big (\frac{k_1^2 + k_2^2}{L^2}\Big )^3} \right| ^2 |\widehat{\theta '_m}({\mathbf {k}})|^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2. \end{aligned}$$
(4.19)

By (4.11), we see that

$$\begin{aligned} \Vert \Delta _h w_m\Vert _2^2 = \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3} \right| ^2 |\widehat{\theta '_m}(\mathbf {k})|^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2. \end{aligned}$$
(4.20)

Also, using (4.11), one has

$$\begin{aligned} \Vert \partial _z w_m\Vert _2^2&= \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2 k_3}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3} \right| ^2 |{\widehat{\theta '_m}(\mathbf {k})}|^2 \nonumber \\&= \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^{\frac{3}{2}} k_3}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3} \right| ^2 \left| \frac{k_1^2+k_2^2}{L^2} \right| | {\widehat{\theta '_m}(\mathbf {k})}|^2 \nonumber \\&\le \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{k_1^2+k_2^2}{L^2} \right| | {\widehat{\theta '_m}(\mathbf {k})}|^2 = \text {Ra}^2 \Vert \nabla _h \theta '_m\Vert _2^2. \end{aligned}$$
(4.21)

Moreover, applying Young’s inequality, we obtain

$$\begin{aligned} \Vert \partial _z^{\frac{2}{3}}w_m\Vert _2^2 = \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k}\in {\mathbb {Z}}^3\\ k_1^2 + k_2^2 \not =0 \end{array} } \left| \frac{ |k_3|^{2/3} \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^2}{k_3^2 + (\frac{k_1^2+k_2^2}{L^2})^3} \right| ^2 |{\widehat{\theta '_m}(\mathbf {k})}|^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2. \end{aligned}$$
(4.22)

Let us fix an arbitrary time \(T>0\).

Owing to estimates (4.16)–(4.18), (4.20) and (4.21) and using (4.10)–(4.12), we conclude

$$\begin{aligned}&\theta _m' \;\; \text {is uniformly bounded in} \;\; L^{\infty }(0,T;L^2(\Omega ))\cap L^2(0,T;H^1_h(\Omega )); \end{aligned}$$
(4.23)
$$\begin{aligned}&\partial _z \overline{\theta _m} \;\; \text {is uniformly bounded in} \;\; L^2((0,2\pi ) \times (0,T)); \end{aligned}$$
(4.24)
$$\begin{aligned}&{\mathbf {u}}_m, \; w_m \;\; \text {are uniformly bounded in} \;\; L^2(0,T;H^1(\Omega )); \end{aligned}$$
(4.25)
$$\begin{aligned}&\Delta _h \omega _m \;\; \text {is uniformly bounded in} \;\; L^2(\Omega \times (0,T)); \end{aligned}$$
(4.26)
$$\begin{aligned}&\Delta _h {\mathbf {u}}_m, \; \Delta _h w_m, \; \partial _z \psi _m \;\; \text {are uniformly bounded in} \;\;\nonumber \\&\quad L^{\infty }(0,T;L^2(\Omega )) \cap L^2(0,T;H^1_h(\Omega )) . \end{aligned}$$
(4.27)

Therefore, on a subsequence, we have the following weak convergences as \(m\rightarrow \infty \):

$$\begin{aligned}&\theta _m' \rightarrow \theta ', \; \Delta _h {\mathbf {u}}_m \rightarrow \Delta _h {\mathbf {u}}, \; \Delta _h w_m \rightarrow \Delta _h w, \; \partial _z \psi _m \rightarrow \psi _z\;\nonumber \\&\quad \text {weakly}^* \;\text {in} \; L^{\infty }(0,T;L^2(\Omega )); \end{aligned}$$
(4.28)
$$\begin{aligned}&\theta _m' \rightarrow \theta ', \; \Delta _h {\mathbf {u}}_m {\rightarrow } \Delta _h {\mathbf {u}}, \; \Delta _h w_m {\rightarrow } \Delta _h w, \; \partial _z \psi _m {\rightarrow } \psi _z\; \; \text {weakly in} \; L^2(0,T;H^1_h(\Omega )); \end{aligned}$$
(4.29)
$$\begin{aligned}&{\mathbf {u}}_m\rightarrow {\mathbf {u}}, \; w_m\rightarrow w \;\; \text {weakly in} \;\; L^2(0,T;H^1(\Omega )); \end{aligned}$$
(4.30)
$$\begin{aligned}&\Delta _h \omega _m \rightarrow \Delta _h \omega \;\; \text {weakly in} \;\; L^2(\Omega \times (0,T)); \end{aligned}$$
(4.31)
$$\begin{aligned}&\partial _z \overline{\theta _m} \rightarrow \overline{\theta }_z \;\; \text {weakly in} \;\; L^2((0,2\pi ) \times (0,T)). \end{aligned}$$
(4.32)

Using these weak convergences and inequality (4.16), we obtain the energy inequality:

$$\begin{aligned} \frac{1}{2} \Vert \theta '(t)\Vert _2^2 + \int _0^t \Vert \nabla _h \theta '(s)\Vert _2^2 \mathrm{d}s + 4\pi ^2 L^2 \int _0^t \int _0^{2\pi } \left| \frac{\partial \overline{\theta }(z,s)}{\partial z}\right| ^2 \mathrm{d}z \mathrm{d}s \le \frac{1}{2} \Vert \theta '_0\Vert _2^2, \end{aligned}$$
(4.33)

for all \(t\in [0,T]\).

Also, using these weak convergences, we can pass to the limit for the linear equations (4.1) and (4.2) in the Galerkin approximation system to obtain

$$\begin{aligned}&\frac{\partial \psi }{\partial z} = \text {Ra} \, \theta ' + \Delta _h w, \;\; \text {in} \;\; L^2(0,T;H^1_h(\Omega )) \cap L^{\infty }(0,T;L^2(\Omega )), \end{aligned}$$
(4.34)
$$\begin{aligned}&- \frac{\partial w}{\partial z} = \Delta _h \omega , \;\; \text {in} \;\; L^2(\Omega \times (0,T)). \end{aligned}$$
(4.35)

Then, we can use the same calculations as (4.17)–(4.22), to derive that

$$\begin{aligned}&\Vert \Delta _h {\mathbf {u}}\Vert _2^2 + \Vert \Delta _h w\Vert ^2_2 \le 2\text {Ra}^2\Vert \theta '\Vert _2^2 \;\; \text {and} \;\; \Vert {\mathbf {u}}_z\Vert _2^2 + \Vert w_z\Vert ^2_2 \le 2\text {Ra}^2 \Vert \nabla _h \theta '\Vert _2^2, \end{aligned}$$
(4.36)
$$\begin{aligned}&\Vert \partial _z^{\frac{2}{3}}{\mathbf {u}}\Vert _2^2 + \Vert \partial _z^{\frac{2}{3}}w\Vert ^2_2 \le 2\text {Ra}^2 \Vert \theta '\Vert _2^2, \end{aligned}$$
(4.37)

for all \(t\in [0,T]\).

Passage to the limit

In order to pass to the limit for the nonlinear equation (4.3) in the Galerkin approximation system, we shall derive certain strong convergence, besides the already known weak convergences (4.28) and (4.32). To this purpose, one has to find a uniform bound for the time derivative \(\partial _t \theta '_m\), in a certain function space. From Eq. (4.3), we know

$$\begin{aligned} \frac{\partial \theta '_m}{\partial t} = -P_m({\mathbf {u}}_m \cdot \nabla _h \theta '_m) - P_m(w_m \frac{\partial \overline{\theta _m}}{\partial z}) + \Delta _h \theta '_m. \end{aligned}$$
(4.38)

We aim to find a uniform bound for each term on the right-hand side of (4.38).

For any \(\varphi \in L^2(0,T;H^1_h(\Omega ))\), applying identity (3.4) and Lemma 3.1, we estimate

$$\begin{aligned}&\int _0^T \int _{\Omega } [P_m({\mathbf {u}}_m \cdot \nabla _h \theta '_m)] \varphi \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t = -\int _0^T \int _{\Omega } ({\mathbf {u}}_m \cdot \nabla _h P_m \varphi ) \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \Vert \nabla _h {\mathbf {u}}_m\Vert ^{1/2} \left( \Vert \mathbf{u}_m\Vert _2 + \Vert \partial _z {\mathbf {u}}_m\Vert _2 \right) ^{1/2} \Vert \theta '_m\Vert _2^{1/2} \Vert \nabla _h \theta '_m\Vert _2^{1/2} \Vert \nabla _h P_m \varphi \Vert _2 \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \Vert \theta '_m\Vert _2 \Vert \nabla _h \theta '_m\Vert _2 \Vert \nabla _h \varphi \Vert _2 \mathrm{d}t \nonumber \\&\quad \le C \Vert \theta _0'\Vert _2 \left( \int _0^T \Vert \nabla _h \theta '_m(t)\Vert _2^2 \mathrm{d}t\right) ^{1/2} \left( \int _0^T \Vert \nabla _h \varphi (t)\Vert _2^2 \mathrm{d}t\right) ^{1/2} \nonumber \\&\quad \le C \Vert \theta _0'\Vert _2^2 \Vert \varphi \Vert _{L^2(0,T;H^1_h(\Omega ))}, \end{aligned}$$
(4.39)

where we have used estimates (4.16), (4.17) and (4.18).

Hence

$$\begin{aligned} \Vert P_m ({\mathbf {u}}_m \cdot \nabla _h \theta '_m)\Vert _{L^2(0,T;(H^1_h(\Omega ))')} \le C \Vert \theta _0'\Vert _2^2. \end{aligned}$$
(4.40)

Furthermore, for any function \(\phi \in L^2(\Omega \times (0,T))\),

$$\begin{aligned}&\int _0^T \int _{\Omega } [P_m(w_m \frac{\partial \overline{\theta _m}}{\partial z})] \phi \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t {=}\int _0^T \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} w_m (P_m \phi ) \mathrm{d}x\,\mathrm{d}y\Big ) \frac{\partial \overline{\theta _m}}{\partial z} \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le \int _0^T \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |w_m|^2 \mathrm{d}x\,\mathrm{d}y\Big )^{1/2} \Big (\int _{[0,2\pi L]^2} |P_m \phi |^2 \mathrm{d}x\,\mathrm{d}y\Big )^{1/2} \Big |\frac{\partial \overline{\theta _m}}{\partial z} \Big | \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \Big [\sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |w_m|^2 \mathrm{d}x\,\mathrm{d}y \Big )^{1/2}\Big ] \Vert P_m \phi \Vert _2 \Big ( \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big )^{1/2} \mathrm{d}t \nonumber \\&\quad \le C\int _0^T ( \Vert w_m\Vert _2 + \Vert \partial _z^{\frac{2}{3}} w_m\Vert _2 ) \Vert \phi \Vert _2 \left( \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\right) ^{1/2} \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \Vert \theta '_m\Vert _2 \Vert \phi \Vert _2 \left( \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\right) ^{1/2} \mathrm{d}t \nonumber \\&\quad \le C \Vert \theta _0'\Vert _2 \left( \int _0^T\Vert \phi \Vert _2^2 \mathrm{d}t\right) ^{1/2} \left( \int _0^T \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z \mathrm{d}t\right) ^{1/2} \nonumber \\&\quad \le C \Vert \theta _0'\Vert _2^2 \Vert \phi \Vert _{L^2(\Omega \times (0,T))}, \end{aligned}$$
(4.41)

where we have used Lemma 3.3 and estimates (4.16), (4.20) and (4.22).

Thus,

$$\begin{aligned} \left\| P_m\left( w_m \frac{\partial \overline{\theta _m}}{\partial z}\right) \right\| _{L^2(\Omega \times (0,T))} \le C\Vert \theta _0\Vert _2^2. \end{aligned}$$
(4.42)

Also, it is easy to verify that \(\Vert \Delta _h \theta _m'\Vert _{L^2(0,T;(H^1_h(\Omega ))')} \le \Vert \theta _0'\Vert _2\). Therefore, owing to (4.38), (4.40) and (4.42), we obtain

$$\begin{aligned} \Vert \partial _t \theta _m'\Vert _{L^2(0,T;(H^1_h(\Omega ))')} \le C\Vert \theta _0'\Vert _2^2 +\Vert \theta _0'\Vert _2. \end{aligned}$$
(4.43)

By the uniform bound (4.43), it follows that there exists a subsequence satisfying

$$\begin{aligned} \partial _t \theta _m' \rightarrow \partial _t \theta ' \;\; \text {weakly in} \;\; L^2(0,T;(H^1_h(\Omega ))'). \end{aligned}$$
(4.44)

Using (4.11), one has

$$\begin{aligned}&\Vert \partial _t w_m\Vert _2^2 = \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } |\partial _t \widehat{w_m}(\mathbf{k},t)|^2\nonumber \\&\qquad \qquad = \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } \left| \frac{\Big (\frac{k_1^2+k_2^2}{L^2}\Big )^{2}}{k_3^2 + \Big (\frac{k_1^2+k_2^2}{L^2}\Big )^3} \right| ^2 |{\partial _t \widehat{\theta '_m}(\mathbf {k},t)}|^2 \nonumber \\&\qquad \qquad \le \text {Ra}^2 \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } \left| \frac{L^2}{k_1^2 + k_2^2} \right| ^2 |{\partial _t \widehat{\theta '_m}(\mathbf {k},t)}|^2 \nonumber \\&\quad \le \text {Ra}^2 L^2 \sum _{\begin{array}{c} \mathbf {k} = (k_1, k_2, k_3) \in {\mathbb {Z}}^3 \\ k_1^2+k_2^2 \not = 0 \\ |{\mathbf {k}}| \le m \end{array} } \frac{L^2}{k_1^2 + k_2^2} |{\partial _t \widehat{\theta '_m}(\mathbf {k},t)}|^2 \nonumber \\&\quad \le C \Vert \partial _t \theta '_m\Vert _{(H^1_h(\Omega ))'}^2. \end{aligned}$$
(4.45)

Due to (4.43) and (4.45), we obtain

$$\begin{aligned} \partial _t w_m \;\; \text {is uniformly bounded in} \;\; L^2(\Omega \times (0,T)). \end{aligned}$$
(4.46)

In a similar manner, one can show

$$\begin{aligned} \partial _t {\mathbf {u}}_m \;\; \text {is uniformly bounded in} \;\; L^2(\Omega \times (0,T)). \end{aligned}$$
(4.47)

By virtue of (4.25), (4.46) and (4.47) and using Aubin Compactness Theorem (see, e.g., [16]), one can extract a subsequence such that

$$\begin{aligned} {\mathbf {u}}_m\rightarrow {\mathbf {u}}, \; w_m\rightarrow w \;\; \text {strongly in} \;\; L^2(\Omega \times (0,T)). \end{aligned}$$
(4.48)

Now we can pass to the limit for the nonlinear terms of equation (4.3) in the Galerkin system. Indeed, let \(\eta =\sum _{|{\mathbf {k}}|\le N} \hat{\eta }({\mathbf {k}},t) e_{\mathbf{k}}\) be a trigonometric polynomial with continuous coefficients, where \(e_{{\mathbf {k}}}= \frac{1}{(2\pi )^3 L^2} \exp [{i\left( \frac{k_1x+k_2 y}{L} + k_3 z \right) }]\), \({\mathbf {k}}=(k_1,k_2,k_3) \in {\mathbb {Z}}^3\). Then,

$$\begin{aligned}&\lim _{m\rightarrow \infty }\int _0^T \int _{\Omega } (P_m({\mathbf {u}}_m \cdot \nabla _h \theta '_m)) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t = \lim _{m\rightarrow \infty }\int _0^T \int _{\Omega } ({\mathbf {u}}_m \cdot \nabla _h \theta '_m) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad =\lim _{m \rightarrow \infty } \int _0^T \int _{\Omega } (({\mathbf {u}}_m - {\mathbf {u}}) \cdot \nabla _h \theta '_m) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\qquad + \lim _{m\rightarrow \infty } \int _0^T \int _{\Omega } ({\mathbf {u}} \cdot \nabla _h \theta '_m) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad =\int _0^T \int _{\Omega } ({\mathbf {u}} \cdot \nabla _h \theta ') \eta \mathrm{d}x \mathrm{d}y \mathrm{d}z \mathrm{d}t, \end{aligned}$$
(4.49)

by virtue of the fact that \({\mathbf {u}}_m \rightarrow {\mathbf {u}}\) in \(L^2(\Omega \times (0,T))\), and the fact that \(\nabla _h \theta '_m\) is uniformly bounded in \(L^2(\Omega \times (0,T))\), as well as the fact that \(\nabla _h \theta '_m \rightarrow \nabla _h \theta '\) weakly in \(L^2(\Omega \times (0,T))\).

For the other nonlinear term in (4.3), we have

$$\begin{aligned}&\lim _{m\rightarrow \infty } \int _0^T \int _{\Omega } P_m\Big (w_m \frac{\partial {\overline{\theta _m}}}{\partial z} \Big ) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t = \lim _{m\rightarrow \infty } \int _0^T \int _{\Omega } \Big (w_m \frac{\partial {\overline{\theta _m}}}{\partial z} \Big ) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad = \lim _{m\rightarrow \infty } \int _0^T \int _{\Omega } (w_m -w) \frac{\partial {\overline{\theta _m}}}{\partial z} \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t +\lim _{m\rightarrow \infty } \int _0^T \int _{\Omega } w \frac{\partial {\overline{\theta _m}}}{\partial z} \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad = \int _0^T \int _{\Omega } w \frac{\partial {\overline{\theta }}}{\partial z} \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t, \end{aligned}$$
(4.50)

due to the fact that \(w_m \rightarrow w\) strongly in \(L^2(\Omega \times (0,T))\), and the fact that \(\frac{\partial {\overline{\theta _m}}}{\partial z} \) is uniformly bounded in \(L^2((0,2\pi ) \times (0,T))\), along with the fact that \(\frac{\partial {\overline{\theta _m}}}{\partial z} \rightarrow \frac{\partial {\overline{\theta }}}{\partial z} \) weakly in \(L^2((0,2\pi ) \times (0,T))\).

Therefore, we have

$$\begin{aligned} \int _0^T \int _{\Omega } \left( \frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} - \Delta _h \theta '\right) \eta \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t =0, \end{aligned}$$
(4.51)

for any trigonometric polynomial \(\eta \) with continuous coefficients.

Using similar estimates as (4.39) and (4.41), we can derive \({\mathbf {u}} \cdot \nabla _h \theta ' \in L^2(0,T;(H^1_h(\Omega ))')\) and \( w \frac{\partial \overline{\theta }}{\partial z} \in L^2(\Omega \times (0,T))\), thus \(\frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} - \Delta _h \theta '\in L^2(0,T;(H^1_h(\Omega ))')\). Then, we conclude from (4.51) that

$$\begin{aligned} \frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} - \Delta _h \theta ' = 0, \;\; \text {in} \; L^2(0,T;(H^1_h(\Omega ))'). \end{aligned}$$
(4.52)

Next, we aim to pass to the limit for the nonlinear term \(\frac{\partial (\overline{\theta _m' w_m})}{\partial z}\) in equation (4.4). To this end, we shall first show \(\frac{\partial (\overline{\theta ' w})}{\partial z} \in L^2(0,T; H^{-1}(0,2\pi ))\). Indeed, for any \(\varphi \in L^2(0,T;\dot{H}^1(0,2\pi ))\), we use Lemma 3.3 as well as estimates (4.33) and (4.36), to derive

$$\begin{aligned}&\int _0^{T} \int _0^{2\pi } (\overline{\theta ' w}) \varphi _z \mathrm{d}z \mathrm{d}t = \frac{1}{4\pi ^2 L^2} \int _0^T \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} \theta ' w \mathrm{d}x\,\mathrm{d}y\Big ) \varphi _z \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le C \int _0^T \int _0^{2\pi } \Big (\int _{[0, 2\pi L]^2} |\theta '|^2 \mathrm{d}x\,\mathrm{d}y \Big )^{1/2} \Big (\int _{[0, 2\pi L]^2} |w|^2 \mathrm{d}x\,\mathrm{d}y \Big )^{1/2} |\varphi _z| \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |w|^2 \mathrm{d}x\,\mathrm{d}y\Big )^{1/2} \Vert \theta '\Vert _2 \Big (\int _0^{2\pi } |\varphi _z|^2 \mathrm{d}z\Big )^{1/2} \mathrm{d}t \nonumber \\&\quad \le C \Vert \theta '_0\Vert _2 \int _0^T (\Vert w\Vert _2 + \Vert w_z\Vert _2) \Big (\int _0^{2\pi } |\varphi _z|^2 \mathrm{d}z\Big )^{1/2} \mathrm{d}t \nonumber \\&\quad \le C \Vert \theta '_0\Vert _2 \int _0^T \Vert \nabla _h \theta '\Vert _2 \Big (\int _0^{2\pi } |\varphi _z|^2 \mathrm{d}z\Big )^{1/2} \mathrm{d}t \nonumber \\&\quad \le C \Vert \theta '_0\Vert _2 \Big ( \int _0^T \Vert \nabla _h \theta '\Vert _2^2 \mathrm{d}t \Big )^{1/2} \Big ( \int _0^T \int _0^{2\pi } |\varphi _z|^2 \mathrm{d}z \mathrm{d}t \Big )^{1/2} \nonumber \\&\quad \le C \Vert \theta '_0\Vert _2^2 \Vert \varphi \Vert _{L^2(0,T;\dot{H}^1(0,2\pi ))}. \end{aligned}$$

It follows that \(\frac{\partial (\overline{\theta ' w})}{\partial z} \in L^2(0,T; H^{-1}(0,2\pi ))\) and \(\Vert \frac{\partial (\overline{\theta ' w})}{\partial z}\Vert _{L^2(0,T; H^{-1}(0,2\pi ))} \le C \Vert \theta '_0\Vert _2^2\).

Now we take a test function \(\xi =\sum _{0<|j| \le N} \hat{\xi }(j,t) e^{ij z}\) where Fourier coefficients \(\hat{\xi }(t,j)\) are continuous in t. By using the fact that \(w_m \rightarrow w\) strongly in \(L^2(\Omega \times (0,T))\) and that \(\theta _m' \rightarrow \theta '\) weakly in \(L^2(\Omega \times (0,T))\), we derive

$$\begin{aligned}&\lim _{m\rightarrow \infty }\int _0^T \int _0^{2\pi } \frac{\partial (\overline{\theta _m' w_m})}{\partial z} \xi \mathrm{d}z \mathrm{d}t = - \lim _{m\rightarrow \infty }\int _0^T \int _0^{2\pi } (\overline{\theta _m' w_m}) \xi _z \mathrm{d}z \mathrm{d}t \nonumber \\&\quad = -\lim _{m\rightarrow \infty } \frac{1}{4\pi ^2 L^2}\int _0^T \int _{\Omega } \theta _m' (w_m -w) \xi _z \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\qquad - \lim _{m\rightarrow \infty } \frac{1}{4\pi ^2 L^2}\int _0^T \int _{\Omega } \theta _m' w \xi _z \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad =- \frac{1}{4\pi ^2 L^2}\int _0^T \int _{\Omega } \theta ' w \xi _z \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \mathrm{d}t = - \int _0^T \int _0^{2\pi } (\overline{\theta ' w}) \xi _z \mathrm{d}z \mathrm{d}t \nonumber \\&\quad = \int _0^T \left\langle \frac{\partial (\overline{\theta ' w})}{\partial z}, \xi \right\rangle _{{H^{-1}(0,2\pi )} \times \dot{H}^1(0,2\pi )} \mathrm{d}t. \end{aligned}$$
(4.53)

Also, since \(\partial _z \overline{\theta _m} \rightarrow \overline{\theta }_z\) weakly in \(L^2((0,2\pi ) \times (0,T))\), we obtain

$$\begin{aligned}&\lim _{m\rightarrow \infty }\int _0^T \int _0^{2\pi } \frac{\partial ^2 \overline{\theta _m}}{\partial z^2} \xi \mathrm{d}z \mathrm{d}t = - \lim _{m\rightarrow \infty }\int _0^T \int _0^{2\pi } \frac{\partial \overline{\theta _m}}{\partial z} \xi _z \mathrm{d}z \mathrm{d}t \nonumber \\&\quad = -\int _0^T \int _0^{2\pi } \frac{\partial \overline{\theta }}{\partial z} \xi _z \mathrm{d}z \mathrm{d}t = \int _0^T \left\langle \frac{\partial ^2 \overline{\theta }}{\partial z^2}, \xi \right\rangle _{{H^{-1}(0,2\pi )} \times \dot{H}^1(0,2\pi )} \mathrm{d}t.\end{aligned}$$
(4.54)

Because \(\frac{\partial (\overline{\theta '_m w_m})}{\partial z} = \frac{\partial ^2 \overline{\theta _m}}{\partial z^2}\) and due to (4.53) and (4.54), we obtain

$$\begin{aligned} \int _0^T \left\langle \frac{\partial (\overline{\theta ' w})}{\partial z} - \frac{\partial ^2 \overline{\theta }}{\partial z^2} , \xi \right\rangle _{{H^{-1}(0,2\pi )} \times \dot{H}^1(0,2\pi )} \mathrm{d}t =0. \end{aligned}$$
(4.55)

Then, since \(\frac{\partial (\overline{\theta ' w})}{\partial z}\) and \(\frac{\partial ^2 \overline{\theta }}{\partial z^2}\) both belong to \(L^2(0,T; H^{-1}(0,2\pi ))\), we conclude that

$$\begin{aligned} \frac{\partial (\overline{\theta ' w})}{\partial z} = \frac{\partial ^2 \overline{\theta }}{\partial z^2}, \;\; \text {in} \; L^2(0,T; H^{-1}(0,2\pi )) \end{aligned}$$
(4.56)

In sum, because of (4.34) and (4.35), (4.52) and (4.56), we have obtained a weak solution for system (1.12)–(1.16) on [0, T], in the sense of Definition 2.1. Then, thanks to the energy inequality (4.33), the solution can be extended to a global weak solution on \([0,\infty )\). This completes the proof for the global existence of weak solutions for system (1.12)–(1.16).

Energy identity

Let \(T>0\). Since

$$\begin{aligned} \frac{\partial \theta '}{\partial t} \in L^2(0,T;(H^1_h(\Omega ))') \;\; \text {and} \;\; \theta ' \in L^2(0,T;H^1_h(\Omega )), \end{aligned}$$

then according to Lemma 2.1 on page 176 of Temam’s book [16], we obtain \(\theta '\in C([0,T];L^2(\Omega ))\) and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \theta '\Vert _2^2 = 2 \left\langle \frac{\partial \theta '}{\partial t}, \theta ' \right\rangle _{(H^1_h(\Omega ))' \times H^1_h(\Omega )}. \end{aligned}$$
(4.57)

Therefore, we can take scalar product of (4.52) with \(\theta '\) to obtain

$$\begin{aligned} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \theta '(t)\Vert _2^2 + \Vert \nabla _h \theta ' (t)\Vert _2^2 + 4\pi ^2 L^2 \int _0^{2\pi } |\partial _z \overline{\theta }(z,t)|^2 \mathrm{d}z = 0, \;\; \text {for all} \; t\in [0,T]. \end{aligned}$$
(4.58)

Integrating (4.58) over [0, t] yields the energy identity (2.7).

Decay of the solution

Since \(\overline{\theta '}=0\), then the Poincaré inequality (3.3) shows that \( \frac{1}{L^2} \Vert \theta ' (t)\Vert _2^2 \le \Vert \nabla _h \theta ' (t)\Vert _2^2\). Therefore, it follows from (4.58) that

$$\begin{aligned} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \theta '(t)\Vert _2^2 + \frac{1}{L^2} \Vert \theta ' (t)\Vert _2^2 \le 0, \;\; \text {for all} \; t\ge 0. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \theta '(t)\Vert _2^2 \le e^{-\frac{2}{L^2} t} \Vert \theta '_0\Vert _2^2, \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(4.59)

Then, due to (4.36) and (4.59), we obtain

$$\begin{aligned} \Vert \Delta _h {\mathbf {u}}(t)\Vert _2^2 + \Vert \Delta _h w(t)\Vert _2^2 \le C e^{-\frac{2}{L^2} t} \Vert \theta '_0\Vert _2^2, \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(4.60)

Due to the Poincaré inequality (3.3), we have also \(\Vert {\mathbf {u}}(t)\Vert _2^2 + \Vert w(t)\Vert _2^2 \le C e^{-\frac{2}{L^2} t} \Vert \theta '_0\Vert _2^2\).

Next, taking the scalar product of Eq. (4.56) and \(\overline{\theta }\) in the duality between \(H^{-1}(0,2\pi )\) and \(\dot{H}^1(0,2\pi )\) yields

$$\begin{aligned} \int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z = \int _0^{2\pi } (\overline{\theta ' w})\overline{\theta }_z \mathrm{d}z \le \frac{1}{2} \int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z + \frac{1}{2} \int _0^{2\pi } \left| \overline{\theta ' w} \right| ^2 \mathrm{d}z, \end{aligned}$$
(4.61)

where we use the Cauchy–Schwarz inequality and the Young’s inequality.

By (4.61), we have

$$\begin{aligned}&\int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z \le \int _0^{2\pi } \left| \overline{\theta ' w} \right| ^2 \mathrm{d}z \le C\int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |\theta '|^2 \mathrm{d}x\,\mathrm{d}y\Big ) \nonumber \\&\qquad \times \Big (\int _{[0,2\pi L]^2} |w|^2 \mathrm{d}x\,\mathrm{d}y\Big ) \mathrm{d}z \nonumber \\&\quad \le C \sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |w|^2 \mathrm{d}x \mathrm{d}y\Big ) \Vert \theta '\Vert _2^2 \le C (\Vert w\Vert _2^2 + \Vert \partial _z^{\frac{2}{3}} w\Vert _2^2) \Vert \theta '\Vert _2^2 \nonumber \\&\quad \le C \Vert \theta '\Vert _2^4 \le C e^{-\frac{4}{L^2} t} \Vert \theta '_0\Vert _2^4, \end{aligned}$$

where we have used Lemma 3.3 and (4.37) as well as the decay estimate (4.59). Since \(\int _0^{2\pi } \overline{\theta } \mathrm{d}z =0\), then we can use the Poincaré inequality to conclude

$$\begin{aligned} \int _0^{2\pi } |\overline{\theta }|^2 \mathrm{d}z \le C\int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z \le C e^{-\frac{4}{L^2} t} \Vert \theta '_0\Vert _2^4. \end{aligned}$$

Uniqueness of weak solutions and continuous dependence on initial data

This section is devoted to proving the uniqueness of weak solutions. Assume there are two weak solutions \((\theta '_1, \overline{\theta _1}, {\mathbf {u}}_1,w_1)\) and \((\theta '_2, \overline{\theta _2}, \mathbf{u}_2,w_2)\) on [0, T], in the sense of Definition 2.1. Set \(\theta '=\theta _1'-\theta _2'\), \(\overline{\theta }=\overline{\theta _1} - \overline{\theta _2}\), \(\theta =\theta _1 - \theta _2\), \({\mathbf {u}}={\mathbf {u}}_1-{\mathbf {u}}_2\), \(w=w_1-w_2\), \(\psi =\psi _1 - \psi _2\) and \(\omega = \omega _1 - \omega _2\). Therefore,

$$\begin{aligned}&\frac{\partial \psi }{\partial z} = \text {Ra}\,\theta ' + \Delta _h w, \;\; \text {in} \;\; L^2(0,T; H^1_h(\Omega )) \cap C([0,T];L^2(\Omega )), \end{aligned}$$
(4.62)
$$\begin{aligned}&- \frac{\partial w}{\partial z} = \Delta _h \omega , \;\; \text {in} \;\; L^2(\Omega \times (0,T)), \end{aligned}$$
(4.63)
$$\begin{aligned}&\frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta '_1 + {\mathbf {u}}_2 \cdot \nabla _h \theta ' + w\frac{\partial \overline{\theta _1}}{\partial z} + w_2 \frac{\partial \overline{\theta }}{\partial z}= \Delta _h \theta ', \;\; \text {in} \;\; L^2(0,T;(H^1_h(\Omega ))'), \end{aligned}$$
(4.64)
$$\begin{aligned}&\frac{\partial (\overline{\theta ' w_2})}{\partial z} + \frac{\partial (\overline{\theta '_1 w})}{\partial z} = \frac{\partial ^2 {\overline{\theta }}}{\partial z^2}, \;\; \text {in} \;\; L^2(0,T;H^{-1}(0,2\pi )), \end{aligned}$$
(4.65)

and \(\omega = \nabla _h \times {\mathbf {u}}\), \(\psi = \Delta _h^{-1} \omega \).

Due to the linear equations (4.62) and (4.63) and using the same calculations as (4.17)–(4.22), we have

$$\begin{aligned}&\Vert \Delta _h {\mathbf {u}}\Vert _2^2 + \Vert \Delta _h w\Vert ^2_2 \le 2\text {Ra}^2\Vert \theta '\Vert _2^2 \;\; \text {and} \;\; \Vert {\mathbf {u}}_z\Vert _2^2 + \Vert w_z\Vert ^2_2 \le 2\text {Ra}^2 \Vert \nabla _h \theta '\Vert _2^2, \end{aligned}$$
(4.66)
$$\begin{aligned}&\Vert \partial _z^{\frac{2}{3}}{\mathbf {u}}\Vert _2^2 + \Vert \partial _z^{\frac{2}{3}}w\Vert ^2_2 \le 2\text {Ra}^2 \Vert \theta '\Vert _2^2. \end{aligned}$$
(4.67)

Thanks to (4.57), we can take the scalar product of Eq. (4.64) and \(\theta '\) in the duality between \((H^1_h(\Omega ))'\) and \(H^1_h(\Omega )\) to obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \Vert \theta '(t)\Vert _2^2 + \Vert \nabla _h \theta '(t)\Vert _2^2 \nonumber \\&\quad =-\int _{\Omega } ({\mathbf {u}} \cdot \nabla _h \theta '_1) \theta ' \mathrm{d}x \mathrm{d}y \mathrm{d}z - \int _{\Omega } w\frac{\partial \overline{\theta _1}}{\partial z} \theta ' \mathrm{d}x\,\mathrm{d}y \mathrm{d}z - \int _{\Omega } w_2 \frac{\partial \overline{\theta }}{\partial z} \theta ' \mathrm{d}x\,\mathrm{d}y \mathrm{d}z, \end{aligned}$$
(4.68)

where we have used the fact that \(\int _{\Omega } \left( {\mathbf {u}}_2 \cdot \nabla _h \theta ' \right) \theta ' \mathrm{d}x\,\mathrm{d}y \mathrm{d}z =0\) since \(\nabla _h \cdot {\mathbf {u}}_2=0\).

Next, we estimate each term on the right-hand side of (4.68). By employing identity (3.4) and Lemma 3.1 as well as estimate (4.66), we deduce

$$\begin{aligned}&-\int _{\Omega } ({\mathbf {u}} \cdot \nabla _h \theta '_1) \theta ' \mathrm{d}x\,\mathrm{d}y \mathrm{d}z = \int _{\Omega } ({\mathbf {u}} \cdot \nabla _h \theta ') \theta '_1 \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \nonumber \\&\quad \le C\Vert \nabla _h {\mathbf {u}}\Vert _2^{1/2} \left( \Vert {\mathbf {u}}\Vert _2 + \Vert {\mathbf {u}}_z\Vert _2 \right) ^{1/2} \Vert \nabla _h \theta '\Vert _2 \Vert \theta _1'\Vert _2^{1/2} \Vert \nabla _h \theta _1'\Vert _2^{1/2} \nonumber \\&\quad \le C \Vert \theta '\Vert _2^{1/2} \Vert \nabla _h \theta '\Vert _2^{3/2} \Vert \theta _1'\Vert _2^{1/2} \Vert \nabla _h \theta _1'\Vert _2^{1/2} \nonumber \\&\quad \le \frac{1}{2} \Vert \nabla _h \theta '\Vert _2^2 + C \Vert \theta '\Vert _2^2 \Vert \theta _1'\Vert _2^2 \Vert \nabla _h \theta _1'\Vert _2^2, \end{aligned}$$
(4.69)

where we use Hölder’s inequality.

To treat the second integral on the right-hand of (4.68), we apply the Cauchy–Schwarz inequality to deduce

$$\begin{aligned}&\int _{\Omega } \big |w\frac{\partial \overline{\theta _1}}{\partial z} \theta ' \big | \mathrm{d}x\,\mathrm{d}y \mathrm{d}z = \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |w \theta '| \mathrm{d}x\,\mathrm{d}y\Big ) | \partial _z {\overline{\theta _1}} | \mathrm{d}z \nonumber \\&\quad \le \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |w(x,y,z)|^2 \mathrm{d}x \mathrm{d}y\Big )^{1/2} \Big (\int _{[0,2\pi L]^2} \nonumber \\&\qquad \times |\theta '(x,y,z)|^2 \mathrm{d}x \mathrm{d}y\Big )^{1/2} |\partial _z {\overline{\theta _1}(z)}| \mathrm{d}z \nonumber \\&\quad \le \Big [\sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |w(x,y,z)|^2 \mathrm{d}x\,\mathrm{d}y\Big )^{1/2} \Big ] \Vert \theta '\Vert _2 \Big (\int _0^{2\pi } |\partial _z {\overline{\theta _1}(z)}|^2 \mathrm{d}z\Big )^{1/2} \nonumber \\&\quad \le C(\Vert w\Vert _2+ \Vert \partial _z^{\frac{2}{3}}w\Vert _2) \Vert \theta '\Vert _2 \Big (\int _0^{2\pi } |\partial _z {\overline{\theta _1}}|^2 \mathrm{d}z\Big )^{1/2} \nonumber \\&\quad \le C \Vert \theta '\Vert _2^2 \Big (\int _0^{2\pi } |\partial _z {\overline{\theta _1}}|^2 \mathrm{d}z\Big )^{1/2}, \end{aligned}$$
(4.70)

where we have used Lemma 3.3 and estimates (4.66) and (4.67).

Next, we deal with the third integral on the right-hand side of (4.68). Indeed, from (4.65), we have \(\frac{\partial (\overline{\theta ' w_2})}{\partial z}=\frac{\partial ^2 {\overline{\theta }}}{\partial z^2} - \frac{\partial (\overline{\theta '_1 w})}{\partial z}\) in \(L^2(0,T;H^{-1}(0,2\pi ))\), and thus

$$\begin{aligned}&- \frac{1}{4 \pi ^2 L^2}\int _{\Omega } w_2 \frac{\partial \overline{\theta }}{\partial z} \theta ' \mathrm{d}x\,\mathrm{d}y \mathrm{d}z =-\int _0^{2\pi } (\overline{w_2 \theta '}) \frac{\partial \overline{\theta }}{\partial z} \mathrm{d}z \nonumber \\&\quad = \left\langle \frac{\partial ({\overline{w_2 \theta '} }) }{\partial z}, \overline{\theta } \right\rangle _{H^{-1}(0,2\pi ) \times \dot{H}^1(0,2\pi )} = \left\langle \frac{\partial ^2 {\overline{\theta }}}{\partial z^2} - \frac{\partial (\overline{\theta '_1 w})}{\partial z}, \overline{\theta } \right\rangle _{H^{-1}(0,2\pi ) \times \dot{H}^1(0,2\pi )} \nonumber \\&\quad = -\int _0^{2\pi } \left| \overline{\theta }_z\right| ^2 \mathrm{d}z + \int _0^{2\pi } (\overline{\theta '_1 w}) \overline{\theta }_z \mathrm{d}z \nonumber \\&\quad \le -\int _0^{2\pi } \left| \overline{\theta }_z\right| ^2 \mathrm{d}z + \frac{1}{2}\int _0^{2\pi } \left| \overline{\theta '_1 w}\right| ^2 \mathrm{d}z + \frac{1}{2} \int _0^{2\pi } \left| \overline{\theta }_z\right| ^2 \mathrm{d}z \nonumber \\&\quad \le -\frac{1}{2} \int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z + \frac{1}{2}\int _0^{2\pi } \left| \overline{\theta '_1 w}\right| ^2 \mathrm{d}z. \end{aligned}$$
(4.71)

Using the Cauchy–Schwarz inequality and Lemma 3.3 as well as estimate (4.67), we have

$$\begin{aligned}&\int _0^{2\pi } \left| \overline{\theta '_1 w}\right| ^2 \mathrm{d}z \le C\int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |\theta '_1|^2 \mathrm{d}x \mathrm{d}y\Big ) \Big (\int _{[0,2\pi L]^2} |w|^2 \mathrm{d}x\,\mathrm{d}y\Big ) \mathrm{d}z \nonumber \\&\quad \le C\Big [\sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |w|^2 \mathrm{d}x\,\mathrm{d}y\Big ) \Big ] \Vert \theta '_1\Vert _2^2 \le C \Big (\Vert w\Vert _2^2+\Vert \partial _z^{\frac{2}{3}}w\Vert _2^2 \Big ) \Vert \theta '_1\Vert _2^2 \nonumber \\&\quad \le C \Vert \theta '\Vert _2^2 \Vert \theta '_1\Vert _2^2. \end{aligned}$$
(4.72)

Then, combining (4.71) and (4.72) gives

$$\begin{aligned} - \int _{\Omega } w_2 \frac{\partial \overline{\theta }}{\partial z} \theta ' \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \le - 2\pi ^2 L^2 \int _0^{2\pi } \left| \overline{\theta }_z\right| ^2 \mathrm{d}z + C \Vert \theta '\Vert _2^2 \Vert \theta '_1\Vert _2^2. \end{aligned}$$
(4.73)

Using (4.69), (4.70) and (4.73), we infer from (4.68) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \Vert \theta '\Vert _2^2 + \Vert \nabla _h \theta '\Vert _2^2 + 4 \pi ^2 L^2 \int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z \nonumber \\&\quad \le C \Vert \theta '\Vert _2^2 \Big (\Vert \theta _1'\Vert _2^2 \Vert \nabla _h \theta _1'\Vert _2^2 + \int _0^{2\pi } |\partial _z {\overline{\theta _1}}|^2 \mathrm{d}z + \Vert \theta '_1\Vert _2^2+1 \Big ), \end{aligned}$$
(4.74)

for all \(t\in [0,T]\).

By virtue of the energy identity (2.7), we know that

$$\begin{aligned} \frac{1}{2}\Vert \theta '_1(t)\Vert _2^2 + \int _0^t \Vert \nabla _h \theta '_1(s) \Vert _2^2 \mathrm{d}s + 4\pi ^2 L^2 \int _0^t \int _0^{2\pi } |\partial _z {\overline{\theta _1}(z,s)}|^2 \mathrm{d}z \mathrm{d}s = \frac{1}{2}\Vert \theta '_1(0)\Vert _2^2, \end{aligned}$$
(4.75)

for all \(t\in [0,T]\). Thanks to Grönwall’s inequality and estimate (4.75), we derive from (4.74) that

$$\begin{aligned} \Vert \theta '(t)\Vert _2^2&\le \Vert \theta '(0)\Vert _2^2 \, \nonumber \\&\quad \times \exp \Big [{C \int _0^t \Big (\Vert \theta _1'(s)\Vert _2^2 \Vert \nabla _h \theta _1'(s)\Vert _2^2 + \int _0^{2\pi } |\partial _z {\overline{\theta _1}(z,s)}|^2 \mathrm{d}z + \Vert \theta '_1(s)\Vert _2^2+1\Big ) \mathrm{d}s }\Big ] \nonumber \\&\le \Vert \theta '(0)\Vert _2^2 \, \exp \left[ C (\Vert \theta _1'(0)\Vert _2^4 + \Vert \theta _1'(0)\Vert _2^2 + t) \right] , \end{aligned}$$
(4.76)

for all \(t\in [0,T]\). As a result, if \(\theta _1'(0)=\theta _2'(0)\), i.e., \(\theta '(0)=\theta _1'(0)- \theta _2'(0)=0\), then (4.76) implies that \(\Vert \theta '(t)\Vert _2^2 =0\) for all \(t\in [0,T]\), namely \(\theta _1'(t)=\theta _2'(t)\) for all \(t\in [0,T]\).

In addition, taking advantage of estimate (4.66) and the Poincaré inequality (3.3), we know that \(\Vert \mathbf{u}(t)\Vert _2^2 + \Vert w(t)\Vert _2^2 \le C \Vert \theta '(t)\Vert _2^2=0\) for all \(t\in [0,T]\), thus \({\mathbf {u}}_1(t)={\mathbf {u}}_2(t)\) and \(w_1(t)=w_2(t)\) for all \(t\in [0,T]\).

Furthermore, we take the scalar product of Eq. (4.65) and \(-\overline{\theta }\) in the duality between \(H^{-1}(0,2\pi )\) and \(\dot{H}^1(0,2\pi )\), then

$$\begin{aligned} \int _0^{2\pi } |\overline{\theta }_z|^2 \mathrm{d}z = \int _0^{2\pi } \left[ (\overline{\theta ' w_2}) + (\overline{\theta '_1 w})\right] \overline{\theta }_z \mathrm{d}z =0, \;\; \text {for all} \; t\in [0,T], \end{aligned}$$

since \(\theta '(t)=w(t)=0\) for all \(t\in [0,T]\). Thus, \(\overline{\theta }_z(t)=0\) for all \(t\in [0,T]\). That is,

$$\begin{aligned} \partial _z \overline{\theta _1}(t) = \partial _z \overline{\theta _2}(t), \;\; \text {for all}\,\, t\in [0,T]. \end{aligned}$$
(4.77)

From (4.77), we know that \(\overline{\theta _1}(z,t) = \overline{\theta _2}(z,t) + C(t)\) for all \(z\in [0,2\pi ]\), and for all \(t\in [0,T]\). Then, since \(\overline{\theta _1}\) and \(\overline{\theta _2}\) both have average zero over \([0,2\pi ]\), i.e., \(\int _0^{2\pi } \overline{\theta _1} \mathrm{d}z = \int _0^{2\pi } \overline{\theta _2} \mathrm{d}z =0\), it follows that \(C(t)=0\) for all \(t\in [0,T]\). Consequently, \(\overline{\theta _1}(z,t) = \overline{\theta _2}(z,t)\) for all \(z\in [0,2\pi ]\), and for all \(t\in [0,T]\).

Since both \(\theta _1'=\theta _2'\) and \(\overline{\theta _1}=\overline{\theta _2}\) are valid, then \(\theta _1= \theta _1'+ \overline{\theta _1}= \theta _2'+ \overline{\theta _2}= \theta _2\) for all \(t\in [0,T]\).

This completes the proof for the uniqueness of weak solutions.

Furthermore, by using (4.76), it is easy to obtain the continuous dependence on initial data stated in Theorem 2.2.

Strong solutions

In this section, we prove the global well-posedness of strong solutions for system (1.12)–(1.16). The uniqueness of strong solutions follows from the uniqueness of weak solutions. It remains to show the existence of global strong solutions, when the initial value \(\theta _0' \in H^1(\Omega )\).

Existence of strong solutions

We use the method of Galerkin approximation. Let us consider the Galerkin system (4.1)–(4.4) and perform energy estimate as follows.

Estimate from linear equations (4.1) and (4.2)

Recall the following estimate from Sect. 4.1.2.

$$\begin{aligned}&\Vert \Delta _h {\mathbf {u}}_m\Vert _2^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2, \;\;\;\; \Vert \Delta _h w_m\Vert _2^2\le \text {Ra}^2 \Vert \theta '_m\Vert _2^2; \end{aligned}$$
(5.1)
$$\begin{aligned}&\Vert \partial _z {\mathbf {u}}_m\Vert _2^2 \le \text {Ra}^2 \Vert \nabla _h \theta '_m\Vert _2^2, \;\;\;\; \Vert \partial _z w_m\Vert _2^2 \le \text {Ra}^2 \Vert \nabla _h \theta '_m\Vert _2^2; \end{aligned}$$
(5.2)
$$\begin{aligned}&\Vert \partial _z^{\frac{2}{3}} {\mathbf {u}}_m\Vert _2^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2, \;\;\;\; \Vert \partial _z^{\frac{2}{3}} w_m\Vert _2^2 \le \text {Ra}^2 \Vert \theta '_m\Vert _2^2. \end{aligned}$$
(5.3)

Corresponding to (5.2), we can also derive

$$\begin{aligned}&\Vert \nabla _h \partial _z {\mathbf {u}}_m\Vert _2^2 \le \text {Ra}^2 \Vert \Delta _h \theta '_m\Vert _2^2, \;\;\;\; \Vert \nabla _h \partial _z w_m\Vert _2^2 \le \text {Ra}^2 \Vert \Delta _h \theta '_m\Vert _2^2; \end{aligned}$$
(5.4)
$$\begin{aligned}&\Vert \partial _{zz} {\mathbf {u}}_m\Vert _2^2 \le \text {Ra}^2 \Vert \nabla _h \partial _z \theta '_m\Vert _2^2, \;\;\;\; \Vert \partial _{zz} w_m\Vert _2^2 \le \text {Ra}^2 \Vert \nabla _h \partial _z \theta '_m\Vert _2^2. \end{aligned}$$
(5.5)

Estimate for \(\Vert \theta '_m\Vert _2^2\)

This is exactly the same as the energy estimate performed in Sect. 4.1.2. Therefore, by (4.16), we have

$$\begin{aligned}&\frac{1}{2} \Vert \theta '_m(t)\Vert _2^2 + \int _0^t \Vert \nabla _h \theta '_m \Vert _2^2 \mathrm{d}s + 4\pi ^2 L^2 \int _0^t \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z \mathrm{d}s \nonumber \\&\quad = \frac{1}{2} \Vert \theta '_m(0)\Vert _2^2 \le \frac{1}{2} \Vert \theta '_0\Vert _2^2, \end{aligned}$$
(5.6)

for all \(t\ge 0\).

Estimate for \(\Vert \nabla _h \theta '_m\Vert _2^2\)

Taking the \(L^2\) inner product of (4.3) with \(-\Delta _h \theta '_m\) yields

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \Vert \nabla _h \theta '_m\Vert _2^2 + \Vert \Delta _h \theta '_m\Vert _2^2 \nonumber \\&\quad = \int _{\Omega } [P_m ({\mathbf {u}}_m \cdot \nabla _h \theta '_m )] \Delta _h \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z +\int _{\Omega } \Big [P_m\Big (w_m \frac{\partial \overline{\theta _m}}{\partial z}\Big )\Big ] \Delta _h \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z . \end{aligned}$$
(5.7)

Thanks to Lemma 3.1, we have

$$\begin{aligned}&\int _{\Omega } [P_m ({\mathbf {u}}_m \cdot \nabla _h \theta '_m )] \Delta _h \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z = \int _{\Omega } ({\mathbf {u}}_m \cdot \nabla _h \theta '_m ) \Delta _h \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \nonumber \\&\quad \le C \Vert \nabla _h {\mathbf {u}}_m\Vert _2^{1/2} \left( \Vert {\mathbf {u}}_m\Vert _2 + \Vert \partial _z {\mathbf {u}}_m\Vert _2 \right) ^{1/2} \Vert \nabla _h \theta '_m\Vert _2^{1/2} \Vert \Delta _h \theta '_m\Vert _2^{3/2} \nonumber \\&\quad \le \frac{1}{4} \Vert \Delta _h \theta '_m\Vert _2^2 + C\Vert \nabla _h {\mathbf {u}}_m\Vert _2^2 \left( \Vert {\mathbf {u}}_m\Vert _2^2 + \Vert \partial _z {\mathbf {u}}_m\Vert _2^2 \right) \Vert \nabla _h \theta '_m\Vert _2^2 \nonumber \\&\quad \le \frac{1}{4} \Vert \Delta _h \theta '_m\Vert _2^2 + C \Vert \theta '_m\Vert _2^2 \Vert \nabla _h \theta '_m\Vert _2^4 , \end{aligned}$$
(5.8)

where we have used estimates (5.1) and (5.2).

Next, using the Cauchy–Schwarz inequality, we estimate

$$\begin{aligned}&\int _{\Omega } \Big [P_m\Big (w_m \frac{\partial \overline{\theta _m}}{\partial z}\Big )\Big ] \Delta _h \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z = \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} w_m \Delta _h \theta '_m \mathrm{d}x\,\mathrm{d}y \Big ) \frac{\partial \overline{\theta _m}}{\partial z} \mathrm{d}z \nonumber \\&\quad \le \int _0^{2\pi } \Big ( \int _{[0,2\pi L]^2} |w_m|^2 \mathrm{d}x \mathrm{d}y \Big )^{1/2} \Big ( \int _{[0,2\pi L]^2} |\Delta _h \theta '_m|^2 \mathrm{d}x\,\mathrm{d}y \Big )^{1/2} \Big |\frac{\partial \overline{\theta _m}}{\partial z}\Big | \mathrm{d}z \nonumber \\&\quad \le \Big [\sup _{z\in [0,2\pi ]} \Big ( \int _{[0,2\pi L]^2} |w_m|^2 \mathrm{d}x\,\mathrm{d}y \Big )^{1/2} \Big ] \Vert \Delta _h \theta '_m\Vert _2 \Big (\int _0^{2\pi } \Big |\frac{\partial \overline{\theta _m}}{\partial z}\Big |^2 \mathrm{d}z\Big )^{1/2}, \nonumber \\&\quad \le C\Big (\Vert w_m\Vert _2+ \Vert \partial _z^{\frac{2}{3}} w_m\Vert _2 \Big ) \Vert \Delta _h \theta '_m\Vert _2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z \Big )^{1/2} \nonumber \\&\quad \le \frac{1}{4} \Vert \Delta _h \theta '_m\Vert _2^2 + C \Big (\Vert w_m\Vert _2^2+ \Vert \partial _z^{\frac{2}{3}} w_m\Vert _2^2 \Big ) \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big ) \nonumber \\&\quad \le \frac{1}{4} \Vert \Delta _h \theta '_m\Vert _2^2 + C \Vert \theta '_m\Vert _2^2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big ), \end{aligned}$$
(5.9)

where we have used Lemma 3.3 and estimates (5.1) and (5.3).

By applying estimates (5.8) and (5.9) into (5.7), we infer

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \nabla _h \theta '_m\Vert _2^2 + \Vert \Delta _h \theta '_m\Vert _2^2&\le C\Vert \theta '_m\Vert _2^2 \Vert \nabla _h \theta '_m\Vert _2^4 + C \Vert \theta '_m\Vert _2^2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big ). \end{aligned}$$
(5.10)

Applying Grönwall’s inequality to (5.10) and using estimate (5.6), we obtain

$$\begin{aligned} \Vert \nabla _h \theta '_m (t)\Vert _2^2&\le e^{ C \Vert \theta '_0\Vert _2^2 \int _0^t \Vert \nabla _h \theta '_m\Vert _2^2 \mathrm{d}s } \Big ( \Vert \nabla _h \theta '_0\Vert _2^2 + C\Vert \theta '_0\Vert _2^2 \int _0^t \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z \mathrm{d}s \Big ) \nonumber \\&\le e^{C \Vert \theta '_0\Vert _2^4} (\Vert \nabla _h \theta '_0\Vert _2^2 + C\Vert \theta '_0\Vert _2^4 ), \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(5.11)

Also, by integrating (5.10) over [0, t], and using (5.6) and (5.11), one has

$$\begin{aligned} \int _0^t \Vert \Delta _h \theta '_m(s)\Vert _2^2 \mathrm{d}s \le C(\Vert \nabla _h \theta '_0\Vert _2^2), \;\; \text {for all} \; t\ge 0, \end{aligned}$$
(5.12)

where \(C(\Vert \nabla _h \theta '_0\Vert _2^2)\) is a constant depending on \(\Vert \nabla _h \theta '_0\Vert _2^2\) but independent of time.

Estimate for \(\Vert \partial _z \theta '_m\Vert _2^2\)

Differentiating (4.3) with respect to z gives

$$\begin{aligned}&\partial _t \partial _z \theta '_m + P_m(\partial _z {\mathbf {u}}_m \cdot \nabla _h \theta '_m) + P_m({\mathbf {u}}_m \cdot \nabla _h \partial _z \theta '_m) \nonumber \\&\quad +P_m(\partial _z w_m \partial _z \overline{\theta _m}) + P_m(w_m \partial _{zz} \overline{\theta _m})= \Delta _h \partial _z \theta '_m. \end{aligned}$$
(5.13)

Taking the \(L^2\) inner product of (5.13) with \(\partial _{z}\theta '_m\), we obtain

$$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \partial _z \theta '_m\Vert _2^2 + \Vert \nabla _h \partial _z \theta '_m\Vert _2^2 = - \int _{\Omega } \left( \partial _z {\mathbf {u}}_m \cdot \nabla _h \theta '_m\right) \partial _z \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \nonumber \\&\quad - \int _{\Omega } \partial _z w_m (\partial _z \overline{\theta _m}) \partial _z \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z - \int _{\Omega } w_m (\partial _{zz}\overline{\theta _m}) \partial _z \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z, \end{aligned}$$
(5.14)

where we have used \(\int _{\Omega } \left( {\mathbf {u}}_m \cdot \nabla _h \partial _z \theta '_m\right) \partial _z \theta '_m \mathrm{d}x\,\mathrm{d}y \mathrm{d}z=0\) since \(\nabla _h \cdot {\mathbf {u}}_m=0\).

We evaluate every integral on the right-hand side of (5.14). First, by using Lemma 3.1 as well as estimate (5.4) and (5.5), we have

$$\begin{aligned}&\int _{\Omega } |\left( \partial _z {\mathbf {u}}_m \cdot \nabla _h \theta '_m\right) \partial _z \theta '_m| \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \nonumber \\&\quad \le C \Vert \nabla _h \partial _z {\mathbf {u}}_m\Vert _2^{1/2} \Vert \partial _{zz}{\mathbf {u}}_m\Vert _2^{1/2} \Vert \nabla _h \theta '_m\Vert _2^{1/2} \Vert \Delta _h \theta '_m\Vert _2^{1/2} \Vert \partial _z \theta '_m\Vert _2 \nonumber \\&\quad \le C \Vert \Delta _h \theta '_m\Vert _2 \Vert \nabla _h \partial _z \theta '_m\Vert _2^{1/2} \Vert \nabla _h \theta '_m\Vert _2^{1/2} \Vert \partial _z \theta '_m\Vert _2 \nonumber \\&\quad \le \frac{1}{4} \Vert \nabla _h \partial _z \theta '_m\Vert _2^2 + \Vert \nabla _h \theta '_m\Vert _2^2 + C \Vert \Delta _h \theta '_m\Vert _2^2 \Vert \partial _z \theta '_m\Vert _2^2. \end{aligned}$$
(5.15)

Applying the Cauchy–Schwarz inequality yields

$$\begin{aligned}&\int _{\Omega } |\partial _z w_m (\partial _z \overline{\theta _m}) \partial _z \theta '_m| \mathrm{d}x\,\mathrm{d}y \mathrm{d}z \nonumber \\&\quad \le \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |\partial _z w_m|^2 \mathrm{d}x \mathrm{d}y\Big )^{1/2} \Big (\int _{[0,2\pi L]^2} |\partial _z\theta '_m|^2 \mathrm{d}x \mathrm{d}y\Big )^{1/2} |\partial _z \overline{\theta _m}| \mathrm{d}z \nonumber \\&\quad \le \Big [\sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |\partial _z w_m|^2 \mathrm{d}x\,\mathrm{d}y\Big )^{1/2} \Big ] \Vert \partial _z \theta '_m\Vert _2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big )^{1/2}, \nonumber \\&\quad \le C \Vert \partial _{zz}w_m\Vert _2 \Vert \partial _z \theta '_m\Vert _2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z \Big )^{1/2} \nonumber \\&\quad \le C \Vert \nabla _h \partial _z \theta '_m\Vert _2 \Vert \partial _z \theta '_m\Vert _2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big )^{1/2} \nonumber \\&\quad \le \frac{1}{4} \Vert \nabla _h \partial _z \theta '_m\Vert _2^2 + C \Vert \partial _z \theta '_m\Vert _2^2 \Big (\int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z\Big ), \end{aligned}$$
(5.16)

where we have used Lemma 3.3 and estimate (5.5).

Now we estimate the third integral on the right-hand side of (5.14). Since \(\partial _z (\overline{\theta '_m w_m})= \partial _{zz}\overline{\theta _m}\) due to (4.4), we have

$$\begin{aligned}&- \frac{1}{4\pi ^2 L^2} \int _{\Omega } w_m (\partial _{zz}\overline{\theta _m}) \partial _z \theta '_m \mathrm{d}x \mathrm{d}y \mathrm{d}z = - \frac{1}{4 \pi ^2 L^2} \int _{\Omega } w_m [\partial _z (\overline{\theta '_m w_m})] \partial _z \theta '_m \mathrm{d}x \mathrm{d}y \mathrm{d}z \nonumber \\&\quad = - \int _0^{2\pi } (\overline{w_m \partial _z\theta '_m}) [\partial _z (\overline{ \theta '_m w_m})] \mathrm{d}z \nonumber \\&\quad = - \int _0^{2\pi } (\overline{w_m \partial _z \theta '_m}) \left[ \overline{(\partial _z \theta '_m) w_m} + \overline{\theta '_m (\partial _z w_m)} \right] \mathrm{d}z \nonumber \\&\quad = - \int _0^{2\pi } (\overline{w_m \partial _z \theta '_m})^2 \mathrm{d}z - \int _0^{2\pi } (\overline{w_m \partial _z \theta '_m}) (\overline{\theta '_m \partial _z w_m}) \mathrm{d}z \nonumber \\&\quad \le -\frac{1}{2} \int _0^{2\pi } (\overline{w_m \partial _z \theta '_m})^2 \mathrm{d}z + \frac{1}{2} \int _0^{2\pi } (\overline{\theta '_m \partial _z w_m})^2 \mathrm{d}z, \end{aligned}$$
(5.17)

where the last inequality is due to the Cauchy–Schwarz inequality and the Young’s inequality.

By using the Cauchy–Schwarz inequality, and Lemma 3.3 as well as estimate (5.2), then

$$\begin{aligned}&\int _0^{2\pi } (\overline{\theta '_m \partial _z w_m})^2 \mathrm{d}z \le C\int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |\theta '_m|^2 \mathrm{d}x \mathrm{d}y \Big ) \Big (\int _{[0,2\pi L]^2} |\partial _z w_m|^2 \mathrm{d}x \mathrm{d}y\Big ) \mathrm{d}z \nonumber \\&\quad \le C \Big [\sup _{z\in [0,2\pi ]} \Big (\int _{[0,2\pi L]^2} |\theta '_m|^2 \mathrm{d}x \mathrm{d}y \Big ) \Big ] \Vert \partial _z w_m\Vert _2^2 \le C \left( \Vert \theta '_m\Vert _2^2 + \Vert \partial _z \theta '_m\Vert _2^2 \right) \Vert \nabla _h \theta '_m\Vert _2^2. \end{aligned}$$
(5.18)

Substituting (5.18) into (5.17) gives

$$\begin{aligned}&- \frac{1}{4 \pi ^2 L^2} \int _{\Omega } w_m (\partial _{zz}\overline{\theta _m}) \partial _z \theta '_m \mathrm{d}x \mathrm{d}y \mathrm{d}z \nonumber \\&\quad \le -\frac{1}{2} \int _0^{2\pi } (\overline{w_m \partial _z\theta '_m})^2 \mathrm{d}z + C \left( \Vert \theta '_m\Vert _2^2 + \Vert \partial _z \theta '_m\Vert _2^2 \right) \Vert \nabla _h \theta '_m\Vert _2^2. \end{aligned}$$
(5.19)

Using (5.15), (5.16) and (5.19), we infer from (5.14) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \Vert \partial _z \theta '_m\Vert _2^2 + \Vert \nabla _h \partial _z \theta '_m\Vert _2^2 \nonumber \\&\quad \le C \Vert \partial _z \theta '_m\Vert _2^2 \left[ \Vert \Delta _h \theta '_m\Vert _2^2 + \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z \right] +C (\Vert \theta '_m\Vert _2^2+1) \Vert \nabla _h \theta '_m\Vert _2^2. \end{aligned}$$
(5.20)

Due to (5.6) and (5.11) and (5.12), we can use Grönwall’s inequality on (5.20) to obtain

$$\begin{aligned} \Vert \partial _z \theta '_m(t)\Vert _2^2&\le e^{\int _0^t (\Vert \Delta _h \theta '_m\Vert _2^2 + \int _0^{2\pi } |\partial _z \overline{\theta _m}|^2 \mathrm{d}z) \mathrm{d}s } \Big (\Vert \partial _z \theta _0'\Vert _2^2 + C (\Vert \theta '_0\Vert _2^2+1) \int _0^t \Vert \nabla _h \theta '_m\Vert _2^2 \mathrm{d}s \Big ) \nonumber \\&\quad \le C(\Vert \theta '_0\Vert _{H^1(\Omega )}), \;\; \text {for all} \; t\ge 0, \end{aligned}$$
(5.21)

where \(C(\Vert \theta '_0\Vert _{H^1(\Omega )})\) is a constant depending on \(\Vert \theta '_0\Vert _{H^1(\Omega )}\) but independent of t. In addition, integrating (5.20) over [0, t] gives

$$\begin{aligned} \int _0^t \Vert \nabla _h \partial _z \theta '_m\Vert _2^2 \mathrm{d}s \le C(\Vert \theta '_0\Vert _{H^1(\Omega )}), \;\; \text {for all} \; t\ge 0. \end{aligned}$$
(5.22)

On account of (5.6), (5.11) and (5.21), we conclude that

$$\begin{aligned} \Vert \theta '_m(t)\Vert _{H^1(\Omega )}^2 \le C(\Vert \theta '_0\Vert _{H^1(\Omega )}), \;\; \text {for all} \;t\ge 0, \end{aligned}$$
(5.23)

where \(C(\Vert \theta '_0\Vert _{H^1(\Omega )})\) is a constant depending on \(\Vert \theta '_0\Vert _{H^1(\Omega )}\) but independent of t.

Estimate for \(\int _0^{2\pi }\left| \partial _{zz}\overline{\theta _m}\right| ^2 \mathrm{d}z\)

Since \( \frac{\partial ^2 \overline{\theta _m}}{\partial z^2} = \frac{\partial (\overline{\theta '_m w_m})}{\partial z} \), we have

$$\begin{aligned}&\int _0^{2\pi } \Big |\frac{\partial ^2 \overline{\theta _m}}{\partial z^2} \Big |^2 \mathrm{d}z = \int _0^{2\pi } \Big |\frac{\partial (\overline{\theta '_m w_m})}{\partial z}\Big |^2 \mathrm{d}z = \int _0^{2\pi } \Big | \overline{\frac{\partial \theta '_m}{\partial z} w_m} + \overline{\theta '_m \frac{\partial w_m}{\partial z} } \Big |^2 \mathrm{d}z \nonumber \\&\quad \le C \int _0^{2\pi } \Big | \int _{[0,2\pi L]^2} \frac{\partial \theta '_m}{\partial z} w_m \mathrm{d}x \mathrm{d}y \Big |^2 \mathrm{d}z + C \int _0^{2\pi } \Big | \int _{[0,2\pi L]^2} \theta '_m \frac{\partial w_m}{\partial z} \mathrm{d}x \mathrm{d}y \Big |^2 \mathrm{d}z \nonumber \\&\quad \le C \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} \Big |\frac{\partial \theta '_m}{\partial z}\Big |^2 \mathrm{d}x \mathrm{d}y\Big ) \Big (\int _{[0,2\pi L]^2} | w_m |^2 \mathrm{d}x \mathrm{d}y\Big ) \mathrm{d}z \nonumber \\&\qquad + C \int _0^{2\pi } \Big (\int _{[0,2\pi L]^2} |\theta '_m|^2 \mathrm{d}x \mathrm{d}y\Big ) \Big (\int _{[0,2\pi L]^2} \Big | \frac{\partial w_m}{\partial z} \Big |^2 \mathrm{d}x \mathrm{d}y\Big ) \mathrm{d}z \nonumber \\&\quad \le C \Big (\sup _{z\in [0,2\pi ]} \int _{[0,2\pi L]^2} |w_m|^2 \mathrm{d}x \mathrm{d}y\Big ) \Vert \partial _z \theta '_m\Vert _2^2 \nonumber \\&\qquad + C \Big (\sup _{z\in [0,2\pi ]} \int _{[0,2\pi L]^2} |\theta '_m|^2 \mathrm{d}x \mathrm{d}y\Big ) \Vert \partial _z w_m\Vert _2^2 \nonumber \\&\quad \le C (\Vert w_m\Vert _2^2+ \Vert \partial _z w_m\Vert _2^2) \Vert \partial _z \theta '_m\Vert _2^2 + C (\Vert \theta '_m\Vert _2^2+ \Vert \partial _z \theta '_m\Vert _2^2) \Vert \partial _z w_m\Vert _2^2 \nonumber \\&\quad \le C \Vert \nabla _h \theta '_m\Vert _2^2 \Vert \partial _z \theta '_m\Vert _2^2 + C (\Vert \theta '_m\Vert _2^2+ \Vert \partial _z \theta '_m\Vert _2^2) \Vert \nabla _h \theta '_m\Vert _2^2, \end{aligned}$$
(5.24)

for all \(t\ge 0\), where we have used Lemma 3.3 and estimates (5.1) and (5.2).

Applying uniform bound (5.23) to estimate (5.24) yield

$$\begin{aligned} \int _0^{2\pi } \Big |\frac{\partial ^2 \overline{\theta _m}}{\partial z^2}(z,t) \Big |^2 \mathrm{d}z \le C(\Vert \theta '_0\Vert _{H^1}), \;\; \text {for all} \; t\ge 0, \end{aligned}$$
(5.25)

where \(C(\Vert \theta '_0\Vert _{H^1(\Omega )})\) is a constant depending on \(\Vert \theta '_0\Vert _{H^1(\Omega )}\) but independent of t.

Passage to the limit

Let \(T>0\). In order to pass to the limit for nonlinear terms in the Galerkin system as \(m\rightarrow \infty \), we shall show that \(\partial _t \theta '_m\) is uniformly bounded in \(L^2(\Omega \times (0,T))\).

In fact, for any function \(\varphi \in L^{4/3}(0,T;L^2(\Omega ))\), using Lemma 3.1, one has

$$\begin{aligned}&\int _0^T \int _{\Omega } [P_m({\mathbf {u}}_m \cdot \nabla _h \theta '_m)] \varphi \mathrm{d}x \mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \Vert \nabla _h {\mathbf {u}}_m\Vert ^{1/2} \left( \Vert {\mathbf {u}}_m\Vert _2 + \Vert \partial _z {\mathbf {u}}_m\Vert _2 \right) ^{1/2} \Vert \nabla _h \theta '_m\Vert _2^{1/2} \Vert \Delta _h \theta '_m\Vert _2^{1/2} \Vert \varphi \Vert _2 \mathrm{d}t \nonumber \\&\quad \le C\int _0^T \Vert \theta _m'\Vert ^{1/2} \Vert \nabla _h \theta '_m\Vert _2 \Vert \Delta _h \theta '_m\Vert _2^{1/2} \Vert \varphi \Vert _2 \mathrm{d}t \nonumber \\&\quad \le C(\Vert \nabla _h \theta '_0\Vert _2) \int _0^T \Vert \Delta _h \theta '_m\Vert _2^{1/2} \Vert \varphi \Vert _2 \mathrm{d}t \nonumber \\&\quad \le C(\Vert \nabla _h \theta '_0\Vert _2) \left( \int _0^T \Vert \Delta _h \theta '_m\Vert _2^2 \mathrm{d}t\right) ^{1/4} \left( \int _0^T \Vert \varphi \Vert _2^{4/3} \mathrm{d}t \right) ^{3/4} \nonumber \\&\quad \le C(\Vert \nabla _h \theta '_0\Vert _2) \Vert \varphi \Vert _{L^{4/3}(0,T;L^2(\Omega ))}, \end{aligned}$$
(5.26)

where we have used estimate (5.1) and (5.2) and (5.11) and (5.12). Here, \(C(\Vert \nabla _h \theta '_0\Vert _2)\) is a constant depending on \(\Vert \nabla _h \theta '_0\Vert _2\) but independent of t.

We infer from (5.26) that

$$\begin{aligned} \Vert P_m({\mathbf {u}}_m \cdot \nabla _h \theta '_m)\Vert _{L^4(0,T;L^2(\Omega ))} \le C(\Vert \nabla _h \theta '_0\Vert _2). \end{aligned}$$
(5.27)

Then, using (4.3), (4.42), (5.12) and (5.27), we obtain

$$\begin{aligned} \Vert \partial _t \theta '_m\Vert _{L^2(\Omega \times (0,T))} \le C(\Vert \nabla _h \theta '_0\Vert _2) . \end{aligned}$$
(5.28)

Therefore, on a subsequence, we have

$$\begin{aligned} \partial _t \theta _m' \rightarrow \partial _t \theta ' \;\; \text {weakly in} \; L^2(\Omega \times (0,T)), \; \text {as} \; m\rightarrow \infty . \end{aligned}$$
(5.29)

In addition, due to uniform bounds (5.23) and (5.28), and thanks to the Aubin’s compactness theorem (see, e.g., [16]), the following strong convergence holds for a subsequence of \(\{\theta '_m\}\):

$$\begin{aligned} \theta _m' \rightarrow \theta ' \;\; \text {in} \; L^2(\Omega \times (0,T)), \; \text {as} \; m\rightarrow \infty . \end{aligned}$$
(5.30)

Also, using (4.12), we see that

$$\begin{aligned} \Vert {\mathbf {u}}_m -{\mathbf {u}}_n\Vert _2^2 \le C\Vert \theta '_m - \theta '_n\Vert _2^2, \;\; \text {for any} \; n,m \in {\mathbb {N}}. \end{aligned}$$
(5.31)

We infer from (5.30) and (5.31) that \(\{{\mathbf {u}}_m\}\) is a Cauchy sequence in \(L^2(\Omega \times (0,T))\), then

$$\begin{aligned} {\mathbf {u}}_m \rightarrow {\mathbf {u}} \;\; \text {in} \;\;L^2(\Omega \times (0,T)), \; \text {as} \; m\rightarrow \infty . \end{aligned}$$
(5.32)

Analogously, we deduce

$$\begin{aligned} w_m \rightarrow w \;\; \text {in} \;\;L^2(\Omega \times (0,T)), \; \text {as} \; m\rightarrow \infty . \end{aligned}$$
(5.33)

Hence, we can use the same argument as in Sect. 4.1.3 to pass to the limit \(m\rightarrow \infty \) for the Galerkin system (4.1)–(4.4) to derive

$$\begin{aligned} \frac{\partial \theta '}{\partial t} + {\mathbf {u}} \cdot \nabla _h \theta ' + w \frac{\partial \overline{\theta }}{\partial z} -\Delta _h \theta '=0 \;\; \text {in} \;\; L^2(\Omega \times (0,T)). \end{aligned}$$
(5.34)

Next, we shall pass to the limit for Eq. (4.4) in the Galerkin system as \(m\rightarrow \infty \). By virtue of (5.25), \(\int _0^{2\pi } \left| \frac{\partial ^2 \overline{\theta _m} }{\partial z^2}(z,t) \right| ^2 \mathrm{d}z \le C(\Vert \theta '_0\Vert _{H^1})\), for all \(t\ge 0\), namely the sequence \(\{\frac{\partial ^2 \overline{\theta _m}}{\partial z^2}\}\) is uniformly bounded in \(L^{\infty }(0,T;L^2(0,2\pi ))\), which implies, on a subsequence,

$$\begin{aligned} \frac{\partial ^2 \overline{\theta _m}}{\partial z^2} \rightarrow \frac{\partial ^2 \overline{\theta }}{\partial z^2} \;\; \text {weakly}^* \hbox { in} \;L^{\infty }(0,T;L^2(0,2\pi )), \; \text {as} \; m\rightarrow \infty . \end{aligned}$$
(5.35)

In order to pass to the limit for the nonlinear term \(\frac{\partial (\overline{\theta _m' w_m})}{\partial z}\) in equation (4.4), we take a test function \(\xi =\sum _{|j|<N} \hat{\xi }(t,j) e^{ij z}\) where Fourier coefficients \(\hat{\xi }(t,j)\), \(|j|\le N\), are continuous in t. Consider

$$\begin{aligned}&\left| \int _0^T \int _0^{2\pi } \left( \frac{\partial (\overline{\theta _m' w_m})}{\partial z} - \frac{\partial (\overline{\theta ' w})}{\partial z}\right) \xi \mathrm{d}z \mathrm{d}t \right| = \left| \int _0^T \int _0^{2\pi } (\overline{\theta _m' w_m}- \overline{\theta ' w}) \xi _z \mathrm{d}z \mathrm{d}t \right| \nonumber \\&\quad \le \left| \int _0^T \int _0^{2\pi } (\overline{(\theta _m'-\theta ') w_m}) \xi _z \mathrm{d}z \mathrm{d}t \right| + \left| \int _0^T \int _0^{2\pi } (\overline{\theta ' (w_m-w)}) \xi _z \mathrm{d}z \mathrm{d}t \right| \nonumber \\&\quad \le C\Vert \xi _z\Vert _{L^{\infty }((0,2\pi )\times (0,T))} \int _0^T \int _{\Omega } \left( \left| (\theta _m'-\theta ') w_m\right| + \left| \theta ' (w_m-w)\right| \right) \mathrm{d}x \mathrm{d}y \mathrm{d}z \mathrm{d}t \nonumber \\&\quad \le C\Vert \xi _z\Vert _{L^{\infty }((0,2\pi )\times (0,T))} \big (\Vert \theta '_m - \theta '\Vert _{L^2(\Omega \times (0,T))} \Vert w_m\Vert _{L^2(\Omega \times (0,T))} \nonumber \\&\qquad + \Vert w_m - w \Vert _{L^2(\Omega \times (0,T))} \Vert \theta '\Vert _{L^2(\Omega \times (0,T))}\big ), \end{aligned}$$
(5.36)

where we have used the Cauchy–Schwarz inequality.

Then, since \(\Vert w_m\Vert _2 \le C\Vert \theta '_0\Vert \) for all m on [0, T] and since \(\theta '_m \rightarrow \theta '\), \(w_m \rightarrow w\) in \(L^2(\Omega \times (0,T))\), we can let \(m\rightarrow \infty \) in (5.36) to obtain

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _0^T \int _0^{2\pi } \frac{\partial (\overline{\theta _m' w_m})}{\partial z} \xi \mathrm{d}z \mathrm{d}t = \int _0^T \int _0^{2\pi } \frac{\partial (\overline{\theta ' w})}{\partial z} \xi \mathrm{d}z \mathrm{d}t. \end{aligned}$$
(5.37)

On account of (5.35) and (5.37), we pass to the limit for Eq. (4.4) in the Galerkin approximate system, and arrive at

$$\begin{aligned} \int _0^T \int _0^{2\pi } \left( \frac{\partial (\overline{\theta ' w})}{\partial z} - \frac{\partial ^2 \overline{\theta }}{\partial z^2}\right) \xi \mathrm{d}z \mathrm{d}t =0, \end{aligned}$$
(5.38)

for any test function \(\xi =\sum _{|j|<N} \hat{\xi }(t,j) e^{ij z}\) with continuous Fourier coefficients.

Similarly to (5.24), we derive that \(\frac{\partial (\overline{\theta ' w})}{\partial z} \in L^{\infty }(0,T;L^2(0,2\pi ))\). Also \(\frac{\partial ^2 \overline{\theta }}{\partial z^2} \in L^{\infty }(0,T;L^2(0,2\pi ))\) from (5.35). Hence, \(\frac{\partial (\overline{\theta ' w})}{\partial z} - \frac{\partial ^2 \overline{\theta }}{\partial z^2} \in L^{\infty }(0,T;L^2(0,2\pi ))\). Then, we infer from (5.38) that

$$\begin{aligned} \frac{\partial (\overline{\theta ' w})}{\partial z} = \frac{\partial ^2 \overline{\theta }}{\partial z^2} \;\;\text {in} \;\; L^{\infty }(0,T;L^2(0,2\pi )). \end{aligned}$$

This completes the proof for the existence of a global strong solution in the sense of Definition 2.3.

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Acknowledgements

The work of E.S.T. was supported in part by the Einstein Stiftung Berlin, through the Einstein Visiting Fellow Program.

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Correspondence to Edriss S. Titi.

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Dedicated to Professor Matthias Hieber on the occasion of his 60th birthday

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Cao, C., Guo, Y. & Titi, E.S. Global well-posedness for a rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00644-4

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Keywords

  • Rayleigh–Bénard convection
  • Rotational fluid flow
  • Cyclonic and anticyclonic coherent structures
  • Incompressible
  • Infinite Prandtl number
  • Global well-posedness
  • Weak solutions
  • Strong solutions

Mathematics Subject Classification

  • 35A01
  • 35A02
  • 35Q35
  • 35K55