On the Well-posedness of Reduced $3D$ Primitive Geostrophic Adjustment Model with Weak Dissipation

TL;DR

This paper establishes well-posedness results for a reduced 3D geostrophic model with weak dissipation, demonstrating global solutions, regularization convergence, and blow-up criteria.

Contribution

It proves well-posedness for the reduced 3D primitive geostrophic model with weak dissipation and analyzes the regularized model's convergence and blow-up conditions.

Findings

Proved local and global well-posedness for the model.

Established convergence of Voigt α-regularized solutions.

Derived criteria for finite-time blow-up.

Abstract

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization.

Equations

$$u_t+u{u}_x+w{u}_z-f_{0}v+p_x=0,$$

$$v_t+u{v}_x+w{v}_z+f_{0}u=0,$$

$$p_z+T=0,$$

$$u_x+w_z=0,$$

$$T_t+u{T}_x+w{T}_z=0,$$

$$u_t-\nu u_{zz}+u{u}_x+w{u}_z-f_{0}v+p_x=0,$$

$$v_t-\nu v_{zz}+u{v}_x+w{v}_z+f_{0}u=0,$$

$$p_z+T=0,$$

$$u_x+w_z=0,$$

$$T_t-\kappa \Delta T+u{T}_x+w{T}_z=0.$$

$$u_t+u{u}_x+w{u}_z+p_x=0,$$

$$p_z=0,$$

$$u_x+w_z=0,$$

$$u_t+u{u}_x+w{u}_z+p_x-\nu u_{zz}=0,$$

$$p_z=0,$$

$$u_x+w_z=0,$$

$$u_t+u{u}_x+w{u}_z+{ϵ}_{1}u-f_{0}v+p_x-\nu u_{zz}=0,$$

$$v_t+u{v}_x+w{v}_z+{ϵ}_{1}v+f_{0}u-\nu v_{zz}=0,$$

$${ϵ}_{2}w+p_z+T=0,$$

$$u_x+w_z=0,$$

$$T_t-\kappa \Delta T+u{T}_x+w{T}_z=0,$$

$$(u_z,v_z,w,T){|}_{z=0,H}=0,$$

$$u,v,w,T\text{are periodic in}x\text{with period}1.$$

$$(u-{\alpha }^2{u}_{zz}{)}_t-\nu u_{zz}+u{u}_x+w{u}_z+{ϵ}_{1}u-f_{0}v+p_x=0,$$

$$(v-{\alpha }^2{v}_{zz}{)}_t-\nu v_{zz}+u{v}_x+w{v}_z+{ϵ}_{1}v+f_{0}u=0,$$

$${ϵ}_{2}w+p_z+T=0,$$

$$u_x+w_z=0,$$

$$T_t-\kappa \Delta T+u{T}_x+w{T}_z=0,$$

$${\int }_{{\mathbb{T}}^2}|fgh|dxdz\le C\Vert f{\Vert }_{L^2}\Vert g{\Vert }_{L^2}^{\frac{1}{2}}(\Vert g{\Vert }_{L^2}^{\frac{1}{2}}+\Vert g_z{\Vert }_{L^2}^{\frac{1}{2}})\Vert h{\Vert }_{L^2}^{\frac{1}{2}}(\Vert h{\Vert }_{L^2}^{\frac{1}{2}}+\Vert h_x{\Vert }_{L^2}^{\frac{1}{2}}).$$

$$\Vert \varphi {\Vert }_{L^{\infty }(0,1)}\le C\left(\Vert \varphi {\Vert }_{L^2(0,1)}+\Vert \varphi {\Vert }_{L^2(0,1)}^{\frac{1}{2}}\Vert {\varphi }_x{\Vert }_{L^2(0,1)}^{\frac{1}{2}}\right).$$

$$\displaystyle\hskip-49.14307pt\int_{\mathbb{T}^{2}}|fgh|(x,z)dxdz\leq C\int_{0}^{1}\left[\Big{(}\int_{0}^{1}|f(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|g(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}\Big{(}\sup_{0\leq x\leq 1}|h(x,z)|\Big{)}\right]dz$$

$$\displaystyle\hskip 10.84006pt\leq C\int_{0}^{1}\Bigg{\{}\Big{(}\int_{0}^{1}|f(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|g(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}$$

$$\displaystyle\hskip 61.42993pt\left[\Big{(}\int_{0}^{1}|h(x,z)|^{2}dx\Big{)}^{\frac{1}{4}}\Big{(}\int_{0}^{1}|h_{x}(x,z)|^{2}dx\Big{)}^{\frac{1}{4}}+\Big{(}\int_{0}^{1}|h(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}\right]\Bigg{\}}dz$$

$$\displaystyle\hskip 10.84006pt\leq C\|f\|_{L^{2}}\sup_{0\leq z\leq 1}\Big{(}\int_{0}^{1}|g(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}\|h\|_{L^{2}}^{\frac{1}{2}}(\|h\|_{L^{2}}^{\frac{1}{2}}+\|h_{x}\|_{L^{2}}^{\frac{1}{2}}).$$

$$\displaystyle\hskip-49.14307pt\sup_{0\leq z\leq 1}\Big{(}\int_{0}^{1}|g(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}\leq C\Big{(}\int_{0}^{1}\sup_{0\leq z\leq 1}|g(x,z)|^{2}dx\Big{)}^{\frac{1}{2}}$$

$$\displaystyle\hskip 21.68121pt\leq C\left(\int_{0}^{1}\left[\Big{(}\int_{0}^{1}|g(x,z)|^{2}dz\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|g_{z}(x,z)|^{2}dz\Big{)}^{\frac{1}{2}}+\int_{0}^{1}|g(x,z)|^{2}dz\right]dx\right)^{\frac{1}{2}}$$

$$\le C\Vert g{\Vert }_{L^2}^{\frac{1}{2}}(\Vert g{\Vert }_{L^2}^{\frac{1}{2}}+\Vert g_z{\Vert }_{L^2}^{\frac{1}{2}}).$$

$$\Vert f{\Vert }_{L^{\infty }}\le C{\left(\Vert f{\Vert }_{H^1}^2+\Vert f_{xz}{\Vert }_{L^2}^2\right)}^{\frac{1}{2}}.$$

$$\Vert f{\Vert }_{L^{\infty }}\le \sum _{k\in {\mathbb{Z}}^2}|\widehat{f_k}|=\sum _{k\in {\mathbb{Z}}^2}\frac{|\widehat{f_k}|(1+k_1^2+k_2^2+k_1^2{k}_2^2{)}^{\frac{1}{2}}}{(1+k_1^2+k_2^2+k_1^2{k}_2^2{)}^{\frac{1}{2}}}$$

$$\displaystyle\hskip-49.14307pt\leq\left(\sum\limits_{k\in\mathbb{Z}^{2}}|\hat{f_{k}}|^{2}(1+k_{1}^{2}+k_{2}^{2}+k_{1}^{2}k_{2}^{2})\right)^{\frac{1}{2}}\left(\sum\limits_{k\in\mathbb{Z}^{2}}\frac{1}{(1+k_{1}^{2})(1+k_{2}^{2})}\right)^{\frac{1}{2}}\leq C\Big{(}\|f\|_{H^{1}}^{2}+\|f_{xz}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{2}}<\infty.$$

$$(u,v,T){|}_{t=0}=(u_0,v_0,T_0).$$

$$u,v,w,p\text{and}T\text{are periodic in}x\text{and}z\text{with period 1};$$

$$u,v,p\text{are even in}z,\text{and}w,T\text{are odd in z};$$

$$(u,v,T){|}_{t=0}=(u_0,v_0,T_0).$$

$$\mathcal{H}:=\{(u,v,w,p,T)|u,v,w,p\text{and}T\text{are spatially periodic in both}x\text{and}z\text{variables with period 1},$$

Full text

MSC Subject Classifications: 35A01, 35B44, 35Q35, 35Q86, 76D03, 86-08, 86A10.

Keywords: primitive geostrophic adjustment model; regularization; blow-up criterion

1. Introduction

It is commonly believed that the dynamics of ocean and atmosphere adjusts itself toward a geostrophic balance. The following reduced $3D$ primitive geostrophic adjustment model is a main type diagnostic model for studying geostrophic adjustment (cf. e.g., [28], [34], [52]):

[TABLE]

where the velocity field $(u,v,w)$, the temperature $T$ and the pressure $p$ are the unknown functions of the variable $(x,z,t)$, i.e., they depend only on two spatial variables, and $f_{0}$ is the Coriolis parameter. System (1)–(5) is reduced from the $3D$ inviscid primitive equations model by assuming that the flow is independent of the $y$ variable. This system has been a standard framework to study geostrophic adjustment of frontal anomalies in a rotating continuously stratified fluid of strictly rectilinear fronts and jets (cf. e.g., [2], [27], [28], [33], [34], [36], [52], [54] and references therein).

The first systematically mathematical studies of the viscous primitive equations (PEs) were carried out in the 1990s by Lions–Temam–Wang [45, 46, 47], where they considered the PEs with both full viscosities and full diffusivity, and established the global existence of weak solutions. The uniqueness of weak solutions in the $3D$ viscous case is still an open problem, while the weak solutions in $2D$ turn out to be unique, see Bresch, Guillén-González, Masmoudi and Rodríguez-Bellido [5]. Concerning the strong solutions for the $2D$ case, the local existence result was established by Guillén-González, Masmoudi and Rodríguez-Bellido [30], while the global existence for $2D$ case was achieved by Bresch, Kazhikhov and Lemoine in [6], and by Temam and Ziane in [60]. The global existence of strong solutions for $3D$ case was established by Cao and Titi in [16] and later by Kobelkov in [35], see also the subsequent articles of Kukavica and Ziane [40, 41], for different boundary conditions, as well as Hieber and Kashiwabara [32] for some progress towards relaxing the smoothness on the initial data by using the semigroup method. On the other hand, it has already been proven that smooth solutions to the inviscid $2D$ or $3D$ PEs, with or without coupling to the temperature equation, can develop singularities in finite time, see Cao et al. [8] and Wong [61]. Motivated by physical considerations, it is of great interest to investigate the global well-posedness, finite-time blow-up, or even ill-posedness of the PEs with only partial viscosities or partial diffusivity. There has been several works concerning the mathematical study of these models. The global existence and uniqueness of strong solutions for the PEs with full viscosities and with either horizontal or vertical diffusivity have been established by Cao–Titi [18] and Cao–Li–Titi [9, 10]. Concerning partial viscosities, strong solutions are shown to be unique and exist globally in time for the PEs with only horizontal viscosity and only horizontal diffusivity for any initial data in $H^{2}$ by Cao–Li–Titi [11] (see Cao–Li–Titi [12] for some generalization of the result in [11]), and for the PEs with only horizontal viscosity and only vertical diffusivity by Cao–Li–Titi [13]. On the other hand, there is no results concerning the well-posedness or finite-time blow-up for the PEs with only vertical viscosity, even for the two-dimensional case. In this paper, we are interested in this situation. More specifically, we are interested in system (1)–(5) with only vertical viscosity and full diffusivity:

[TABLE]

The main mathematical difficulty to prove well-posedness of system (6)–(10) is the lack of control over the horizontal derivatives. The situation with only vertical viscosity, i.e. system (6)–(10), is worse than the situation with only horizontal viscosity, for which the global well-posedness was established. Indeed, system (1)–(5) and system (6)–(10) turn out to be ill-posed. By considering system (1)–(5) and system (6)–(10) without the Coriolis force, the velocity in $y$ direction and the temperature, i.e., by setting $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$, we end up with the so-called $2D$ hydrostatic Euler equations:

[TABLE]

and $2D$ hydrostatic Navier-Stokes equations:

[TABLE]

correspondingly. The linear ill-posedness of system (11)–(13), about certain shear-flows, has been established by Renardy in [53]. Furthermore, the author of [53] has also indicated, without providing details, that one should be able to show the linear ill-posedness of system (14)–(16) in any Sobolev space by means of matched asymptotics. On the other hand, the nonlinear ill-posedness of system (11)–(13) has been established by Han-Kwan and Nguyen in [31], where they built an abstract framework to show the hydrostatic Euler equations are ill-posed in any Sobolev space. One might be able to argue that the main reason for the ill-posedness in these is again due the lack of control over the horizontal derivatives. From a mathematical perspective, system (14)–(16) is reminiscent of the 2D Prandtl system, in the upper half space. The ill-posedness in Sobolev spaces of Prandtl system was established by Gérard-Varet and Dormy [24], and by Gérard-Varet and Nguyen [26]. In [22] , the author established finite-time blow-up of solutions of Prandtl system. On the other hand, the only well-posedness results of hydrostatic Euler equations and Prandtl system can be obtained by assuming either real analyticity or some special structures on the initial data [3, 4, 29, 37, 38, 39, 48, 49, 50]. Recently, the authors in [25] showed the local well-posedness for system (14)–(16) with convex initial data in Gevrey class, which is slightly larger than analyticity class.

All the results and discussions above suggests that, in order to prove the well-posedness for system (6)–(10) in Sobolev spaces, without assuming any special structures, some additional horizontal dissipation terms are necessary. In the derivation of system (14)–(16) from the Navier-Stokes equations, only vertical viscosity is survived. This suggests that the strong dissipation, i.e., horizontal viscosity, is not always expected to exist. Instead, we will consider some weaker dissipation. Inspired by Samelson and Vallis [55], and Salmon [56], by introducing the linear (Rayleigh-like friction) damping in both horizontal momentum equation and vertical hydrostatic approximation to system (6)–(10), we consider the following model:

[TABLE]

in the horizontal channel $\{(x,z):0\leq z\leq H,x\in\mathbb{R}\}$, with the following boundary conditions:

[TABLE]

Here $\epsilon_{1}$ and $\epsilon_{2}$ are positive constants representing the linear (Rayleigh-like friction) damping coefficients, and $\nu$ is positive constant which stands for the vertical viscosity of the horizontal momentum equations. Since it is believed that the adjustment time is not very long, friction may be assumed to have negligible influence on the subsequent development of the flow. However, in this article, we show that the friction plays important roles in the well-posedness of the adjustment system. More specifically, the damping term $\epsilon_{2}w$ in (19) plays an important role as a weak dissipation in horizontal direction through $w_{z}$ and the incompressibility (20). Unlike the case with strong horizontal dissipation, i.e., with horizontal viscosity, the global well-posedness of system (17)–(21) is still a difficult task. However, in this paper, we are able to prove the local well-posedness and global well-posedness with small initial data. We refer the reader to [17] where the authors established the global well-posedness of the $3D$ Salmon’s planetary geostrophic oceanic dynamics model which involves similar damping term in the hydrostatic equations.

In order to study the possible finite-time blow-up of system (17)–(21), and to give a reliable numerical model/scheme to system (17)–(21), we also study the Voigt $\alpha$-regularization of system (17)–(21), with the regularization only in the $z$ variable. More specifically, we consider the following model:

[TABLE]

in the horizontal channel $\{(x,z):0\leq z\leq H,x\in\mathbb{R}\}$, with boundary conditions (22). We will show the global well-posedness of system (23)–(27) with $\nu=0$. The same results hold for $\nu>0$. Based on this, we show the convergence of the strong solution of system (23)–(27) to the corresponding solution of system (17)–(21) on the interval of existence of the latter, as $\alpha\rightarrow 0$. Furthermore, we derive a criterion for finite-time blow-up of system (17)–(21) based on this Voigt $\alpha$-regularization. For more details of Voigt $\alpha$-regularization of the $3D$ Euler equations, we refer the reader to [19, 42, 43].

The paper is organized as follows. In section 2, we introduce some notations and collect some preliminary results which will be used in the rest of this paper. In section 3, we show the local well-posedness and global well-posedness with small initial data of system (17)–(21). In section 3.3, we will see by assuming $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$, we are reduced to the 2D hydrostatic Navier-Stokes equations (14)–(16) with damping, for which we can obtain local well-posedness by requiring less on the initial conditions. In section 4, we show the global well-posedness of system (23)–(27). Again, we provide the similar result by requiring less on the initial conditions for the case when $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$. In section 5, we show the convergence of the strong solution of system (23)–(27) to the corresponding solution of system (17)–(21) on the interval of existence of the latter, as $\alpha\rightarrow 0$, with $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$. The reason by assuming $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$ is for sake of simplicity, and we can get the same convergence result without this assumption. Finally, in section 6, we derive criterion for finite-time blow-up of system (17)–(21), with $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$, based on this Voigt $\alpha$-regularization.

2. Preliminaries

In this section, we introduce some notations and collect some preliminary results which will be used in the rest of this paper. For domain $\Omega\subset\mathbb{R}^{2}$, we denote by $L^{p}(\Omega)$, for $p\geq 1$, the Lebesgue space of functions $f$ satisfying $\int_{\Omega}|f|^{p}dxdz<\infty$, and denote the corresponding norm by $\|f\|_{L^{p}}:=\|f\|_{L^{p}(\Omega)}=(\int_{\Omega}|f|^{p}dxdz)^{\frac{1}{p}}$. For the space $L^{2}(\Omega)$, we denote its inner product by $(\cdot,\cdot)$. Similarly, for $m\geq 1$ an integer, we denote by $H^{m}(\Omega)=W^{m,2}(\Omega)$ the Sobolev space of functions $f$ satisfying $\sum\limits_{|\alpha|\leq m}\|D^{\alpha}f\|^{2}_{L^{2}}<\infty$, and denote the corresponding norm by $\|f\|_{H^{m}}:=(\sum\limits_{|\alpha|\leq m}\|D^{\alpha}f\|_{L^{2}}^{2})^{\frac{1}{2}}$. Given time $\mathcal{T}>0$, we denote by $L^{p}(0,\mathcal{T};X)$ the space of functions $f:[0,T]\rightarrow X$ satisfying $\int_{0}^{\mathcal{T}}\|f(t)\|_{X}^{p}dt<\infty$, where $X$ is a Banach space and $\|\cdot\|_{X}$ represents its norm. Similarly, we denote by $C([0,\mathcal{T}];X)$ the space of continuous functions $f:[0,T]\rightarrow X$. We write $L^{p}(0,\mathcal{T};L^{2})$ and $L^{p}(0,\mathcal{T};H^{m})$ instead of $L^{p}(0,\mathcal{T};L^{2}(\Omega))$ and $L^{p}(0,\mathcal{T};H^{m}(\Omega))$, respectively, for simplicity. When $\Omega=\mathbb{T}^{2}$ is the unit two-dimensional flat torus, we denote by $L^{2}(\mathbb{T}^{2}),H^{m}(\mathbb{T}^{2})$ the set of all periodic functions in $x\in\mathbb{R}$ and $z\in\mathbb{R}$ with period 1, which have bounded $L^{2}(\mathbb{T}^{2})$ norm or $H^{m}(\mathbb{T}^{2})$ norm, respectively. We start by recalling the following:

Lemma 1**.**

(cf. [14]) Assume that $f,g,h,g_{z},h_{x}\in L^{2}(\mathbb{T}^{2})$. Then

[TABLE]

Proof.

First, recall that by one-dimensional Agmon’s inequality (or Gagliardo–Nirenberg interpolation inequality), for $\phi\in H^{1}(0,1)$, one has

[TABLE]

Therefore, by Hölder’s inequality and Agmon’s inequality (28),

[TABLE]

By Minkowski’s inequality, Agmon’s inequality (28), and Hölder inequality,

[TABLE]

Inserting (30) to (29) yields the desired inequality. ∎

Next we prove the following:

Lemma 2**.**

Assume that $f\in H^{1}(\mathbb{T}^{2})$ and $f_{xz}\in L^{2}(\mathbb{T}^{2})$. Then $f\in L^{\infty}(\mathbb{T}^{2})$. Moreover,

[TABLE]

Proof.

Let $\{\hat{f_{k}}\}_{k\in\mathbb{Z}^{2}}$ be the Fourier coefficients of $f$. By Cauchy–Schwarz inequality, we have

[TABLE]

Therefore, $f\in L^{\infty}(\mathbb{T}^{2})$. ∎

We also need the following Aubin-Lions theorem.

Proposition 3**.**

*(Aubin-Lions Lemma, cf. Simon [57] Corollary 4) Assume that X, B and Y are three Banach spaces, with $X\hookrightarrow\hookrightarrow B\hookrightarrow Y$. Then it holds that

*(i) If $\mathcal{F}$ is a bounded subset of $L^{p}(0,\mathcal{T};X)$, where $1\leq p<\infty$, and $\mathcal{F}_{t}:=\{\frac{\partial f}{\partial t}|f\in\mathcal{F}\}$ is bounded in $L^{1}(0,\mathcal{T};Y)$, then $\mathcal{F}$ is relative compact in $L^{p}(0,\mathcal{T};B)$.

(ii) If $\mathcal{F}$ is a bounded subset of $L^{\infty}(0,\mathcal{T};X)$ and $\mathcal{F}_{t}$ is bounded in $L^{q}(0,\mathcal{T};Y)$, where $q>1$, then $\mathcal{F}$ is relative compact in $C([0,\mathcal{T}];B)$.

3. Well-posedness of system (17)–(21)

In this section we study the system (17)–(21) in the horizontal channel $\{(x,z):0\leq z\leq H,x\in\mathbb{R}\}$. We complement this system with the boundary conditions (22) and the initial condition

[TABLE]

In particular, without loss of generality, we choose $H=\frac{1}{2}$. Instead of considering this physical problem, in this section, we consider another problem, that is, system (17)–(21) in the unit two dimensional torus $\mathbb{T}^{2}$, subject to the following symmetric boundary conditions and initial conditions:

[TABLE]

The periodicity and symmetry are valid due to the fact that the periodic subspace $\mathcal{H}$, give by

[TABLE]

[TABLE]

is invariant under the evolution system (17)–(21). After solving this problem in the flat torus, the solution restricted on original horizontal channel $\{(x,z):0\leq z\leq\frac{1}{2},x\in\mathbb{R}\}$ will solve the original physical problem with corresponding boundary conditions (22) and initial conditions (31).

3.1. Reformulation of The Problem.

First, let us reformulate the system (17)–(21) by deriving equations for $w,p_{x}$ and $p_{z}$ in terms of $u,v$ and $T$. For the sake of simplicity, we drop the argument $t$ in functions when there is no confusion. First, from (20) and by boundary condition (33), i.e., $w(x,0)=0$, we have

[TABLE]

From (19) and (35), we have

[TABLE]

Next, we will derive equation for $p_{x}$. Notice that since $w(x,0)=w(x,1)=0$, from (35), one has the compatibility condition

[TABLE]

Let us denote by $c(t):=\int_{0}^{1}u(x,z,t)dz$ and $d(x,t):=\int_{0}^{1}v(x,z,t)dz$. Integrating (17) with respect to $z$ over $(0,1)$, using boundary condition (32) and (33), one has:

[TABLE]

By integration by parts and using (20), (32) and (33), we get

[TABLE]

Integrating (38) with respect to $x$ over $(0,1)$, using compatibility condition (37), we have

[TABLE]

Thanks to (32), we have

[TABLE]

Plugging (39) back into (38) yields

[TABLE]

Next, from (35) and (36), we have

[TABLE]

where $p_{s}(x)=p(x,0)$ is the pressure at $z=0$. By differentiating (41) with respect to $x$, and integrating respect to $z$ over $(0,1)$, by virtue of (40), we have

[TABLE]

Therefore, by differentiating (41) with respect to $x$, and using (42), we have

[TABLE]

By virtue of (35), (36) and (43), and since $p$ is determined up to a constant, the unknowns for system (17)–(21) are only $(u,v,T)$. Therefore, we reformulate system (17)–(21) to the following system:

[TABLE]

with $w,p_{x},p_{z}$ defined by:

[TABLE]

In this section, we are interested in system (44)–(46) with (47)–(49) in the unit two dimensional torus $\Omega=\mathbb{T}^{2}$, subject to the following symmetry boundary conditions and initial conditions:

[TABLE]

It’s worth mentioning again that our system (44)–(46) with (47)–(49) satisfies the compatibility condition (37). By virtue of (47)–(49) and (50), (51), we obtain that $w,p$ also satisfy the symmetry conditions:

[TABLE]

From (47) and (49), and by differentiating (47) with respect to $z$, we have

[TABLE]

Therefore, we conclude system (44)–(46) with (47)–(49) and subjects to (50)–(52) is equivalent to original system (17)–(21) subjects to (32)–(34).

3.2. Local Well-posedness

In this section, we will show the local regularity of strong solutions to the system (44)–(46) with (47)–(49), subjects to boundary and initial conditions (50)–(52). First, we give the definition of strong solution to system (44)–(46) with (47)–(49).

Definition 4**.**

*Suppose that $u_{0},v_{0},T_{0},\partial_{x}u_{0},\partial_{x}v_{0},\partial_{x}T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (50) and (51), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Given time $\mathcal{T}>0$, we say $(u,v,T)$ is a strong solution to system (44)–(46) with (47)–(49), subjects to (50)–(52), on the time interval $[0,\mathcal{T}]$, if

*(i) u, v and T satisfy the symmetry conditions (50) and (51);

(ii) u, v and T have the regularities

[TABLE]

(iii) u, v and T satisfy system (44)–(46) in the following sense:

[TABLE]

with $w,p_{x},p_{z}$ defined by (47)–(49), and fulfill the initial condition (52).

We have the following result concerning the existence and uniqueness of strong solutions to system (44)–(46) with (47)–(49), subjects to (50)–(52), on $\mathbb{T}^{2}\times(0,\mathcal{T})$, for some positive time $\mathcal{T}$.

Theorem 5**.**

Suppose that $u_{0},v_{0},T_{0},\partial_{x}u_{0},\partial_{x}v_{0},\partial_{x}T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (50) and (51), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Then there exists some time $\mathcal{T}>0$ such that there exists a unique strong solution $(u,v,T)$ of system (44)–(46) with (47)–(49), subjects to (50)–(52), on the interval $[0,\mathcal{T}]$. Moreover, the unique strong solution $(u,v,T)$ depends continuously on the initial data.

In section 3.2.1, we establish the existence of solutions to system (44)–(46) with (47)–(49) by employing the standard Galerkin approximation procedure. In section 3.2.2, we establish formal a priori estimates for the solutions of system (44)–(46) with (47)–(49). These estimates can be justified rigorously by deriving them first to the Galerkin approximation system and then passing to the limit using the Aubin-Lions compactness theorem. In section 3.2.3, we establish the uniqueness of strong solutions, and its continuous dependence on the initial data.

3.2.1. Galerkin approximating system.

In this section, we employ the standard Galerkin approximation procedure. Let

[TABLE]

[TABLE]

and

[TABLE]

Observe that functions in $\mathcal{E}$ and $\mathcal{O}$ are even and odd with respect to $z$ variable, respectively. Moreover, $\mathcal{E}$ and $\mathcal{O}$ are closed subspace of $L^{2}(\mathbb{T}^{2})$, orthogonal to each other and consist of real valued functions. For any $m\in{\mathbb{N}}$, denote by

[TABLE]

the finite-dimensional subspaces of $\mathcal{E}$ and $\mathcal{O}$, respectively. For any function $f\in L^{2}(\mathbb{T}^{2})$, we denote by $\hat{f_{j}}=(f,\bar{\phi_{j}})$ and $\tilde{f_{j}}=(f,\bar{\psi_{j}})$, and we write $P_{m}f:=\sum_{|k|\leq m}\hat{f_{k}}\phi_{k}$ and $\Pi_{m}f:=\sum_{|k|\leq m}\tilde{f_{k}}\psi_{k}$. Then $P_{m}$ and $\Pi_{m}$ are the orthogonal projections from $L^{2}(\mathbb{T}^{2})$ to $\mathcal{E}_{m}$ and $\mathcal{O}_{m}$, respectively. Now let

[TABLE]

and consider the following Galerkin approximation system for our model (44)–(46), with (47)–(49):

[TABLE]

with $w_{m},\partial_{x}p_{m},\partial_{z}p_{m}$ defined by:

[TABLE]

subjects to the following initial conditions:

[TABLE]

Observe that the definitions of $w_{m},\partial_{x}p_{m}$ and $\partial_{z}p_{m}$ are inspired by (47)–(49). Moreover, notice that $(\partial_{x}p_{m})_{z}(x,z)=-\partial_{x}T_{m}(x,z)+\epsilon_{2}\int_{0}^{z}\partial^{2}_{xx}u_{m}(x,s)ds=(\partial_{z}p_{m})_{x}(x,z)$, and hence (63) and (64) are compatible. For each $m\geq 1$, the Galerkin approximation, system (59)–(61), together with (62)–(64) corresponds to a first order system of ordinary differential equations, in the coefficients $a_{k},b_{k}$ and $c_{k}$ for $0\leq|k|\leq m$, with quadratic nonlinearity. Therefore, by the theory of ordinary differential equations, there exists some $t_{m}>0$ such that system (59)–(61) together with (62)–(64) admit a unique solution $(u_{m},v_{m},T_{m})$ on the interval $[0,t_{m}]$. Observe that from (65), we have $a_{k}(0),b_{k}(0),c_{k}(0)\in\mathbb{C}$ satisfying $a_{-k_{1},k_{2}}(0)=a_{k_{1},k_{2}}^{*}(0),b_{-k_{1},k_{2}}(0)=b_{k_{1},k_{2}}^{*}(0)$, and $c_{-k_{1},k_{2}}(0)=c_{k_{1},k_{2}}^{*}(0)$. Thanks to the uniqueness of the solutions of the ODE system, we conclude that $a_{-k_{1},k_{2}}(t)=a_{k_{1},k_{2}}^{*}(t),b_{-k_{1},k_{2}}(t)=b_{k_{1},k_{2}}^{*}(t)$, and $c_{-k_{1},k_{2}}(t)=c_{k_{1},k_{2}}^{*}(t)$, for $t\in[0,t_{m}]$. Therefore, $u_{m},v_{m}\in\mathcal{E}_{m}$, and $T_{m}\in\mathcal{O}_{m}$. In the next section, we perform formal a priori estimates for the original system (44)–(46) with (47)–(49). These formal a priori estimates can be justified rigorously by establishing them first to the Galerkin approximation system and then passing to the limit using the Aubin-Lions compactness theorem.

3.2.2. A priori estimates

The constant $C$ appears in the following inequalities may change from line to line. By taking the $L^{2}$-inner product of equation (44) with $u,-\Delta u,\Delta u_{xx}$, equation (45) with $v,-\Delta v,\Delta v_{xx}$, equation (55) with $w,-\Delta w,\Delta w_{xx}$ and equation (46) with $T,-\Delta T,\Delta T_{xx}$, and by integration by parts, thanks to (50) and (53), we have

[TABLE]

By integration by parts, thanks to (50), (53) and (56), we have

[TABLE]

Therefore, the right-hand side of (66) becomes

[TABLE]

Let us denote by

[TABLE]

From (35), by Hölder inequality and Minkowski inequality, we have

[TABLE]

Similarly, one can get

[TABLE]

Let us estimate terms $I_{1}$–$I_{8}$. By integration by parts, using Cauchy–Schwarz inequality, Young’s inequality and Lemma 1, thanks to (50), (53), (56), (68) and (69), we have

[TABLE]

[TABLE]

From the estimates above, (66) becomes

[TABLE]

Therefore, we have $\frac{dY}{dt}\leq CY^{3}$, and this implies that

[TABLE]

Choose

[TABLE]

From above, we have $Y(t)\leq 2Y(0)$ on $[0,\mathcal{T}]$. Plugging it in (70), we have

[TABLE]

Integrating above from 0 to $t$ for any time $t\in[0,\mathcal{T}]$, we obtain

[TABLE]

Therefore, we have

[TABLE]

By virtue of (72) and (68), we have

[TABLE]

Thanks to Lemma 2, from (72), we also have

[TABLE]

3.2.3. Uniqueness of solutions and its continuous dependence on the initial data

In this section, we will show the continuous dependence on the initial data and the uniqueness of the strong solutions. Let $(u_{1},v_{1},w_{1},p_{1},T_{1})$ and $(u_{2},v_{2},w_{2},p_{2},T_{2})$ be two strong solutions of system (44)–(46) with (47)–(49), and initial data $((u_{0})_{1},(v_{0})_{1},(T_{0})_{1})$ and $((u_{0})_{2},(v_{0})_{2},(T_{0})_{2})$, respectively. Denote by $u=u_{1}-u_{2}$, $v=v_{1}-v_{2}$, $w=w_{1}-w_{2}$, $p=p_{1}-p_{2}$, $T=T_{1}-T_{2}.$ It is clear that

[TABLE]

By taking the inner product of equation (75) with $u$, (76) with $v$, (77) with $w$, and (79) with $T$, in $L^{2}(\mathbb{T}^{2})$, and by integration by parts, thanks to (50), (56), and (78), we get

[TABLE]

By integration by parts, using Hölder inequality and Young’s inequality, thanks to (50), (56), and (78), we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From the estimates above, we obtain

[TABLE]

where

[TABLE]

Thanks to (74), we obtain $K\in L^{1}(0,\mathcal{T})$. Therefore, by Gronwall inequality, we obtain

[TABLE]

The above inequality proves the continuous dependence of the solutions on the initial data, and in particular, when $u(t=0)=v(t=0)=T(t=0)=0$, we have $u(t)=v(t)=T(t)=0,$ for all $t\geq 0$. Therefore, the strong solution is unique.

3.3. The Special Case: $\mathbf{f_{0}=0,v\equiv 0}$ and $\mathbf{T\equiv 0}$

In this section, we assume that $f_{0}=0$, $v\equiv 0$ and $T\equiv 0$. In this case, system (17)–(21) will be reduced to:

[TABLE]

Remark 1*.*

There are two reasons why we consider this special case. Firstly, notice that when $\epsilon_{1}=\epsilon_{2}=0$, system (80)–(82) is exactly the hydrostatic Navier-Stokes equations (14)–(16). So we can regard system (80)–(82) as the hydrostatic Navier-Stokes equations with damping. Secondly, as we will see later, we can show the local regularity of strong solution to system (80)–(82) for initial conditions with less regularity. The reason why we need to assume more regularity for initial data to system (44)–(46) is that we need to bound terms which contain $v_{xx}$. For $u_{xx}$, we can use incompressible condition $u_{xx}=-w_{xz}$ to avoid such an issue. Therefore, in the case when we do not have the evolution equation for $v$, we can require less for the initial data.

As in section 3.2, our domain is $\mathbb{T}^{2}$, and the boundary and initial condition are

[TABLE]

Using an analogue argument to that in section 3.1, system (80)–(82) subjects to (83)–(85) is equivalent to the following:

[TABLE]

with $w,p_{x},p_{z}$ defined by:

[TABLE]

subject to the following symmetry boundary condition and initial condition:

[TABLE]

By virtue of (87)–(89) and (90), we obtain that $w,p$ also satisfy the symmetry conditions:

[TABLE]

By virtue of (87) and (89), and by differentiating (87) with respect to $z$, we have

[TABLE]

In this section, we are interested in system (86) with (87)–(89) in the unit two-dimensional flat torus $\mathbb{T}^{2}$, subjects to (90)–(91). First, we give the definition of strong solution to system (86) with (87)–(89).

Definition 6**.**

*Suppose that $u_{0}\in H^{1}(\mathbb{T}^{2})$ satisfies the symmetry conditions (90), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that $\partial^{2}_{xz}u_{0}\in L^{2}(\mathbb{T}^{2})$. Given time $\mathcal{T}>0$, we say u is a strong solution to system (86) with (87)–(89), subjects to (90)–(91), on the time interval $[0,\mathcal{T}]$, if

*(i) u satisfies the symmetry condition (90);

(ii) u has the regularities

[TABLE]

(iii) u satisfies system (86) in the following sense:

[TABLE]

with $w,p_{x},p_{z}$ defined by (87)–(89), and fulfill the initial condition (91).

We have the following result concerning the locally existence and uniqueness of strong solutions to system (86) with (87)–(89), subjects to (90)–(91), on $\mathbb{T}^{2}\times(0,\mathcal{T})$, for some positive time $\mathcal{T}$.

Theorem 7**.**

Suppose that $u_{0}\in H^{1}(\mathbb{T}^{2})$ satisfies the symmetry conditions (90), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that $\partial^{2}_{xz}u_{0}\in L^{2}(\mathbb{T}^{2})$. Then there exists some $\mathcal{T}>0$ such that there is a unique strong solution u of system (86) with (87)–(89), subjects to (90)–(91), on the interval $[0,\mathcal{T}]$. Moreover, the unique strong solution u depends continuously on the initial data.

Proof.

For sake of simplicity, we will only do* a priori* estimates formally here, and we can use Galerkin method, as remarked in section 3.2, to prove the result rigorously. We denote by $\|\cdot\|:=\|\cdot\|_{L^{2}(\mathbb{T}^{2})}$. By taking the inner product of equation (86) with $u$, $-u_{zz}$, and equation (94) with $w$, $-w_{zz}$, in $L^{2}(\mathbb{T}^{2})$, we get

[TABLE]

By integration by parts, thanks to (90), (92) and (95), we have

[TABLE]

Thanks to Gronwall inequality, we obtain

[TABLE]

From the estimates above, we obtain

[TABLE]

for arbitrary $\mathcal{T}>0$. By taking the inner product of equation (86) with $-u_{xx},u_{xxzz}$ and equation (94) with $-w_{xx}$, $w_{xxzz}$ in $L^{2}(\mathbb{T}^{2})$, integrating by parts, thanks to (90) and (92) we get

[TABLE]

By integration by parts, thanks to (90), (92) and (95), we have

[TABLE]

Therefore, we have

[TABLE]

Let us denote by

[TABLE]

[TABLE]

By integration by parts and Lemma 1, using Young’s inequality, thanks to (68), (69), (90), (92) and (95), we have

[TABLE]

From the estimates above and by (96), we have

[TABLE]

Therefore, we have $\frac{dY}{dt}\leq CKY^{2}$, and this implies that

[TABLE]

Let $\mathcal{T}$ be such that

[TABLE]

From above, we will have $Y(t)\leq 2Y(0)$ on $[0,\mathcal{T}]$. Plugging it in (98), we have

[TABLE]

Integrating above from 0 to t for any time $t\in[0,\mathcal{T}]$, we obtain

[TABLE]

From the estimates above, by virtue of (95), (96) and (68), we obtain

[TABLE]

Using Galerkin method, as remarked in section 3.2, we can obtain local existence of strong solution to system (86) with (87)–(89), subjects to (90)–(91). Next, we show the continuous dependence of solutions on the initial data and the uniqueness of the strong solutions. Let $(u_{1},w_{1},p_{1})$ and $(u_{2},w_{2},p_{2})$ be two strong solutions of system (86) with (87)–(89), and initial data $(u_{0})_{1}$ and $(u_{0})_{2}$, respectively. Denote by $u=u_{1}-u_{2},w=w_{1}-w_{2},p=p_{1}-p_{2}.$ It is clear that

[TABLE]

By taking the inner product of equation (101) with $u$, (102) with $w$ in $L^{2}(\mathbb{T}^{2})$, we have

[TABLE]

By integration by parts, thanks to (90), (92) and (95), we have

[TABLE]

Therefore, we have

[TABLE]

From (99) and (100), and by lemma 2, we obtain that $w_{2},(u_{2})_{z}\in L^{2}(0,\mathcal{T};L^{\infty})$. Therefore, using Young’s inequality and Hölder inequality, we have

[TABLE]

[TABLE]

From the estimates above, we obtain

[TABLE]

Thanks to Gronwall inequality, we have

[TABLE]

The above inequality proves the continuous dependence of the solutions on the initial data, and in particular, when $u(t=0)=0$, we have $u(t)=0,$ for all $t\in[0,\mathcal{T}]$. Therefore, the strong solution is unique. ∎

3.4. Global Well-posedness with Small Initial Data

In this section, we will show the following result concerning the global existence and uniqueness of strong solutions to system (44)–(46) with (47)–(49), subjects to boundary and initial conditions (50)–(52), provided that the initial data is small enough.

Theorem 8**.**

Suppose that $u_{0},v_{0},T_{0},\partial_{x}u_{0},\partial_{x}v_{0},\partial_{x}T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (50) and (51), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that

[TABLE]

is small enough, for some $C_{0}>0$ determined in (105). Then for any time $\mathcal{T}>0$, there exists a unique strong solution $(u,v,T)$ of system (44)–(46) with (47)–(49), subjects to (50)–(52), on the interval $[0,\mathcal{T}]$. Moreover, the unique strong solution $(u,v,T)$ depends continuously on the initial data.

Proof.

From Theorem 5, we know there exists time $\mathcal{T}^{*}>0$ such that there is a unique strong solution $(u,v,T)$ of system (44)–(46) with (47)–(49), subjects to (50)–(52), on the interval $[0,\mathcal{T}^{*}]$. Assume the maximal time $\mathcal{T}$ for existence of solution is finite, then it is necessary to have

[TABLE]

We will prove this is not true for any finite time $\mathcal{T}>0$, and therefore $\mathcal{T}=\infty$. First, notice that since $T$ is an odd function with respect to $z$ variable, we have

[TABLE]

By taking the $L^{2}$-inner product of equation (44) with $u,-\Delta u,\Delta u_{xx}$, equation (45) with $v,-\Delta v,\Delta v_{xx}$, equation (55) with $w,-\Delta w,\Delta w_{xx}$ and equation (46) with $C_{0}T,-C_{0}\Delta T,C_{0}\Delta T_{xx}$, in $L^{2}(\mathbb{T}^{2})$, by integration by parts, thanks to (50), (53) and (56), we have

[TABLE]

We denote by:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By integration by parts, using Poincaré inequality, Young’s inequality and Lemma 1, thanks to (50), (53), (56), (68), (69) and (103), we have

[TABLE]

[TABLE]

where, thanks to (103), we apply Poincaré inequality to obtain the last inequality.

[TABLE]

From the estimates above, we obtain

[TABLE]

Therefore, we obtain

[TABLE]

Choose

[TABLE]

Observe that if $Y_{0}<\min(\frac{\nu^{2}}{C^{2}},\frac{\epsilon_{1}^{2}}{C^{2}},\frac{\epsilon_{2}^{2}}{4C^{2}},\frac{\kappa^{2}}{4C^{2}})$, there exists $t^{*}>0$ such that $\frac{dY}{dt}\leq 0$ on $[0,t^{*}]$, and hence $Y(t)\leq Y_{0}$ for $t\in[0,t^{*}]$, and in particular, $Y(t^{*})<\min(\frac{\nu^{2}}{C^{2}},\frac{\epsilon_{1}^{2}}{C^{2}},\frac{\epsilon_{2}^{2}}{4C^{2}},\frac{\kappa^{2}}{4C^{2}})$. Thus we can repeat this procedure to arbitrary time $t>0$ to get $Y(t)\leq Y_{0}<\min(\frac{\nu^{2}}{C^{2}},\frac{\epsilon_{1}^{2}}{C^{2}},\frac{\epsilon_{2}^{2}}{4C^{2}},\frac{\kappa^{2}}{4C^{2}})$ for all time. This implies the required bound for the global in time existence of strong solution. ∎

4. Global Well-posedness of system (23)–(27)

In this section we study system (23)–(27) with $\nu=0$. Similar as in section 3.1, our domain is $\mathbb{T}^{2}$, and the boundary conditions and initial condition are

[TABLE]

Using analogue argument as in section 3.1, system (23)–(27) subjects to (106)–(108) is equivalent to the following:

[TABLE]

with $w,p_{x},p_{z}$ defined by (47)–(49), subject to the symmetry boundary conditions and initial conditions (50)–(52). We also have (55) and (56), for which we repeat here:

[TABLE]

In this section, we are interested in system (109)–(111) with (47)–(49), subjects to (50)–(52). We will show the global regularity of strong solution to system (109)–(111) with (47)–(49), subject to (50)–(52). First, we give the definition of strong solution to system (109)–(111) with (47)–(49), subject to (50)–(52).

Definition 9**.**

*Suppose that $u_{0},v_{0}\in H^{2}(\mathbb{T}^{2})$ and $T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (50) and (51), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that $\partial^{3}_{xxz}u_{0},\partial^{3}_{xxz}v_{0}\in L^{2}(\mathbb{T}^{2})$. Given time $\mathcal{T}>0$, we say $(u,v,T)$ is a strong solution to the system (109)–(111) with (47)–(49), subjects to (50)–(52), on the time interval $[0,\mathcal{T}]$, if

*(i) u, v and T satisfy the symmetry conditions (50) and (51);

(ii) u, v and T have the regularities

[TABLE]

(iii) u, v and T satisfy system (44)–(46) in the following sense:

[TABLE]

with $w,p_{x},p_{z}$ defined by (47)–(49), and fulfill the initial condition (52).

We have the following result concerning the existence and uniqueness of strong solutions to system (109)–(111) with (47)–(49), subjects to (50)–(52), on $\mathbb{T}^{2}\times(0,\mathcal{T})$, for any positive time $\mathcal{T}$.

Theorem 10**.**

Suppose that $u_{0},v_{0}\in H^{2}(\mathbb{T}^{2})$ and $T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (50) and (51), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that $\partial^{3}_{xxz}u_{0},\partial^{3}_{xxz}v_{0}\in L^{2}(\mathbb{T}^{2})$. Let time $\mathcal{T}>0$. Then there exists a unique strong solution $(u,v,T)$ of system (109)–(111) with (47)–(49), subjects to (50)–(52), on the interval $[0,\mathcal{T}]$. Moreover, the unique strong solution $(u,v,T)$ depends continuously on the initial data.

Proof.

We will only do* a priori* estimates formally here, and we can use Galerkin method as in section 3.2 to prove the result rigorously. By taking the $L^{2}$-inner product of equation (109) with $u$, equation (110) with $v$, equation (112) with $w$ and equation (111) with $T$, in $L^{2}(\mathbb{T}^{2})$, by integration by parts, thanks to (50), we get

[TABLE]

By integration by parts, thanks to (50), (53) and (113), we have

[TABLE]

By Cauchy–Schwarz inequality and Young’s inequality, we have

[TABLE]

As a result of the above, we have

[TABLE]

Thanks to Gronwall inequality, we obtain

[TABLE]

Consequently, we have

[TABLE]

for arbitrary $\mathcal{T}>0$. By taking the $L^{2}$-inner product of equation (109) with $-u_{zz}$ and equation (112) with $-w_{zz}$ in $L^{2}(\mathbb{T}^{2})$, by integration by parts, thanks to (50) and (53), we get

[TABLE]

By integration by parts, thanks to (50), (53) and (113), we have

[TABLE]

By integration by parts, using Cauchy–Schwarz inequality and Young’s inequality, thanks to (50) and (53), we have

[TABLE]

As a result of the above, we have

[TABLE]

Thanks to Gronwall inequality, we obtain

[TABLE]

By virtue of (114) and the above, we have

[TABLE]

for arbitrary $\mathcal{T}>0$. By taking the $L^{2}$-inner product of equation (109) with $-u_{xx}$ and equation (112) with $-w_{xx}$, in $L^{2}(\mathbb{T}^{2})$, by integration by parts, thanks to (50) and (53), we get

[TABLE]

By integration by parts, thanks to (50), (53) and (113), we have

[TABLE]

By integration by parts, using Cauchy–Schwarz inequality and Young’s inequality, thanks to (50), (53) and (113), we have

[TABLE]

By integration by parts, using Young’s inequality and Lemma 1, thanks to (50), (53) and (113), we have

[TABLE]

From the estimates above, we have

[TABLE]

By Gronwall inequality, we obtain

[TABLE]

By virtue of (114), (115) and the above, we have

[TABLE]

for arbitrary $\mathcal{T}>0$. By virtue of (116), (68) and (69), we have

[TABLE]

for arbitrary $\mathcal{T}>0$. By taking the $L^{2}$-inner product of equation (110) with $-\Delta v$ in $L^{2}(\mathbb{T}^{2})$, and by integration by parts, thanks to (50), we have

[TABLE]

By integration by parts, using Cauchy–Schwarz inequality and Lemma 1, thanks to (50), (53) and (113), we have

[TABLE]

As a result of the above and by Young’s inequality, we conclude

[TABLE]

Thanks to Gronwall inequality, we obtain

[TABLE]

By virtue of (114), (115), (116) and the above, we have

[TABLE]

for arbitrary $\mathcal{T}>0$. By taking the $L^{2}$-inner product of equation (111) with $-\Delta T$ in $L^{2}(\mathbb{T}^{2})$, and by integration by parts, thanks to (50), we have

[TABLE]

By Lemma 1 and Young’s inequality, thanks to (113), we have

[TABLE]

As a result of the above we conclude

[TABLE]

Thanks to Gronwall inequality, we obtain

[TABLE]

By virtue of (114), (115), (116), (117) and the above, we have

[TABLE]

By taking the $L^{2}$-inner product of equation (109) with $u_{xxxx}$, equation (110) with $v_{xxxx}$, and equation (112) with $w_{xxxx}$ in $L^{2}(\mathbb{T}^{2})$, and by integration by parts, thanks to (50) and (53), we get

[TABLE]

By integration by parts, thanks to (50), (53) and (113), , we have

[TABLE]

By integration by parts, using Young’s inequality and Lemma 1, thanks to (50), (53) and (113), we have

[TABLE]

and

[TABLE]

By integration by parts, using Cauchy-Schwarz inequality and Young’s inequality, thanks to (50) and (53), we have

[TABLE]

As a result of the above, we conclude

[TABLE]

By Gronwall inequality, we obtain

[TABLE]

From (114), (115), (116), (117), (118), (119) and the above, we have

[TABLE]

for arbitrary $\mathcal{T}>0$. By virtue of (120), thanks (68) and (69), we have

[TABLE]

for arbitrary $\mathcal{T}>0$. Using standard Galerkin method as in section 3.2, we can establish the existence result, and we omit the details here. Next, we show the continuous dependence on the initial data and the the uniqueness of the strong solutions. Let $(u_{1},v_{1},w_{1},p_{1},T_{1})$ and $(u_{2},v_{2},w_{2},p_{2},T_{2})$ be two strong solutions of the system (109)–(111) with (47)–(49), and initial data $((u_{0})_{1},(v_{0})_{1},(T_{0})_{1})$ and $((u_{0})_{2},(v_{0})_{2},(T_{0})_{2})$, respectively. Denote by $u=u_{1}-u_{2},v=v_{1}-v_{2},w=w_{1}-w_{2},p=p_{1}-p_{2},T=T_{1}-T_{2}.$ Thanks to (112) and (113), it is clear that

[TABLE]

By taking the inner product of equation (122) with $u$, (123) with $v$, (124) with $w$, and (126) with $T$ in $L^{2}(\mathbb{T}^{2})$, and by integration by parts, thanks to (50), (113) and (125), we get

[TABLE]

By integration by parts, using Hölder inequality and Young’s inequality, thanks to (50), (113) and (125) and Lemma 1,

[TABLE]

From the estimates above, we obtain

[TABLE]

where

[TABLE]

Using lemma 2, thanks to (119) and (120), we obtain $K\in L^{1}(0,\mathcal{T})$. Therefore, by Gronwall inequality, we obtain

[TABLE]

The above inequality proves the continuous dependence of the solutions on the initial data, and in particular, when $u(t=0)=v(t=0)=T(t=0)=0$, we have $u(t)=v(t)=T(t)=0,$ for all $t\geq 0$. Therefore, the strong solution is unique. ∎

In the case when $f_{0}=0$ and $v\equiv 0$, our system (23)–(27) will be reduced to the following system:

[TABLE]

in $\mathbb{T}^{2}$. We impose similar boundary and initial conditions for this system:

[TABLE]

Using analogue argument as in section 3.1, the system (129)–(132) subjects to (133)–(135) is equivalent to the following:

[TABLE]

with $w,p_{x},p_{z}$ defined by:

[TABLE]

We are interested in the system (136)–(137) with (138)–(140) in the unit two dimensional torus $\mathbb{T}^{2}$, subject to the following symmetry boundary conditions and initial conditions:

[TABLE]

We have the global well-posedness for system (136)–(137) with (138)–(140), for initial condition with less regularity. i.e., for $u_{0},\partial_{z}u_{0}\in H^{1}$ and $T_{0}\in{H^{1}}$. Let us give the definition of strong solution first.

Definition 11**.**

*Suppose that $u_{0}\in H^{1}(\mathbb{T}^{2})$ and $T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (141) and (142), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that $\partial_{z}u_{0}\in H^{1}$. Given time $\mathcal{T}>0$, we say $(u,T)$ is a strong solution to the system (136)–(137) with (138)–(140), subjecto to (141)–(143), on the time interval $[0,\mathcal{T}]$, if

*(i) u and T satisfy the symmetry conditions (141) and (142);

(ii) u and T have the regularities

[TABLE]

(iii) u, T satisfy system (136)–(137) in the following sense:

[TABLE]

with $w,p_{x},p_{z}$ defined by (138)–(140), and fulfill the initial condition (143).

Based on theorem 10, we have the following theorem on the existence and uniqueness of strong solutions to system (136)–(137) with (138)–(140), subject to (141)–(143), on $\mathbb{T}^{2}\times(0,\mathcal{T})$, for any positive time $\mathcal{T}$. The proof is similar as theorem 10, and we omit the details here.

Theorem 12**.**

Suppose that $u_{0}\in H^{1}(\mathbb{T}^{2})$ and $T_{0}\in H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (141) and (142), with the compatibility condition $\int_{0}^{1}\partial_{x}u_{0}dz=0$. Moreover, suppose that $\partial_{z}u_{0}\in H^{1}$. Given time $\mathcal{T}>0$. Then there exists a unique strong solution $(u,T)$ of the system (136)–(137) with (138)–(140), subject to (141)–(143), on the interval $[0,\mathcal{T}]$. Moreover, the unique strong solution $(u,T)$ depends continuously on the initial data. Same result holds when $T\equiv 0$.

Remark 2*.*

The reason why we need to assume more regularity for the initial data to system (109)–(111) is that we need a bound for $\|(v_{2})_{x}\|_{L^{\infty}}$ appears in (127). If we do not have the evolution equation in $v$, we can require less for the initial data.

5. Convergence

In this section, we will prove the convergence of the strong solution of the following system

[TABLE]

subjects to the following symmetric boundary conditions and initial condition

[TABLE]

to the strong solution of system (80)–(82) subjects to (83)–(85), as $\alpha\rightarrow 0$.

Remark 3*.*

The global well-posedness of system (144)–(146) subjects to (147)–(149) can be easily obtained as in section 4. Moreover, as indicated in the last part of section 4, we only need to assume that $u^{\alpha}_{0}$ and $u^{\alpha}_{0z}\in H^{1}(\mathbb{T}^{2})$ since we do not have the evolution equation in $v^{\alpha}$.

Theorem 13**.**

Suppose that $u_{0},\{u_{0}^{\alpha}\}_{0<\alpha\leq 1}\subset H^{1}(\mathbb{T}^{2})$ satisfy the symmetry conditions (83)–(84) and (147)–(148), with the compatibility conditions $\int_{0}^{1}\partial_{x}u_{0}dz=0$ and $\int_{0}^{1}\partial_{x}u_{0}^{\alpha}dz=0$, for $\forall\;0<\alpha\leq 1$, and suppose that $\partial^{2}_{xz}u_{0}\in L^{2}(\mathbb{T}^{2}),\{\partial_{z}u_{0}^{\alpha}\}_{0<\alpha\leq 1}\subset H^{1}(\mathbb{T}^{2})$. Moreover, suppose there exists some constant $M>0$ such that the following uniform bound for initial data holds:

[TABLE]

Let $\mathcal{T}>0$ be such that $u$ is the strong solution of system (80)–(82) on $[0,\mathcal{T}]$ with initial data $u_{0}$. Let $u^{\alpha}$ be the strong solution to system (144)–(146) on $[0,\mathcal{T}]$ with initial data $u_{0}^{\alpha}$. If $u_{0}^{\alpha}\rightarrow u_{0}$ in $L^{2}$, as $\alpha\rightarrow 0$, then $u^{\alpha}\rightarrow u$ in $L^{\infty}(0,\mathcal{T};L^{2})$, and $u_{z}^{\alpha}\rightarrow u_{z}$ in $L^{2}(0,\mathcal{T};L^{2})$, as $\alpha\rightarrow 0$.

Proof.

Let us first derive the uniform bounds of some norms of the strong solution $u^{\alpha}$. By taking the $L^{2}$-inner product of equation (144) with $u^{\alpha},-u^{\alpha}_{zz}$, and equation (145) with $w^{\alpha},-w^{\alpha}_{zz}$, in $L^{2}(\mathbb{T}^{2})$, and by integration by parts, thanks to (147), we get

[TABLE]

By integration by parts, thanks to (146) and (147), we have

[TABLE]

As a result of the above, we have

[TABLE]

Thanks to Gronwall inequality, we obtain

[TABLE]

for $t\in[0,\mathcal{T}]$. Thanks to the uniform bound for initial data (150), we have

[TABLE]

where $C(M)$ is a constant depending on $M$, but not on $\alpha$. Now subtracting (80)–(81) from (144)–(145), we obtain

[TABLE]

For simplicity, we denote $\|\cdot\|:=\|\cdot\|_{L^{2}}$ from now on. By taking the inner product of equation (152) with $u^{\alpha}-u$ and equation (153) with $w^{\alpha}-w$, by integration by parts, and using (82) and (146), we get

[TABLE]

By integration by parts, using Hölder inequality and Young’s inequality, thanks to (82), (83), (146) and (147), we have

[TABLE]

From all the estimates above, we obtain

[TABLE]

Let us denote by

[TABLE]

Therefore, we obtain

[TABLE]

Notice that the constant C appears above may change from line to line, and may depend on $\nu$, $\epsilon_{2}$, and $\mathbb{T}^{2}$, but not on $\alpha$. Thanks to Gronwall inequality, we obtain

[TABLE]

By virture of the regularity of strong solution to system (80)–(82) as stated in Definition 6, and the uniform bound (151), using Lemma 2, we have $F,G\in L^{1}(0,\mathcal{T})$. By virtue of uniform bound (151), we have $\alpha^{2}H(t)\rightarrow 0$, as $\alpha\rightarrow 0$. Since $u_{0}^{\alpha}\rightarrow u_{0}$ in $L^{2}$, and thanks to (150), we have $u^{\alpha}\rightarrow u$ in $L^{\infty}(0,\mathcal{T};L^{2})$, $u^{\alpha}_{z}\rightarrow u_{z}$ in $L^{2}(0,\mathcal{T};L^{2})$, and $w^{\alpha}\rightarrow w$ in $L^{2}(0,\mathcal{T};L^{2})$, as $\alpha\rightarrow 0$.

∎

6. Blow-up Criterion

In this section we give a blow-up criterion for system (80)–(82) subjects to (83)–(85). The following result follows the idea in [42].

Theorem 14**.**

With the same assumptions in Theorem 13, and take $u_{0}^{\alpha}=u_{0}$ for all $\alpha$. Suppose there exists some time $\mathcal{T}^{*}<\infty$ such that

[TABLE]

then the solution for system (80)–(82) blows up on $[0,\mathcal{T}^{*}]$.

Proof.

Assume the solution for system (80)–(82) will not blow up on $[0,\mathcal{T}^{*}]$, then $u\in L^{\infty}(0,\mathcal{T}^{*};H^{1})$ and $\partial_{t}u\in L^{2}(0,\mathcal{T}^{*};L^{2})$. By taking the inner product of equation (80) with u and equation (81) with w in $L^{2}(\mathbb{T}^{2})$, by integration by parts and thanks to (82) and (83), we have

[TABLE]

Integrating (156) from 0 to $t$ for $t\in[0,\mathcal{T}^{*}]$, we have

[TABLE]

On the other hand, using analogue argument for system (144)–(146), we have

[TABLE]

From (154) and thanks to the fact that $u_{0}^{\alpha}=u_{0}$, for any $t\in[0,\mathcal{T}^{*}]$, we have

[TABLE]

since $H^{1/2}(t)$ is monotonically increasing. By virtue of (157), we know $\|u_{0}\|\geq\|u(t)\|\geq\|u(\mathcal{T}^{*})\|$ for any $t\in[0,\mathcal{T}^{*}]$. Therefore, we can take $\alpha<\frac{\|u(\mathcal{T}^{*})\|}{CH^{1/2}(\mathcal{T}^{*})}$ to guarantee the right hand side of (159) is positive. Take square on (159), we obtain

[TABLE]

Subtracting (158) from (157), we have

[TABLE]

Combining (161) with (160), we obtain

[TABLE]

By Cauchy–Schwarz inequality and Hölder inequality, thanks to (99)–(100) and the uniform bound (151), we have the estimate for the last term in (162):

[TABLE]

Plugging this back into (162), we have

[TABLE]

By virtue of Theorem 13, the right hand side of (163) is independent of $t$, and it converges to 0 as $\alpha\rightarrow 0$. Therefore, by taking $\limsup\limits_{\alpha\rightarrow 0^{+}}\sup\limits_{t\in[0,\mathcal{T}^{*}]}$ on both hand sides of (163), we obtain

[TABLE]

which contradicts to (155). ∎

Remark 4*.*

By considering the convergence for the whole system, i.e., the convergence of the strong solution of system (23)–(27) to the corresponding solution of system (17)–(21), we can establish similar blow-up criterion for system (17)–(21).

Acknowledgments

The work of E.S.T. was supported in part by the Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation.