The regularized Boussinesq equations with partial dissipations in dimension two

1.   Introduction

The Boussinesq equations describe the influence of the convection (or convection-diffusion) phenomenon in a viscous or inviscid fluid, which are used as modelling many geophysical flows, such as atmospheric fronts and ocean circulations [18,22]. The standard two-dimensional Boussinesq equations read as:

where $u = (u_1(x,y,t),u_2(x,y,t))$ denotes the fluid velocity vector field, $P = P(x,y, t)$ is the scalar pressure, $\theta = \theta (x,y,t)$ is the scalar temperature, the nonnegative $\mu$ and $\kappa$ denote respectively the viscosity and the thermal diffusivity, $e_2 = (0,1)$ is the unit vector, while $u_0$ and $\theta_0$ are the given initial velocity and initial temperature respectively, with $\nabla\cdot u_0 = 0$. In order to model anisotropic flows with different diffusion properties in the horizontal and vertical directions, the system (1) can be generalized to the following form:

where $u_1$ and $u_2$ are the horizontal and vertical components of $u$. In fact, when $\mu_1 = \mu_2 = \nu_1 = \nu_2$ and $\kappa_1 = \kappa_2$, the system (2) reduces to the system (1). When all six parameters are zero, the system (2) reduces to the inviscid Boussinesq equations. Yet, whether or not the solution of the inviscid Boussinesq can develop finite time singularities remains open since the 2D inviscid Boussinesq equations are identical to the Euler equations for 3D axisymmetric swirling flows [19]. The global regularity has been obtained for the case that all four parameters are positive [5]. On the other hand, for the intermediate cases between two extreme cases (namely, the full dissipation case and the inviscid case), the Boussiesq equations (2) with partial viscosities or partial diffusivity have been attracting much more attention in the recent few years and important progress has been made (see, for example [7,8,9,13,14,15,23] and the references therein).

Most of the kinetic energy in turbulent fluid flows lies in the large scales, whereas the mathematical and computational difficulties lie in understanding the dynamical interaction between the significantly wide range of relevant scales in this multiscale phenomenon. To overcome this obstacle, much effort is being made to produce reliable turbulence models which parameterize the effect of the active small scales in terms of the large scales. Over the last years, the viscous Camassa-Holm equations have been proposed as a subgrid turbulence model (also known as the Lagrangian-averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$) model). Motivated by the remarkable performance of LANS-$\alpha$ model as a closure model of turbulence in infinite channels and pipes, whose solutions give excellent agreement with empirical data for a wide range of large Reynolds numbers, the alpha subgrid scale models of turbulence have been extensively studied in recent years [11,12,21].

An extension of the LANS-$\alpha$ model to the Boussinesq equations is given in [24], which is named as the Lagrangian averaged Boussinesq equations:

where $e_2 = (0,1)$ and $e_3 = (0,0,1)$, $\nabla\cdot v_0 = \nabla\cdot u_0 = 0$ and the constant $\alpha>0$ is a length scale parameter which represents the width of the filter. In [24], the inviscid case (namely, $\mu = \kappa = 0$) to the system (3) is considered and the regularity criterion $\nabla u\in L^1(0,T; \dot{B}_{\infty,\infty}^0(\mathbb{R}^n))$ ($n = 2,3$) is obtained. As mentioned above that the system (2) is the generalized version of(1), the 2D Lagrangian averaged Boussinesq equations in (3) can be generalized to the following form, where we call it the anisotropic Lagrangian averaged Boussinesq equations (also named as Boussinesq-$\alpha$ equations):

where $v = (v_1,v_2)$ and $u = (u_1,u_2)$ with $\nabla\cdot v_0 = \nabla\cdot u_0 = 0$. When $\mu_1 = \mu_2 = \nu_1 = \nu_2$, $\kappa_1 = \kappa_2$ and $\alpha\searrow 0^{+}$, the system (4) reduces to (1) in dimension two.

In this paper, we consider the anisotropic Lagrangian averaged Boussinesq equations with partial dissipations. Firstly, we study the Cauchy problem of the Boussinesq-$\alpha$ model with vertical dissipation in the horizontal regularized velocity equation and horizontal dissipation in the vertical regularized velocity equation, namely

We obtain the global existence of smooth solution to the above system (5), which is stated as follows.

Theorem 1.1. Consider the 2D Boussinesq-$\alpha$ system (5). Assume $(v_0,\theta_0)\in H^3(\mathbb{R}^2)\times H^3(\mathbb{R}^2)$ with $\nabla\cdot v_0 = \nabla\cdot u_0 = 0$. Then the system (5) has a unique global classical solution $(v,\theta)$ satisfying

for any given $T>0$.

Remark 1. As $\alpha\searrow 0^{+}$, the Lagrangian averaged Boussinesq equations (3) (namely, the Boussinesq-$\alpha$) reduce to the standard Boussinesq equations (1). And the anisotropic Lagrangian averaged Boussinesq equations (5) reduce to the anisotropic Boussinesq equations (2) with $\mu_1 = \nu_2 = 0$, $\mu_2>0$ and $\nu_1>0$, namely,

which is considered in [1] and the global regularity of classical solution is established. From this standpoint, our result in Theorem 1.1 can be viewed as an improvement for the regularization version of the Boussinesq equations to the anisotropic Boussinesq equations.

Remark 2. For the proof of Theorem 1.1, we adapt the approach of "weakly nonlinear" energy estimate approach introduced by Lei and Zhou in [17].

Next, let us revisit (3). When $\mu = \kappa = 0$, Zhou-Fan [24] proved the following regularity criterion of smooth solution to the inviscid Boussinesq-$\alpha$ system (3) with $\mu = 0$ and $\kappa = 0$:

Now we consider the following 2D anisotropic Boussinesq-$\alpha$ equations with only one dissipation term $\kappa_2\partial_{yy} \theta$:

We establish a Beale-Kata-Majda type regularity criterion to the system (7). More precisely, our second main result in this paper is stated as follows:

Theorem 1.2. Assume $(v_0,\theta_0)\in H^3(\mathbb{R}^2)\times H^3(\mathbb{R}^2)$ with $\nabla\cdot v_0 = \nabla\cdot u_0 = 0$ and let $(v, \theta)$ be a smooth solution to the system (7) on $[0, T_0]$ for some positive time $T_0>0$. If, for $T>T_0$, $\theta$ satisfies

then the solution can be extended to $[0,T]$.

Remark 3. Compared to (6), our result in (8) is dependent of the derivative of $x$ for $\theta$. The reason is that this comes from the contribution of $\partial_{yy}\theta$, which can help lead to the cancellation between the $\partial_y$-type terms.

Remark 4. As $\alpha\searrow 0^{+}$, the model (7) reduces to the corresponding anisotropic Boussinesq system, which was introduced in [1]. Theorem 1.2 can be seen as a generalization of their work.

2.   Preliminaries

In this section, we provide some notation and basic facts used in the proof.

Notation. Throughout the paper, $C$ stands for some real positive constants which may be different in each occurrence and independent of the initial data unless we give some special explanation. For sake of simplicity, we denote $\int_{\mathbb{R}^2}dxdy$ by $\int_{\mathbb{R}^2}dx$ for $(x,y)\in \mathbb{R}^2$ and write $\sum_j$ for $\sum_{j = 1}^{2}$.

Now, we start with the well-known Gagliardo-Nirenberg inequality.

Lemma 2.1. Suppose that $f\in L^{q}(\mathbb{R}^n)\cap W^{m,r}(\mathbb{R}^n), 1\leq q,r\leq\infty$. Then for $0\leq j\leq m,\frac{j}{m}\leq\theta\leq 1, 1\leq p\leq\infty$ and

there exists a constant $C$ such that

Let us introduce the well-known commutator estimates proved by Kato-Ponce [16].

Lemma 2.2. Let $s>0, 1<p<\infty$, and suppose that $f, g\in \mathcal{S}(\mathbb{R}^n)$. Then we have

where $\frac{1}{p} = \frac{1}{p_1}+\frac{1}{q_1} = \frac{1}{p_2}+\frac{1}{q_2}, \Lambda = (-\Delta)^{\frac{1}{2}}$, $\mathcal{S}(\mathbb{R}^n)$ denotes the Schwarz class of rapidly decreasing functions.

We recall the well-known Calderon-Zygmund operators, which will be used to get the control between the gradient of velocity and the vorticity (see [10]).

Lemma 2.3. (Biot-Savart law). There exists a universally positive constant $C$ such that for every $1<p<\infty$, holding

Next, let us recall the following logarithmic Sobolev inequality (see [3,4]).

Lemma 2.4. Let $n\geq 2$ and $s>1+\frac{n}{p}$. The following logarithmic Sobolev embedding theorem holds for all divergence free vector fields $f$ with $f\in L^2(\mathbb{R}^n)\cap \dot{W}^{s,p}(\mathbb{R}^n)$:

We also need the following well-known Osgood lemma [2], which will be a crucial ingredient to establish the global in time a priori estimates in the proof of Theorem 1.1.

Lemma 2.5. Let $\phi$ be a measurable, positive function, $\gamma$ a positive locally integrable function, and $\mu$ a continuous and nondecreasing function. Assume that, for some nonnegative real number $c$, the function $\phi$ satisfies

If $c$ is positive, then we have

If $c$ is zero and $\mu(s)$ satisfies $\int_{0}^{1}\frac{dr}{\mu(r)} = +\infty$, then $\phi = 0$.

The following lemma, which has been firstly introduced in [6] to deal with the 2D MHD equations with partial viscosity, will play an important role in the proof of our second main result.

Lemma 2.6. Assume that $f, g, \partial_y g, h$ and $\partial_x h$ are all in $L^2(\mathbb{R}^2)$. Then,

3.   Proof of Theorem 1.1

Proof. This section is devoted to the proof of Theorem 1.1. For simplicity, without loss of generality, we assume $\mu_2 = \nu_1 = 1$ in what follows.

Firstly, for the third equation of (5), according to the divergence free condition $\nabla\cdot u = 0$, we obtain immediately

for any $t\in [0,\infty]$.

Multiplying both sides of the first and second equations of (5) by $u_1$ and $u_2$ respectively, it is easy to get

where we used the fact:

by the incompressible condition $\nabla\cdot u = 0$.

Integrating in time and using (9), then for any $T>0$ and $t\leq T$, one have from (10) that

where $C = C(T, u_0,\theta_0, \alpha)$.

So we obtain

The first and second equations of (5) can be rewritten as:

Applying the operator $curl = \nabla\times$ to the first equation of (13), and using the fact

then we have

Testing (14) by $\mbox{curl}$ and using the incompressible condition $\nabla\cdot u = 0$, then we get

For the first term $I_1$ in (15), integrating by parts and using $-\Delta v = \mbox{curl}\mbox{curl}v$ and $\partial_x v_1+\partial_y v_2 = 0$,

which implies that

For the second term $I_2$, similarly,

Inserting the above estimates for $I_1$ and $I_2$ into (15), one has

Integrating in time, then it follows from (16) that

Hence, according to Lemma 2.3, at last we have

Here, we should point out that we can't obtain a global bound for the $H^1-$norm of $\theta$ at this stage. In fact, using the Hölder inequality, the incompressible condition $\nabla\cdot u = 0$, Lemma 2.2 and (18), we have

Here we used the following Gagliardo-Nirenberg inequality in Lemma 2.1

in the third inequality of (19). Then we get

Now, we apply "weakly nonlinear" energy estimate approach introduced firstly by Lei and Zhou [17] to obtain the higher global regularity. For any $T>0$, we assume that the solution $(v(t),\theta(t))$ is regular for $t<T$ and show that it remains regular at $t = T$. More precisely, we define

and assume that $\Phi(t)<\infty$ for $t<T$ and show that

It follows from the equation of $\nabla\theta$ that, for any $0\leq s<t$,

Then, choose $T_0$ close enough to $T$ ($T_0<T$) and let $T_0<t<T$, we have from Lemma 2.4 and (12) that

According to (18), we know that

then we can choose $T_0$ close enough to $T$ such that, for small $\varepsilon>0$,

It then follows from (23) that, for $T_0\leq t<T$,

Now applying the operator $\Delta$ on both sides of (14), multiplying the resulting equation by $\Delta\mbox{curl}v$ and integrating over $\mathbb{R}^2$, we have

For the first term $II_1$ in (25), integrating by parts and using $-\Delta v = \mbox{curl}\mbox{curl}v$, one has

Integrating by parts and using the condition $\partial_x v_1+\partial_y v_2 = 0$, we have

and

Inserting (27) and (28) into (26) implies that

Next we estimate the term $II_2$. By the incompressible condition $\nabla\cdot u = 0$, Lemma 2.2 and the Young inequality,

Here we used the Gagliardo-Nirenberg inequality (20) and (18) in the third and fourth inequalities of (30), respectively.

For the term $II_3$, by the Young inequality, we have

Inserting the estimates (29)-(31) for $II_1-II_3$, it gives that

Applying the operator $\nabla^3$ on both sides of the third equation of (5), by the incompressible condition $\nabla\cdot u = 0$, Lemma 2.2 and the Young inequality, one has

where we used the following Gagliardo-Nirenberg inequality of Lemma 2.1 in the first inequality of (33):

Combining (32) with (33), it gives that

Thanks to Lemma 2.3 and Lemma 2.4 and by the definition of $\Phi(t)$, it infers from the above inequality that

where we have taken $\varepsilon$ to be enough small (smaller than $\frac{1}{6}$) and used the assumption $\Phi(t)\geq 1$ (otherwise, if $\Phi(t)<1$, we here have got (21)).

Integrating in time over $(T_0,t)$ and noticing that the function $\Phi(t)$ is monotonically increasing, one can get from (34) that

Using Lemma 2.5, namely the Osgood inequality, one can conclude that

which (21) holds. This completes the proof of Theorem 1.1.

4.   Proof of Theorem 1.2

Proof. This section is devoted to the proof of Theorem 1.2. For simplicity, without loss of generality, we assume $\kappa_2 = 1$ in what follows.

The existence and uniqueness of local smooth solutions can be done without any difficulty as in the case of the Euler (see, e.g., [19]), thus it is sufficient to establish a priori estimates for $(v, \theta)$, namely, for $t\leq T$,

under the condition (8). Here the constant $C$ depends on $T, v_0,\theta_0, M(T)$.

Firstly, multiplying both sides of the first, second and third equations of (7) by $u_1, u_2$ and $\theta$, respectively, it is easy to get that

and

Applying the operator $\mbox{curl} = \nabla\times$ to the first and second equations of (7), namely,

then, one has

where we used the fact that

Then, from (37) and under the condition (8), we have that, for $t\leq T$,

Using Lemma 2.1, Lemma 2.3 and (36), one has

As a consequence, according to (39),

Multiplying both sides of (37) by $\mbox{curl}v$, integrating by parts in $\mathbb{R}^2$ and using the incompressible condition $\nabla\cdot u = 0$, we have

On the other hand, we multiplying the third equation by $\partial_{xx}\theta$ and $\partial_{yy}\theta$, respectively, and adding the resulting equations up, then integrating by parts in $\mathbb{R}^2$ gives that

Using the incompressible condition $\partial_x u_1+\partial_y u_2 = 0$ and integrating by parts, by the Hölder and Young inequalities, one has

For the last term $J_4$ in (42), using Lemma 2.6 and the Young inequality, we have

Inserting the above estimates (43)-(46) into (42), it gives that

Using the Gronwall inequality and according to (8), (35) and (39), it follows from (47) that, for any $t\in [0,T]$,

Next, we establish the $H^3-$norm bound for $(v,\theta)$, namely, for any $t\in [0,T]$,

Applying the operator $\Delta$ to (37), multiplying the resulting equation by $\Delta\mbox{curl}v$, integrating by parts in $\mathbb{R}^2$, we have

Using the incompressible condition $\nabla\cdot u = 0$, the Hölder inequality and Lemma 2.2, one can get

where we used the following Gagliardo-Nirenberg inequality in Lemma 2.1

in the third inequality, and (36), (40) and (48) in the fourth inequality.

For the term $K_2$, we have

Plugging (50) and (51) into (49), one has

Applying the operator $\nabla^3$ to the third equation of (7), multiplying the resulting equation with $\nabla^3\theta$ and integrating over $\mathbb{R}^2$, we have

The term $L_1$ can be written as

Now we bound the terms of $L_1$ one by one. Using the Hölder inequality, we have

Thanks to Lemma 2.6 and the Young inequality, it gives that

Noting that by $\partial_x u_1+\partial_y u_2 = 0$

then, integrating by parts and using the Hölder inequality and Lemma 2.6, one has

and

For the term $M_5$, integrating by parts and using $\partial_x u_1+\partial_y u_2 = 0$ and Lemma 2.6, we have

Using Lemma 2.3, Lemma 2.6, the Hölder inequality and the Young inequality, we have

Inserting the above estimates into (59), we have

For the terms $M_6-M_8$, integrating by parts and using the Hölder inequality and Young inequality and Lemma 2.6, one has

and

It is easy to estimate the term $M_9$ that, by Lemma 2.4, we can get

Inserting the above estimates (55)-(58) and (60)-(64) into (54) gives that

where we used (8), (36), (38), (40) and (48).

The term $L_2$ can be further written as

Due to the existence of the "favorable" derivative $\partial_y$, the estimates of $L_2$ is simpler compared with $L_1$. Thus, we can obtain the estimates similar to $L_1$. Therefore, we omit the details here.

Now, combining (65) and the estimates of (66) with (53), it follows that

This together with the Gronwall inequality yields that

which completes the proof of Theorem 1.2.

Acknowledgments

The authors thank the editor and reviewer for valuable comments and suggestions.