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Global existence of weak solution and regularity criteria for the 2D Bénard system with partial dissipation

Boundary Value Problems20182018:79

https://doi.org/10.1186/s13661-018-0988-9

  • Received: 1 November 2017
  • Accepted: 25 April 2018
  • Published:

Abstract

In this paper, we first write the velocity equation of the Bénard system in its two components, and consider the global weak solution of the resulting 2D Bénard system with partial dissipation, i.e. (1) \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (2) \(\mu _{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (3) \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa _{1}>0\), \(\kappa_{2}=0\); (4) \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (5) \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu _{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (6) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (7) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}>0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (8) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu _{4}>0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\), where \(\mu_{i}\) (\(i=1,2,3,4\)) and \(\kappa_{j}\) (\(j=1,2\)) are the coefficients of dissipation and thermal diffusivity. Furthermore, we establish some regularity criteria for the corresponding system. This work follows the techniques in the paper of Cao and Wu (Adv. Math. 226:1803–1822, 2011).

Keywords

  • Bénard system
  • Weak solution
  • Regularity criteria
  • Partial dissipation
  • Thermal diffusivity

MSC

  • 35Q35
  • 35B45
  • 35B65

1 Introduction

The Bénard system described the Rayleigh–Bénard convective motion in a heated 2D inviscid incompressible fluid under thermal effects (see e.g. [27]). One of the most fundamental problems in fluid dynamics concerning the Bénard system is whether their classical solutions are global regularity for all time or they develop singularities. The Bénard system has been a center of attention to numerous analytical, experimental, and number investigations. The motion of the incompressible Bénard system in \(\mathbb{R}^{2}\) is governed by
$$ \left\{\textstyle\begin{array}{l} \partial_{t}u+(u\cdot\nabla)u+\nabla{\pi}=\mu\Delta{u}+\theta{e_{2}},\\ \partial_{t}\theta+(u\cdot\nabla)\theta=\kappa\Delta\theta+u\cdot {e_{2}},\\ \nabla\cdot{u}=0, \end{array}\displaystyle \right. $$
(1.1)
where the unknown functions u, θ and π denote the 2D velocity field, temperature and pressure, respectively. The constants μ and κ are the coefficients of dissipation and thermal diffusivity. The forcing term \(\theta{e_{2}}\) in the momentum equation describes the acting of the buoyancy force on fluid motion and \(u\cdot{e_{2}}\) models the Rayleigh–Bénard convection in a heated inviscid fluid, and \(e_{2}=(0,1)^{T}\). Physically, (1.1)1 reflects the conservation of momentum, (1.1)2 describes the thermal convection, and in (1.1)3, \(\nabla\cdot {u}=0\) shows the conservation of mass.

In the fluid dynamics area, the Bénard fluid problem is a very classical problem, which has an important significance in convective motion in a heated fluid, such as the case of planar stationary flows. The incompressible Bénard system have attracted the attention of many physicists and mathematicians due to its important physical background, rich phenomena, mathematical complexity and challenges. Neustupa and Siginer [8] proved the existence of a strong-weak solution \((u,\pi,\theta)\) of the steady Bénard problem in a 2D quadrangular cavity, heated/cooled on two opposite sides and thermally insulated on the other sides. The nonlinear Lyapunov stability of the conduction-diffusion solution of the rotating Bénard problem was studied in [9]. Anh and Son [10] studied the 2D Bénard problem in an arbitrary domain (bounded or unbounded) which satisfying the Poincaré inequality with nonhomogeneous boundary conditions and nonautonomous external force and heat source, and the existence of a weak solution to the problem was proved by using the Galerkin method, and showed that the existence of a unique minimal finite-dimensional pullback \(D_{\sigma}\)-attractor for the process associated to the problem. Wu and Xue [11] considered the Cauchy problem of the 2D inviscid Bénard system with fractional diffusivity, and showed that the system had a unique global solution \((u,\theta)\) such that \(u\in{C}^{0,1}(\mathbf{R}_{+},L^{2}(\mathbb{R}^{2}))\), \(\theta\in{C}(\mathbb{R}_{+},L^{2}\cap{B}^{1-\beta}_{\infty,1})\cap {L^{2}_{\mathrm{loc}}}(\mathbb{R}_{+},H^{\frac{\beta}{2}})\). 2D incompressible Bénard system with critical and supercritical dissipation (\(0\leq\alpha\leq{1}\)) in the velocity was studied in [12]. Cheng and Du [13] considered the Cauchy problem of the 2D magnetic Bénard problem with mixed partial viscosity. More precisely, the global well-posedness of the 2D magnetic Bénard problem without thermal diffusivity and with vertical or horizontal magnetic diffusion was obtained. Moreover, the global regularity and some conditional regularity of strong solutions were obtained for the 2D magnetic Bénard problem with mixed partial viscosity. Zhou–Nakamura [14] studied a 2D magnetic Bénard problem with zero thermal conductivity, and showed a global well-posedness result by a well-known property of Hardy space and BMO. As it is demonstrated in reference [15, 16], we showed the global regularity for the two-and-half-dimensional magnetic Bénard system with zero thermal diffusivity by a well-known property of Hardy space and BMO; resorting to the method of the local-in-time analysis, the global regularity for the two-and-half-dimensional magnetic Bénard system with zero thermal diffusivity and horizontal magnetic diffusion as well as vertical magnetic diffusion are also obtained. Moreover, we proved that, as the initial data satisfy \(\|u_{0}\|^{2}_{H^{1}_{\mathbb{R}^{3}}}+\|b_{0}\|^{2}_{H^{1}_{\mathbb{R}^{3}}}+\|\theta _{0}\|^{2}_{H^{1}_{\mathbb{R}^{3}}}\leq\varepsilon\), where ε is a suitably small positive number, then the 3D magnetic Bénard system with mixed partial dissipation, magnetic diffusion and thermal diffusivity admits global smooth solutions. Zhang and Tang [17] studied the global regularity for a special family of axisymmetric solutions to the 3D magnetic Bénard problem.

In particular, if \(\theta=\textit{Const.}\), then system (1.1) reduces to the classical Navier–Stokes system which describes the motion of incompressible viscous fluid flows and has been extensively studied by many authors; see [1824] and the references therein. In addition to this, the reader is referred to [2529] to find more results about the related fluid flow equations.

Recently, Regmi [30] established global weak solution for the 2D MHD system with partial dissipation and vertical diffusion. Cheng and Li [31] established the global weak solutions for the 2D Boussinesq system with mixed partial dissipation and thermal diffusivity. Chen concerned with the 2D system of the incompressible micropolar fluid flows with mixed partial viscosity and angular viscosity, and the global existence and uniqueness of smooth solution was showed in [32]. Yu considered the global regularity to the initial-boundary value problem of the 2D incompressible MHD system with mixed partial dissipation and magnetic diffusion in [33]. Fan et al. considered the global regularity for the 2D liquid crystal model with mixed partial viscosity and global Cauchy problem of 2D generalized MHD equations, respectively, in [34, 35].

Inspired by this work, we consider the following Bénard system in this paper:
$$ \left\{\textstyle\begin{array}{l} \partial_{t}u_{1}+(u\cdot\nabla)u_{1}+\partial_{x}{\pi}=\mu_{1}\partial _{xx}{u_{1}}+\mu_{2}\partial_{yy}{u_{1}},\\ \partial_{t}u_{2}+(u\cdot\nabla)u_{2}+\partial_{y}{\pi}=\mu_{3}\partial _{xx}{u_{2}}+\mu_{4}\partial_{yy}{u_{2}}+\theta,\\ \partial_{t}\theta+(u\cdot\nabla)\theta=\kappa_{1}\partial_{xx}\theta +\kappa_{2}\partial_{yy}\theta+u_{2},\\ \partial_{x}u_{1}+\partial_{x}u_{2}=0, \end{array}\displaystyle \right. $$
(1.2)
where \(u=(u_{1},u_{2})\). System (1.2) is capable of modeling the motion of anisotropic fluids for which the diffusion properties in different directions are different. Additionally, (1.2) allows us to explore the smooth effects of various partial dissipations. Furthermore, we consider the 2D Bénard system (1.2) with partial dissipation in the following eight cases:
  1. (i)

    \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\);

     
  2. (ii)

    \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\);

     
  3. (iii)

    \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\);

     
  4. (iv)

    \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\);

     
  5. (v)

    \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\);

     
  6. (vi)

    \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\);

     
  7. (vii)

    \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}>0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\);

     
  8. (viii)

    \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}>0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\).

     
In this paper, we equip system (1.2) with the following initial data:
$$\begin{gathered} u_{1}(x,y,0)=u^{0}_{1}(x,y),\qquad u_{2}(x,y,0)=u^{0}_{2}(x,y),\\ \theta(x,y,0)=\theta ^{0}(x,y),\quad (x,y)\in\mathbb{R}^{2}.\end{gathered} $$
(1.3)

The plan of this paper is as follows. Firstly, we give two very useful lemmas and establish the global weak solution for the 2D Bénard system with vertical dissipation in the first component of velocity field and horizontal thermal diffusivity in Sect. 2. We shall give the global regularity criteria for weak solution of the 2D Bénard system with vertical dissipation in the first component of velocity field and horizontal thermal diffusivity in Sect. 3. In Sect. 4, we will give the global existence and regularity criteria of weak solution to Bénard system with other cases for partial viscosity and thermal diffusivity.

Notations. We introduce some notations which are used in this paper. For \(1\leq{p}\leq\infty\), \(L^{p}=L^{p}(\mathbb{R}^{2})\) denotes the usual Lebesgue space with the norm \(\|\cdot\|_{L^{p}}\). The usual Sobolev space of order n is defined by \(H^{n}=\{f\in{L^{2}}(\mathbb{R}^{2})|\nabla^{n}f\in{L^{2}}\}\) with the norm \(\|f\|_{H^{n}}= (\|f\|^{2}_{L^{2}}+\|\nabla^{n}f\|^{2}_{L^{2}} )^{\frac{1}{2}}\).

2 Global weak solution for the Bénard system with vertical dissipation and horizontal thermal diffusivity

In this section, we will establish the global weak solution for the 2D Bénard system with vertical dissipation in the first component of velocity field and horizontal thermal diffusivity.

2.1 Preliminaries

In this subsection, we first provide the lemma that bounds a triple-product in terms of the Lebesgue norms of the functions and their directional derivatives; see for example [1, 36, 37]. The following anisotropic Sobolev inequality will play very important roles in proving our main results.

Lemma 2.1

Assume that \(f, g, h, \partial_{x}g, \partial_{y}h\in{L^{2}}(\mathbb {R}^{2})\). Then there exists an absolute constant C such that
$$ \iint_{\mathbb{R}^{2}}|fgh|\,dx\,dy\,dz\leq{C}\|f\|_{L^{2}}\|g \|^{\frac {1}{2}}_{L^{2}}\|\partial_{x}g\|^{\frac{1}{2}}_{L^{2}} \|h\|^{\frac {1}{2}}_{L^{2}}\|\partial_{y}h\|^{\frac{1}{2}}_{L^{2}}. $$
(2.1)

The following simple fact on the boundedness of Riesz transforms will also be used.

Lemma 2.2

(see [1])

Let f be a divergence-free vector field such that \(\nabla{f}\in {L}^{\gamma}\) for \(\gamma\in(1,\infty)\). Then there exists a pure constant \(C>0\) (independent of γ) such that
$$ \|\nabla{f}\|_{L^{\gamma}}\leq\frac{C\gamma^{2}}{\gamma-1}\|\nabla\times {f} \|_{L^{\gamma}}. $$
(2.2)

For simplicity, throughout this paper, we use the same letter C to denote various generic positive constants whose exact values are unimportant and may vary from line to line.

2.2 Global weak solution

Theorem 2.3

Let \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa _{2}=0\). Suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{1}(\mathbb{R}^{2})}\) and \(\partial_{x}u^{0}_{1}+\partial_{y}u^{0}_{2}=0\). Then the problem (1.2)(1.3) admits a global weak solution \((u_{1}, u_{2}, \theta )\), which obeys
$$u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial _{y}u_{1},\partial_{x}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr) $$
for any \(T>0\).

Theorem 2.3 follows from the following two lemmas immediately.

Lemma 2.4

Consider (1.2) with \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\). Let \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in {L^{2}(\mathbb{R}^{2})}\), then, for any \(T>0\) and \(0< t< T\), we have
$$ \|u_{1}\|^{2}_{L^{2}}+\|u_{2} \|^{2}_{L^{2}}+\|\theta\|^{2}_{L^{2}}+ \mu_{2} \int^{t}_{0}\big\| \partial_{y}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau+\kappa_{1} \int^{t}_{0}\big\| \partial_{x}\theta (\tau) \big\| ^{2}_{L^{2}}\,d\tau\leq{C}. $$
(2.3)

Proof

Taking the \(L^{2}\)-inner product of the first three equations in (1.2) with \(u_{1}\), \(u_{2}\) and θ, respectively, integrating the resulting equations by parts over \(\mathbb{R}^{2}\), and using the divergence-free condition \(\partial_{x}u_{1}+\partial_{x}u_{2}=0\), we find after adding them together that
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt} \bigl(\big\| u_{1}(t)\big\| ^{2}_{L^{2}}+\big\| u_{2}(t) \big\| ^{2}_{L^{2}}+\big\| \theta(t)\big\| ^{2}_{L^{2}} \bigr)+\mu_{2}\|\partial_{y}u_{1} \|^{2}_{L^{2}}+\kappa_{1}\| \partial_{x} \theta\|^{2}_{L^{2}} \\ &\quad=2 \iint_{\mathbb{R}^{2}}u_{2}\theta{\,dx\,dy} \\ &\quad\leq C \bigl(\|u_{2}\|^{2}_{L^{2}}+\|\theta \|^{2}_{L^{2}} \bigr). \end{aligned} $$
(2.4)
This, together with the Gronwall’s inequality, gives the desired estimates of (2.3), which implies
$$ \|u_{1},u_{2},\theta\|_{L^{\infty}([0,T);L^{2}(\mathbb{R}^{2}))}+ \mu_{2}\|\partial _{y}u_{1}\|_{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))}+ \kappa_{1}\|\partial_{x}\theta\| _{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))}\leq{C}. $$
(2.5)

We thus complete the proof of Lemma 2.4. □

To obtain the \(H^{1}\)-estimates for \(u_{1}\), \(u_{2}\) and θ, we consider the following equations:
$$ \left\{\textstyle\begin{array}{l} \partial_{t}\omega+(u\cdot\nabla)\omega=\mu_{2}\partial _{yyy}{u_{1}}+\partial_{x}\theta,\\ \partial_{t}\partial_{x}\theta+\partial_{x}[(u\cdot\nabla)\theta]=\kappa _{1}\partial_{xxx}\theta+\partial_{x}u_{2},\\ \partial_{t}\partial_{y}\theta+\partial_{y}[(u\cdot\nabla)\theta]=\kappa _{1}\partial_{xxy}\theta+\partial_{y}u_{2}, \end{array}\displaystyle \right. $$
(2.6)
where the vorticity is \(\omega=\partial_{x}u_{2}-\partial_{y}u_{1}\).

Lemma 2.5

Consider (1.2) with \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\). Let \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in {H^{1}(\mathbb{R}^{2})}\), then, for any \(T>0\) and \(0< t< T\), we have
$$ \begin{aligned}[b] &\|\omega\|^{2}_{L^{2}}+ \|\partial_{x}\theta\|^{2}_{L^{2}}+\| \partial_{y}\theta\| ^{2}_{L^{2}}+\mu_{2} \int^{t}_{0}\big\| \partial_{xy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau+\mu _{2} \int^{t}_{0}\big\| \partial_{yy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau \\ &\quad+\kappa_{1} \int^{t}_{0}\big\| \partial_{xx}\theta(\tau) \big\| ^{2}_{L^{2}}\,d\tau+\kappa _{1} \int^{t}_{0}\big\| \partial_{xy}\theta(\tau) \big\| ^{2}_{L^{2}}\,d\tau\leq{C}. \end{aligned} $$
(2.7)

Proof

Multiplying the first equation of (2.6) by ω and integrating it over \(\mathbb{R}^{2}\), we deduce
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt}\big\| \omega(t)\big\| ^{2}_{L^{2}}+\mu_{2}\|\partial_{xy}u_{1} \| ^{2}_{L^{2}}+\mu_{2}\|\partial_{yy}u_{1} \|^{2}_{L^{2}} \\ &\quad= \iint_{\mathbb{R}^{2}}\partial_{x}\theta\omega{\,dx\,dy} \\ &\quad\leq{C} \bigl(\|\partial_{x}\theta\|^{2}_{L^{2}}+\| \omega\|^{2}_{L^{2}} \bigr), \end{aligned} $$
(2.8)
and hence, using (2.3), (2.5) and Gronwall’s inequality, we know that
$$ \|\omega\|^{2}_{L^{2}}+\mu_{2} \int^{t}_{0}\big\| \partial_{xy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\, d\tau+\mu_{2} \int^{t}_{0}\big\| \partial_{yy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau\leq{C}, $$
(2.9)
which gives
$$ \|\omega\|_{L^{\infty}([0,T);L^{2}(\mathbb{R}^{2}))}+\mu_{2}\|\nabla\partial _{y}u_{1}\|_{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))}\leq{C}. $$
(2.10)
On the other hand, multiplying (2.6)2 and (2.6)3, respectively, by \(\partial_{x}\theta\) and \(\partial_{y}\theta\), and integrating by parts, yields
$$ \frac{1}{2}\frac{d}{dt} \bigl(\big\| \partial_{x} \theta(t)\big\| ^{2}_{L^{2}}+\big\| \partial _{y}\theta(t) \big\| ^{2}_{L^{2}} \bigr)+\kappa_{1}\|\partial_{xx} \theta\| ^{2}_{L^{2}}+\kappa_{1}\|\partial_{xy} \theta\|^{2}_{L^{2}}=I=\sum^{4}_{l=1}I_{l}, $$
(2.11)
where
$$\begin{gathered} I_{1}=- \iint_{\mathbb{R}^{2}}\partial_{x}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{x}\theta{\,dx}\,dy,\qquad I_{2}= \iint_{\mathbb{R}^{2}}\partial_{x}u_{2}\partial _{x}\theta{\,dx}\,dy, \\ I_{3}=- \iint_{\mathbb{R}^{2}}\partial_{y}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{y}\theta{\,dx}\,dy,\qquad I_{4}= \iint_{\mathbb{R}^{2}}\partial_{y}u_{2}\partial _{y}\theta{\,dx}\,dy.\end{gathered} $$
We now estimate the right-hand side of (2.11) term by term. We first write \(I_{1}\) as
$$ \begin{aligned}[b] I_{1}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{x}\theta{\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\partial_{x}\theta{\, dx}\,dy- \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{y}\theta\partial _{x}\theta{\,dx}\,dy \\ ={}&I_{11}+I_{12}. \end{aligned} $$
(2.12)

Here, we have used the fact \(\iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta \partial_{x}\theta{\,dx}\,dy+\iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}\theta \partial_{x}\theta{\,dx}\,dy=0\), which can easily be obtained by integration by parts and the incompressible condition.

With the help of Lemma 2.1, the Cauchy–Schwarz inequality and Young’s inequality, we infer that
$$ \begin{aligned}[b] I_{11}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\partial _{x}\theta{\,dx}\,dy \\ \leq{}&{C}\|\partial_{x}u_{1}\|_{L^{2}}\| \partial_{x}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xx}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy} \theta\|^{\frac {1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{2}\|\partial_{xx} \theta\|^{2}_{L^{2}}+\frac{\kappa _{1}}{8}\|\partial_{xy} \theta\|^{2}_{L^{2}}+C\|\omega\|^{2}_{L^{2}}\| \partial _{x}\theta\|^{2}_{L^{2}}. \end{aligned} $$
(2.13)
Similarly, we obtain
$$ \begin{aligned}[b] I_{12}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{y}\theta\partial _{x}\theta{\,dx}\,dy \\ \leq{}&{C}\|\partial_{x}u_{2}\|_{L^{2}}\| \partial_{y}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy} \theta\|^{\frac {1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{8}\|\partial_{xy} \theta\|^{2}_{L^{2}}+C\|\omega\| ^{2}_{L^{2}} \bigl(\|\partial_{x}\theta\|^{2}_{L^{2}}+\| \partial_{y}\theta\| ^{2}_{L^{2}} \bigr). \end{aligned} $$
(2.14)
\(I_{3}\) can be bounded by
$$\begin{aligned} I_{3}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{y}\theta{\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}(u_{1} \partial_{x}\theta+u_{2}\partial _{y}\theta) \partial_{y}\theta{\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\theta\partial_{y}\theta{\, dx}\,dy- \iint_{\mathbb{R}^{2}}u_{1}\partial_{xy}\theta \partial_{y}\theta{\, dx}\,dy \\ &- \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{y}\theta\partial _{y}\theta{\,dx}\,dy- \iint_{\mathbb{R}^{2}}u_{2}\partial_{yy}\theta \partial_{y}\theta{\,dx}\, dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\theta\partial_{y}\theta{\, dx}\,dy- \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{y}\theta\partial _{y}\theta{\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\theta\partial_{y}\theta{\, dx}\,dy-2 \iint_{\mathbb{R}^{2}}u_{1}\partial_{y}\theta \partial_{xy}\theta{\, dx}\,dy \\ \leq{}&{C}\|\partial_{y}u_{1}\|_{L^{2}}\| \partial_{x}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial _{y}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy} \theta\|^{\frac {1}{2}}_{L^{2}} \\ &+C\|\partial_{xy}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{y}\theta \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{4}\|\partial_{xy}\theta\|^{2}_{L^{2}}+C \bigl(\|\partial _{y}u_{1}\|^{2}_{L^{2}}+ \|u_{1}\|^{2}_{L^{2}}\|\partial_{y}u_{1} \|^{2}_{L^{2}} \bigr) \bigl(\| \partial_{x}\theta \|^{2}_{L^{2}}+\|\partial_{y}\theta \|^{2}_{L^{2}} \bigr). \end{aligned}$$
(2.15)
In what follows, we estimate \(I_{2}\) and \(I_{4}\). Using the Cauchy inequality, we have
$$ I_{2}= \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{x}\theta{\,dx}\,dy\leq {C} \bigl(\|\omega \|^{2}_{L^{2}}+\|\partial_{x}\theta \|^{2}_{L^{2}} \bigr), $$
(2.16)
and
$$ I_{4}= \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{y}\theta{\,dx}\,dy\leq {C} \bigl(\|\omega \|^{2}_{L^{2}}+\|\partial_{y}\theta \|^{2}_{L^{2}} \bigr). $$
(2.17)
Combining the above estimates, we conclude that
$$ \begin{aligned}[b] &\frac{d}{dt} \bigl(\big\| \partial_{x}\theta(t)\big\| ^{2}_{L^{2}}+\big\| \partial_{y}\theta (t)\big\| ^{2}_{L^{2}} \bigr)+ \kappa_{1}\|\partial_{xx}\theta\|^{2}_{L^{2}}+ \kappa_{1}\| \partial_{xy}\theta\|^{2}_{L^{2}} \\ &\quad\leq{C} \bigl(\|\partial_{y}u_{1}\|^{2}_{L^{2}}+ \|\omega\|^{2}_{L^{2}}+\|u_{1}\| ^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}+1 \bigr) \bigl(\|\partial_{x}\theta\| ^{2}_{L^{2}}+\| \partial_{y}\theta\|^{2}_{L^{2}} \bigr)+C\|\omega \|^{2}_{L^{2}}. \end{aligned} $$
(2.18)
Then (2.18), together with Gronwall’s inequality, immediately yields
$$ \|\nabla\theta\|_{L^{\infty}([0,T;L^{2}(\mathbb{R}^{2}))}+\kappa_{1}\|\nabla \partial_{x}\theta\|_{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))}\leq{C}. $$
(2.19)

Hence, we finish the proof of Lemma 2.5. □

We next prove Theorem 2.3 by using the method of vanishing viscosity. To this end, we consider the following regularized problem:
$$ \left\{\textstyle\begin{array}{l} \partial_{t}u^{\varepsilon}_{1}+(u^{\varepsilon}\cdot\nabla)u^{\varepsilon}_{1}+\partial_{x}{\pi^{\varepsilon}}=\varepsilon\partial_{xx}{u^{\varepsilon}_{1}}+\mu_{2}\partial_{yy}{u^{\varepsilon}_{1}},\\ \partial_{t}u^{\varepsilon}_{2}+(u^{\varepsilon}\cdot\nabla)u^{\varepsilon}_{2}+\partial_{y}{\pi^{\varepsilon}}=\varepsilon\partial_{xx}{u^{\varepsilon}_{2}}+\varepsilon\partial_{yy}{u^{\varepsilon}_{2}}+\theta^{\varepsilon},\\ \partial_{t}\theta^{\varepsilon}+(u^{\varepsilon}\cdot\nabla)\theta ^{\varepsilon}=\kappa_{1}\partial_{xx}\theta^{\varepsilon}+\varepsilon\partial _{yy}\theta^{\varepsilon}+u^{\varepsilon}_{2},\\ \nabla\cdot{u^{\varepsilon}}=0, \end{array}\displaystyle \right. $$
(2.20)
with the smooth initial data
$$ u^{\varepsilon}_{1}(x,y,0)=\varphi_{\varepsilon}\ast{u}^{0}_{1},\qquad u^{\varepsilon}_{2}(x,y,0)= \varphi_{\varepsilon}\ast{u}^{0}_{2},\qquad \theta^{\varepsilon}(x,y,0)=\varphi_{\varepsilon}\ast\theta^{0}, $$
(2.21)
where \(u^{\varepsilon}=(u^{\varepsilon}_{1},u^{\varepsilon}_{2})\), “” is the usual convolution operator, and \(\varphi_{\varepsilon}(x,y)=\varepsilon^{-2}\varphi(x/\varepsilon ,y/\varepsilon)\) is the standard mollifier with width ε, which satisfying
$$ \varphi\geq{0}, ~~\varphi\in{C}^{\infty}_{0}\bigl( \mathbb{R}^{2}\bigr),~~ \iint_{\mathbb {R}^{2}}\varphi{\,dx}\,dy=1. $$

Now, an application of the result of Sect. 3 in [32] in p. 934 shows that, for any \(T>0\), there exists a unique global smooth solution \((u^{\varepsilon}_{1},u^{\varepsilon}_{2},\theta^{\varepsilon})\) of (2.20), (2.21) on \(\mathbb{R}^{2}\times(0,T)\) satisfying the global bounds stated in Lemmas 2.42.5 which are uniform in ε. So, by standard compactness arguments, we can extract a subsequence \((u^{\varepsilon _{j}}_{1},u^{\varepsilon_{j}}_{2},\theta^{\varepsilon_{j}})\) and pass to the limit as \(j\rightarrow\infty\) to find that the limit function \((u_{1},u_{2},\theta)\) is indeed a global weak solution of the problem (1.2)–(1.3) with \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\). The uniqueness of the solution \((u_{1},u_{2},\theta)\) satisfying the condition stated in Theorem 2.3 can be proved in a very standard way, and for simplicity we omit the details here. The proof of Theorem 2.3 is therefore completed.

3 Global regularity criteria for weak solution of the Bénard system with vertical dissipation and horizontal thermal diffusivity

The issue of whether the 2D Bénard system always possesses global (in time) classical solutions can be difficult when there is only partial dissipation. Therefore, the goal of this section is to establish two global regularity criteria for the weak solution of the 2D Bénard system with vertical dissipation in the first component of velocity field and horizontal thermal diffusivity.

Theorem 3.1

Let \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa _{2}=0\). Suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{2}(\mathbb{R}^{2})}\) and \(\partial_{x}u^{0}_{1}+\partial_{y}u^{0}_{2}=0\). If the condition holds that
$$ \int^{T}_{0}\big\| \partial_{xx}u_{2}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty, $$
(3.1)
for any fixed \(T>0\), then the problem (1.2)(1.3) admits a global classical solution \((u_{1}, u_{2}, \theta)\), which obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\\quad \partial _{xyy}u_{1}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial_{y}u_{1}, \partial _{x}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr).\end{gathered} $$

Proof

Applying to Eq. (2.6)1 and taking the \(L^{2}\)-inner product with ω, and integrating by parts, we obtain
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt}\big\| \nabla\omega(t)\big\| ^{2}_{L^{2}}+\mu_{2}\|\partial _{xxy}u_{1}\|^{2}_{L^{2}}+2\mu_{2} \|\partial_{xyy}u_{1}\|^{2}_{L^{2}}+ \mu_{2}\|\partial _{yyy}u_{1}\|^{2}_{L^{2}} \\ &\quad=- \iint_{\mathbb{R}^{2}}\nabla\bigl[(u\cdot\nabla)\omega\bigr]\cdot\nabla \omega{\, dx}\,dy+ \iint_{\mathbb{R}^{2}}\nabla\partial_{x}\theta\nabla\omega{\,dx}\,dy. \end{aligned} $$
(3.2)
By the divergence-free condition, we further split the first term of (3.2) into four terms;
$$ \begin{aligned}[b] &- \iint_{\mathbb{R}^{2}}\nabla\bigl[(u\cdot\nabla)\omega\bigr]\cdot\nabla \omega{\, dx}\,dy \\ &\quad= -\iint_{\mathbb{R}^{2}}\partial_{x}u_{1}( \partial_{x}\omega)^{2}\,dx\,dy- \iint _{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{x}\omega\partial_{y}\omega{\,dx}\, dy \\ &\qquad{}- \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\omega\partial_{y}\omega{\, dx}\,dy- \iint_{\mathbb{R}^{2}}\partial_{y}u_{2}( \partial_{y}\omega)^{2}\,dx\,dy. \end{aligned} $$
(3.3)
Differentiating Eqs. (2.6)2 and (2.6)3 with respect to x and y and multiplying the resulting equations by \(\partial _{xx}\theta\) and \(\partial_{yy}\theta\), respectively, we deduce after integrating by parts over \(\mathbb{R}^{2}\) that
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt} \bigl(\big\| \partial_{xx}\theta(t)\big\| ^{2}_{L^{2}}+\big\| \partial_{yy}\theta(t)\big\| ^{2}_{L^{2}} \bigr)+ \kappa_{1}\|\partial_{xxx}\theta\| ^{2}_{L^{2}}+ \kappa_{1}\|\partial_{xyy}\theta\|^{2}_{L^{2}} \\ &\quad=- \iint_{\mathbb{R}^{2}}\partial_{xx}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{xx}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}\partial_{xx}u_{2}\partial _{xx}\theta{\,dx}\,dy \\ &\qquad{}- \iint_{\mathbb{R}^{2}}\partial_{yy}\bigl[(u\cdot\nabla )\theta \bigr]\partial_{yy}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}\partial_{yy}u_{2} \partial_{yy}\theta{\,dx}\,dy. \end{aligned} $$
(3.4)
We now turn to the first and third terms of (3.4). Again we write them out explicitly as
$$\begin{aligned}& \begin{aligned}[b] &- \iint_{\mathbb{R}^{2}}\partial_{xx}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{xx}\theta{\,dx}\,dy\\&\quad= \iint_{\mathbb{R}^{2}}\partial_{x}\bigl[(u\cdot\nabla )\theta \bigr]\partial_{xxx}\theta{\,dx}\,dy \\ &\quad=- \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\partial _{xxx}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta \partial_{xxx}\theta{\,dx}\,dy \\ &\qquad+ \iint_{\mathbb{R}^{2}}\partial _{x}u_{2} \partial_{y}\theta\partial_{xxx}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}\theta \partial_{xxx}\theta{\, dx}\,dy, \end{aligned} \end{aligned}$$
(3.5)
$$\begin{aligned}& \begin{aligned}[b] &- \iint_{\mathbb{R}^{2}}\partial_{yy}\bigl[(u\cdot\nabla)\theta \bigr]\partial _{yy}\theta{\,dx}\,dy\\&\quad= \iint_{\mathbb{R}^{2}}\partial_{y}\bigl[(u\cdot\nabla )\theta \bigr]\partial_{yyy}\theta{\,dx}\,dy \\ &\quad= \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\theta\partial _{yyy}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}u_{1}\partial_{xy}\theta \partial_{yyy}\theta{\,dx}\,dy \\ &\qquad+ \iint_{\mathbb{R}^{2}}\partial _{y}u_{2} \partial_{y}\theta\partial_{yyy}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{yy}\theta \partial_{yyy}\theta{\, dx}\,dy. \end{aligned} \end{aligned}$$
(3.6)
Combining with (3.2)–(3.6) leads to
$$\begin{aligned} \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt} \bigl(\big\| \nabla\omega(t)\big\| ^{2}_{L^{2}}+\big\| \partial _{xx} \theta(t)\big\| ^{2}_{L^{2}}+\big\| \partial_{yy}\theta(t) \big\| ^{2}_{L^{2}} \bigr)+\mu_{2}\| \partial_{xxy}u_{1} \|^{2}_{L^{2}}+2\mu_{2}\|\partial_{xyy}u_{1} \|^{2}_{L^{2}} \\ &\qquad+\mu_{2}\|\partial_{yyy}u_{1}\|^{2}_{L^{2}}+ \kappa_{1}\|\partial_{xxx}\theta\| ^{2}_{L^{2}}+ \kappa_{1}\|\partial_{xyy}\theta\|^{2}_{L^{2}} \\ &\quad=- \iint_{\mathbb{R}^{2}}\partial_{x}u_{1}( \partial_{x}\omega)^{2}\,dx\,dy- \iint _{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{x}\omega\partial_{y}\omega{\,dx}\, dy- \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\omega\partial_{y}\omega{\, dx}\,dy \\ &\qquad- \iint_{\mathbb{R}^{2}}\partial_{y}u_{2}( \partial_{y}\omega)^{2}\,dx\,dy+ \iint _{\mathbb{R}^{2}}\nabla\partial_{x}\theta\nabla\omega{\,dx}\,dy+ \iint_{\mathbb {R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\partial_{xxx}\theta{\,dx}\,dy \\ &\qquad+ \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{y}\theta\partial_{xxx}\theta {\,dx}\,dy+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}\theta\partial _{xxx}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}\partial_{y}u_{1}\partial _{x}\theta\partial_{yyy}\theta{\,dx}\,dy \\ &\qquad+ \iint_{\mathbb{R}^{2}}u_{1}\partial_{xy}\theta \partial_{yyy}\theta{\, dx}\,dy+ \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{y}\theta\partial _{yyy}\theta{\,dx}\,dy+ \iint_{\mathbb{R}^{2}}\partial_{xx}u_{2}\partial _{xx}\theta{\,dx}\,dy \\ &\qquad+ \iint_{\mathbb{R}^{2}}\partial_{yy}u_{2} \partial_{yy}\theta{\,dx}\, dy+ \iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta \partial_{xxx}\theta{\, dx\,dy}+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{yy}\theta \partial_{yyy}\theta {\,dx\,dy} \\ &\quad=J=\sum^{15}_{l=1}J_{l}. \end{aligned} \end{aligned}$$
(3.7)
We now estimate \(J_{1}\) through \(J_{7}\). Applying Lemma 2.1, the Cauchy–Schwarz inequality and Young’s inequality, we have
$$ \begin{aligned}[b] J_{1}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{1}( \partial_{x}\omega)^{2}\,dx\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{1}( \partial_{xx}u_{2})^{2}\,dx\,dy- \iint _{\mathbb{R}^{2}}\partial_{x}u_{1}( \partial_{xy}u_{1})^{2}\,dx\,dy\\ &+2 \iint _{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{xx}u_{2}\partial_{xy}u_{1}\,dx \,dy \\ \leq{}&{C}\|\partial_{x}u_{1}\|_{L^{4}}\| \partial_{xx}u_{2}\|_{L^{2}}\|\partial _{xx}u_{2}\|_{L^{4}} \\ &+C\|\partial_{x}u_{1}\|_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xyy}u_{1}\|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{x}u_{1}\|_{L^{4}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\partial _{xy}u_{1}\|_{L^{2}} \\ \leq{}&{C}\|\partial_{x}u_{1}\|^{\frac{1}{2}}_{L^{2}} \|\nabla\partial_{x}u_{1}\| ^{\frac{1}{2}}_{L^{2}}\| \partial_{xx}u_{2}\|_{L^{2}}\|\partial_{xx}u_{2} \| _{L^{4}} \\ &+C\|\omega\|_{L^{2}}\|\partial_{xy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy}u_{1} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xyy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{x}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \nabla\partial_{x}u_{1}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\partial_{xy}u_{1} \|_{L^{2}} \\ \leq{}&{C}\|\omega\|^{\frac{1}{2}}_{L^{2}}\|\nabla\omega \|^{\frac {3}{2}}_{L^{2}}\|\partial_{xx}u_{2} \|_{L^{4}}+C\|\omega\|_{L^{2}}\|\nabla\omega \|_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}u_{1}\| ^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\omega\|^{\frac{1}{2}}_{L^{2}}\|\nabla\omega\|^{\frac {3}{2}}_{L^{2}} \|\partial_{xx}u_{2}\|_{L^{4}} \\ \leq{}&\frac{\mu_{2}}{16}\|\partial_{xxy}u_{1} \|^{2}_{L^{2}}+\frac{\mu_{2}}{2}\| \partial_{xyy}u_{1} \|^{2}_{L^{2}}+C \bigl(\|\partial_{xx}u_{2} \|_{L^{4}}+\|\omega\| ^{2}_{L^{2}} \bigr)\|\nabla\omega \|^{2}_{L^{2}} \\ &+C\|\partial_{xx}u_{2}\|_{L^{4}}\|\omega \|^{2}_{L^{2}}, \end{aligned} $$
(3.8)
where we have utilized the inequality
$$ \|f\|_{L^{4}}\leq\|f\|^{\frac{1}{2}}_{L^{2}}\| \nabla{f}\|^{\frac{1}{2}}_{L^{2}}. $$
(3.9)
Similarly, invoking Lemma 2.1 and (3.9), the Cauchy–Schwarz inequality and Young’s inequality, the remainder terms can be estimated as follows:
$$\begin{aligned} J_{2}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{x}\omega\partial_{y}\omega {\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{xx}u_{2}\partial_{xy}u_{2}\, dx \,dy+ \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{xy}u_{1}\partial _{xy}u_{2}\,dx \,dy \\ &+ \iint_{\mathbb{R}^{2}}\partial_{x}u_{2}\partial _{xx}u_{2}\partial_{yy}u_{1}\,dx\,dy- \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{xy}u_{1}\partial_{yy}u_{1}\, dx \,dy \\ \leq{}&{C}\|\partial_{x}u_{2}\|_{L^{4}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\partial _{xy}u_{2}\|_{L^{2}} \\ &+C\|\partial_{x}u_{2}\|_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xy}u_{2}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xyy}u_{2}\|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{x}u_{2}\|_{L^{4}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\partial _{yy}u_{1}\|_{L^{2}} \\ &+C\|\partial_{x}u_{2}\|_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{yy}u_{1}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{yyy}u_{1}\|^{\frac{1}{2}}_{L^{2}} \\\leq{}&{C}\|\partial_{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\nabla\partial_{x}u_{2}\| ^{\frac{1}{2}}_{L^{2}}\|\partial_{xx}u_{2} \|_{L^{4}}\|\nabla\omega\|_{L^{2}} \\ &+C\|\omega\|_{L^{2}}\|\nabla\omega\|^{\frac{1}{2}}_{L^{2}}\| \partial _{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\|\nabla \omega\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{x}u_{2}\|^{\frac{1}{2}}_{L^{2}}\| \nabla\partial_{x}u_{2}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\nabla\omega \|_{L^{2}} \\ &+C\|\omega\|_{L^{2}}\|\nabla\omega\|^{\frac{1}{2}}_{L^{2}}\| \partial _{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\|\nabla \omega\|^{\frac{1}{2}}_{L^{2}}\| \partial_{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&{C}\|\omega\|^{\frac{1}{2}}_{L^{2}}\|\nabla\omega \|^{\frac {3}{2}}_{L^{2}}\|\partial_{xx}u_{2} \|_{L^{4}}+C\|\omega\|_{L^{2}}\|\nabla\omega \|_{L^{2}}\| \partial_{xxy}u_{1}\|_{L^{2}} \\ &+{C}\|\omega\|^{\frac{1}{2}}_{L^{2}}\|\nabla\omega\|^{\frac {3}{2}}_{L^{2}} \|\partial_{xx}u_{2}\|_{L^{4}}+C\|\omega \|_{L^{2}}\|\nabla\omega \|_{L^{2}}\|\partial_{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yyy}u_{1}\| ^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\mu_{2}}{8}\|\partial_{xxy}u_{1} \|^{2}_{L^{2}}+\frac{\mu_{2}}{12}\| \partial_{yyy}u_{1} \|^{2}_{L^{2}}+C \bigl(\|\partial_{xx}u_{2} \|_{L^{4}}+\|\omega\| ^{2}_{L^{2}} \bigr)\|\nabla\omega \|^{2}_{L^{2}} \\ &+C\|\partial_{xx}u_{2}\|_{L^{4}}\|\omega \|^{2}_{L^{2}}, \end{aligned}$$
(3.10)
$$\begin{aligned} J_{3}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\omega\partial_{y}\omega {\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{xx}u_{2}\partial_{xy}u_{2}\, dx \,dy+ \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{xx}u_{2}\partial _{yy}u_{1}\,dx \,dy \\ &+ \iint_{\mathbb{R}^{2}}\partial_{y}u_{1}\partial _{xy}u_{1}\partial_{xy}u_{2}\,dx\,dy- \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{xy}u_{1}\partial_{yy}u_{1}\, dx \,dy \\ \leq{}&{C}\|\partial_{y}u_{1}\|_{L^{4}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\partial _{xy}u_{2}\|_{L^{2}}+{C}\|\partial_{y}u_{1} \|_{L^{4}}\|\partial_{xx}u_{2}\|_{L^{4}}\| \partial_{yy}u_{1}\|_{L^{2}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xy}u_{2}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xyy}u_{2}\|^{\frac{1}{2}}_{L^{2}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{yy}u_{1}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{yyy}u_{1}\|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&{C}\|\partial_{y}u_{1}\|^{\frac{1}{2}}_{L^{2}} \|\nabla\partial_{y}u_{1}\| ^{\frac{1}{2}}_{L^{2}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\nabla\omega \|_{L^{2}} \\ &+{C}\|\partial_{y}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \nabla\partial_{y}u_{1}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xx}u_{2}\|_{L^{4}}\|\nabla\omega \|_{L^{2}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\|\nabla\omega \|^{\frac{1}{2}}_{L^{2}}\|\partial _{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\nabla\omega\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\|\nabla\omega \|^{\frac{1}{2}}_{L^{2}}\|\partial _{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\nabla\omega\|^{\frac{1}{2}}_{L^{2}}\| \partial_{yyy}u_{1}\|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&{C}\|\partial_{y}u_{1}\|^{\frac{1}{2}}_{L^{2}} \|\nabla\omega\|^{\frac {3}{2}}_{L^{2}}\|\partial_{xx}u_{2} \|_{L^{4}}+{C}\|\partial_{y}u_{1}\|^{\frac {1}{2}}_{L^{2}} \|\nabla\omega\|^{\frac{3}{2}}_{L^{2}}\|\partial_{xx}u_{2} \| _{L^{4}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\|\nabla\omega \|_{L^{2}}\|\partial_{xxy}u_{1}\| _{L^{2}}+C\| \partial_{y}u_{1}\|_{L^{2}}\|\nabla\omega \|_{L^{2}}\|\partial _{xxy}u_{1}\|^{\frac{1}{2}}_{L^{2}} \|\partial_{yyy}u_{1}\|^{\frac {1}{2}}_{L^{2}} \\ \leq{}&\frac{\mu_{2}}{8}\|\partial_{xxy}u_{1} \|^{2}_{L^{2}}+\frac{\mu_{2}}{12}\| \partial_{yyy}u_{1} \|^{2}_{L^{2}}+C \bigl(\|\partial_{xx}u_{2} \|_{L^{4}}+\|\partial _{y}u_{1}\|^{2}_{L^{2}} \bigr)\|\nabla\omega\|^{2}_{L^{2}} \\ &+C\|\partial_{xx}u_{2}\|_{L^{4}}\| \partial_{y}u_{1}\|^{2}_{L^{2}}, \end{aligned}$$
(3.11)
$$\begin{aligned} J_{4}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{2}( \partial_{y}\omega)^{2}\,dx\,dy \\ ={}&2 \iint_{\mathbb{R}^{2}}{u_{2}}\partial_{xy}u_{2} \partial_{xyy}u_{2}\,dx\, dy+2 \iint_{\mathbb{R}^{2}}u_{2}\partial_{yy}u_{1} \partial_{yyy}u_{1}\,dx\, dy \\ &+2 \iint_{\mathbb{R}^{2}}u_{2}\partial_{xyy}u_{2} \partial_{yy}u_{1}\,dx\,dy+2 \iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}u_{2} \partial_{yyy}u_{1}\,dx\,dy \\ \leq{}&{C}\|\partial_{xyy}u_{2}\|_{L^{2}} \|u_{2}\|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy}u_{2} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}u_{2} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{yyy}u_{1}\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{xyy}u_{2}\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{yyy}u_{1}\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy}u_{2} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}u_{2} \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&{C}\|\partial_{xxy}u_{1}\|_{L^{2}} \|u_{2}\|^{\frac{1}{2}}_{L^{2}}\|\omega\| ^{\frac{1}{2}}_{L^{2}} \|\nabla\omega\|^{\frac{1}{2}}_{L^{2}}\|\partial _{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{yyy}u_{1}\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\omega\| ^{\frac{1}{2}}_{L^{2}}\|\nabla \omega\|^{\frac{1}{2}}_{L^{2}}\|\partial _{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{xxy}u_{1}\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\omega\| ^{\frac{1}{2}}_{L^{2}}\|\nabla \omega\|^{\frac{1}{2}}_{L^{2}}\|\partial _{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ &+{C}\|\partial_{yyy}u_{1}\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\omega\| ^{\frac{1}{2}}_{L^{2}}\|\nabla \omega\|^{\frac{1}{2}}_{L^{2}}\|\partial _{xxy}u_{1} \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{3\mu_{2}}{16}\|\partial_{xxy}u_{1} \|^{2}_{L^{2}}+\frac{3\mu_{2}}{12}\| \partial_{yyy}u_{1} \|^{2}_{L^{2}}+C\|u_{2}\|^{2}_{L^{2}} \|\omega\|^{2}_{L^{2}}\|\nabla \omega\|^{2}_{L^{2}}, \end{aligned}$$
(3.12)
$$\begin{aligned} J_{5}={}& \iint_{\mathbb{R}^{2}}\nabla\partial_{x}\theta\nabla\omega{\,dx}\,dy \leq {C} \bigl(\|\partial_{xx}\theta\|^{2}_{L^{2}}+\|\partial_{yy}\theta\|^{2}_{L^{2}}+\| \nabla\omega\|^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.13)
$$\begin{aligned} J_{6}={}& \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\partial _{xxx}\theta{\,dx}\,dy \\ \leq{}&{C}\|\partial_{xxx}\theta\|_{L^{2}}\|\partial_{x}u_{1} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{x}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xx}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{6}\|\partial_{xxx} \theta\|^{2}_{L^{2}}+C\|\omega\| ^{2}_{L^{2}}\| \nabla\omega\|^{2}_{L^{2}}+C\|\partial_{x}\theta \|^{2}_{L^{2}}\|\partial _{xx}\theta \|^{2}_{L^{2}}, \end{aligned}$$
(3.14)
$$\begin{aligned} J_{7}={}& \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{y}\theta\partial _{xxx}\theta{\,dx}\,dy \\ \leq{}&{C}\|\partial_{xxx}\theta\|_{L^{2}}\|\partial_{x}u_{2} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xx}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{y}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{6}\|\partial_{xxx} \theta\|^{2}_{L^{2}}+C\|\omega\| ^{2}_{L^{2}}\| \nabla\omega\|^{2}_{L^{2}}+C\|\partial_{y}\theta \|^{2}_{L^{2}}\|\partial _{yy}\theta \|^{2}_{L^{2}}, \end{aligned}$$
(3.15)
$$\begin{aligned} J_{8}={}& \iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}\theta \partial_{xxx}\theta{\, dx}\,dy \\ \leq{}&{C}\|\partial_{xxx}\theta\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{6}\|\partial_{xxx} \theta\|^{2}_{L^{2}}+\frac{\kappa _{1}}{14}\|\partial_{xyy} \theta\|^{2}_{L^{2}}+C\|u_{2}\|^{2}_{L^{2}} \|\omega\| ^{2}_{L^{2}} \bigl(\|\partial_{xx}\theta \|^{2}_{L^{2}}+\|\partial_{yy}\theta\| ^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.16)
where we have used the fact \(\|\partial_{xy}\theta\|^{2}_{L^{2}}\leq{C} (\|\partial_{xx}\theta\|^{2}_{L^{2}}+\|\partial_{yy}\theta\|^{2}_{L^{2}} )\);
$$\begin{aligned} J_{9}={}& \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{x}\theta\partial _{yyy}\theta{\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{yy}u_{1} \partial_{x}\theta\partial _{yy}\theta{\,dx}\,dy- \iint_{\mathbb{R}^{2}}\partial_{y}u_{1}\partial _{xy}\theta\partial_{yy}\theta{\,dx}\,dy \\ \leq{}&{C}\|\partial_{x}\theta\|_{L^{2}}\|\partial_{yy}u_{1} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{yy}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac {1}{2}}_{L^{2}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\| \partial_{xy}\theta\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}\theta\|^{\frac{1}{2}}_{L^{2}}\| \partial_{yy}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xyy}\theta\|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&{C}\|\partial_{x}\theta\|_{L^{2}}\|\nabla\omega \|^{\frac{1}{2}}_{L^{2}}\| \partial_{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ &+C\|\partial_{y}u_{1}\|_{L^{2}}\| \partial_{xy}\theta\|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}\theta\|_{L^{2}}\|\partial_{yy}\theta \|^{\frac {1}{2}}_{L^{2}} \\ \leq{}& \frac{\mu_{2}}{12}\|\partial_{yyy}u_{1}\|^{2}_{L^{2}}+ \frac{\kappa_{1}}{7}\| \partial_{xyy}\theta\|^{2}_{L^{2}} \\ & +C \bigl(\|\partial_{x}\theta\|^{2}_{L^{2}}+\| \partial_{y}u_{1}\|^{2}_{L^{2}} \bigr)\bigl(\|\nabla\omega\|^{2}_{L^{2}}+\|\partial_{xx} \theta\|^{2}_{L^{2}}+\| \partial_{yy}\theta \|^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.17)
$$\begin{aligned} J_{10}={}& \iint_{\mathbb{R}^{2}}u_{1}\partial_{xy}\theta \partial_{yyy}\theta {\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{xy}\theta\partial _{yy}\theta{\,dx\,dy}- \iint_{\mathbb{R}^{2}}u_{1}\partial_{xyy}\theta \partial_{yy}\theta{\,dx\,dy} \\ \leq{}&{C}\|\partial_{y}u_{1}\|_{L^{2}}\| \partial_{xy}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xyy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial _{yy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy} \theta\|^{\frac {1}{2}}_{L^{2}} \\ &+C\|\partial_{xyy}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}& \frac{\kappa_{1}}{7}\|\partial_{xyy}\theta\|^{2}_{L^{2}}+C \bigl(\|\partial _{y}u_{1}\|^{2}_{L^{2}}+ \|u_{1}\|^{2}_{L^{2}}\|\partial_{y}u_{1} \|^{2}_{L^{2}} \bigr) \bigl(\|\partial_{xx}\theta \|^{2}_{L^{2}}+\|\partial _{yy}\theta \|^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.18)
$$\begin{aligned} J_{11}={}& \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{y}\theta\partial _{yyy}\theta{\,dx}\,dy \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{yy}u_{2} \partial_{y}\theta\partial _{yy}\theta{\,dx}\,dy- \iint_{\mathbb{R}^{2}}\partial_{y}u_{2}\partial _{yy}\theta\partial_{yy}\theta{\,dx}\,dy \\ \leq{}&{C}\|\partial_{y}\theta\|_{L^{2}}\|\partial_{yy}u_{2} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{yyy}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{yy}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac {1}{2}}_{L^{2}} \\ &+C\|\partial_{yy}\theta\|_{L^{2}}\|\partial_{y}u_{2} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{yy}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{yy}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac {1}{2}}_{L^{2}} \\ \leq{}&{C}\|\partial_{y}\theta\|_{L^{2}}\|\nabla\omega \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ &+C\|\partial_{yy}\theta\|_{L^{2}}\|\omega\|^{\frac{1}{2}}_{L^{2}} \|\partial _{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{yy}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xyy}\theta\|^{\frac{1}{2}}_{L^{2}} \\ \leq{}& \frac{\mu_{2}}{4}\|\partial_{xyy}u_{1}\|^{2}_{L^{2}}+ \frac{\kappa_{1}}{7}\| \partial_{xyy}\theta\|^{2}_{L^{2}} \\ & +C \bigl(\|\partial_{y}\theta\|^{2}_{L^{2}}+\| \omega \|^{\frac{2}{3}}_{L^{2}}\|\partial_{xy}u_{1} \|^{\frac{2}{3}}_{L^{2}} \bigr)\bigl(\|\nabla\omega\|^{2}_{L^{2}}+\|\partial_{yy} \theta\|^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.19)
$$\begin{aligned} J_{12}={}& \iint_{\mathbb{R}^{2}}\partial_{xx}u_{2} \partial_{xx}\theta{\,dx}\, dy\leq{C} \bigl(\|\nabla\omega \|^{2}_{L^{2}}+\|\partial_{xx}\theta\| ^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.20)
$$\begin{aligned} J_{13}={}& \iint_{\mathbb{R}^{2}}\partial_{yy}u_{2} \partial_{yy}\theta{\,dx}\, dy\leq{C} \bigl(\|\nabla\omega \|^{2}_{L^{2}}+\|\partial_{yy}\theta\| ^{2}_{L^{2}} \bigr), \end{aligned}$$
(3.21)
and
$$ \begin{aligned}[b] J_{14}+J_{15}={}& \iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta\partial _{xxx}\theta{\,dx\,dy}+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{yy}\theta \partial_{yyy}\theta{\,dx\,dy} \\ ={}& \iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta \partial_{xxx}\theta{\,dx\, dy}- \iint_{\mathbb{R}^{2}}u_{1}\partial_{yy}\theta\partial _{xyy}\theta{\,dx\,dy} \\ \leq{}&C\|\partial_{xxx}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xx}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xxx}\theta \|^{\frac{1}{2}}_{L^{2}} \\ &+C\|\partial_{xyy}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{8}\|\partial_{xxx}\theta\|^{2}_{L^{2}}+ \frac{\kappa _{1}}{16}\|\partial_{xyy}\theta\|^{2}_{L^{2}}\\ &+C \|u_{1}\|^{2}_{L^{2}}\|\partial _{y}u_{1} \|^{2}_{L^{2}} \bigl(\|\partial_{xx}\theta \|^{2}_{L^{2}}+\|\partial _{yy}\theta \|^{2}_{L^{2}} \bigr). \end{aligned} $$
(3.22)
Hence, inserting the estimates \(J_{1}\)\(J_{15}\) into (3.7), we finally obtain
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl(\big\| \nabla\omega(t)\big\| ^{2}_{L^{2}}+\big\| \partial _{xx} \theta(t)\big\| ^{2}_{L^{2}}+\big\| \partial_{yy}\theta(t) \big\| ^{2}_{L^{2}} \bigr)+\mu_{2}\| \partial_{xxy}u_{1} \|^{2}_{L^{2}}+\mu_{2}\|\partial_{xyy}u_{1} \|^{2}_{L^{2}} \\ &\qquad+\mu_{2}\|\partial_{yyy}u_{1}\|^{2}_{L^{2}}+ \kappa_{1}\|\partial_{xxx}\theta\| ^{2}_{L^{2}}+ \kappa_{1}\|\partial_{xyy}\theta\|^{2}_{L^{2}} \\ &\quad\leq{C} \bigl(\|\partial_{y}u_{1}\|^{2}_{L^{2}}+ \|\omega\|^{2}_{L^{2}}+\|\partial _{x}\theta \|^{2}_{L^{2}}+\|\partial _{y}\theta \|^{2}_{L^{2}}+\|u_{1}\|^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}+\|\omega \|^{\frac{2}{3}}_{L^{2}}\|\partial_{xy}u_{1} \|^{\frac {2}{3}}_{L^{2}} \\ &\qquad+\|u_{2}\|^{2}_{L^{2}} \|\omega\|^{2}_{L^{2}}+\|\partial _{xx}u_{2}\|_{L^{4}}+1 \bigr)\bigl(\|\nabla\omega \|^{2}_{L^{2}}+\|\partial _{xx}\theta \|^{2}_{L^{2}}+\|\partial_{yy}\theta \|^{2}_{L^{2}} \bigr) \\ &\qquad+C\|\partial_{xx}u_{2}\|_{L^{4}} \bigl(\| \partial_{y}u_{1}\|^{2}_{L^{2}}+\|\omega\| ^{2}_{L^{2}} \bigr). \end{aligned}$$
(3.23)
It thus follows from Gronwall’s inequality that
$$ \begin{aligned}[b] &\|\nabla\omega\|^{2}_{L^{2}}+ \|\partial_{xx}\theta\|^{2}_{L^{2}}+\|\partial _{yy}\theta\|^{2}_{L^{2}} +\mu_{2} \int^{t}_{0}\big\| \partial_{xxy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau+\mu_{2} \int^{t}_{0}\big\| \partial_{xyy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau \\ &\quad\quad+\mu_{2} \int^{t}_{0}\big\| \partial_{yyy}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau+\kappa_{1} \int ^{t}_{0}\big\| \partial_{xxx}\theta(\tau) \big\| ^{2}_{L^{2}}\,d\tau+\kappa_{1} \int^{t}_{0}\big\| \partial_{xyy}\theta(\tau) \big\| ^{2}_{L^{2}}\,d\tau\\ &\quad\leq{C}, \end{aligned} $$
(3.24)
which yields
$$ \begin{aligned}[b] &\|\nabla\omega\|_{L^{\infty}([0,T);L^{2}(\mathbb{R}^{2}))}+\| \triangle\theta \|_{L^{\infty}([0,T);L^{2}(\mathbb{R}^{2}))}+\mu_{2}\|\triangle \partial_{y}u_{1}\| _{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))} \\ &\quad+\mu_{2}\|\partial_{xyy}u_{1}\|_{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))}+ \kappa_{1}\| \triangle\partial_{x}\theta\|_{L^{2}([0,T);L^{2}(\mathbb{R}^{2}))} \leq{C}. \end{aligned} $$
(3.25)

Therefore, we have completed the proof of Theorem 3.1. □

Furthermore, we establish another regularity criterion to the 2D Bénard system with vertical dissipation and horizontal thermal diffusivity.

Theorem 3.2

Let \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa _{2}=0\). Given a positive time \(T\in(0,\infty)\). Assume that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{2}(\mathbb{R}^{2})}\) and \(\nabla\cdot{u^{0}}=0\). Let \((u_{1}, u_{2}, \theta)\) be the solution of (1.2)(1.3). If
$$ \int^{T}_{0}\|\partial_{x}u \|^{2}_{L^{\infty}}\,dt< \infty $$
(3.26)
for some \(T>0\), then \(\|(u_{1},u_{2},\theta)\|_{H^{2}}\) is finite on \([0,T]\).

We now give the proof of Theorem 3.2, we first prove the global \(H^{1}\) bound for \((u_{1},u_{2},\theta)\), we present the proof of the main theorem secondly.

• Global \(H^{1}\)-bound for \((u_{1},u_{2},\theta)\)

Proposition 3.3

Let \((u^{0}_{1}, u^{0}_{2}, \theta^{0})\in{H}^{2}(\mathbb{R}^{2})\) and let \((u_{1},u_{2},\theta)\) be the corresponding solution of (1.2)(1.3). Then \((u_{1},u_{2},\theta)\) obeys the following global \(L^{2}\)-bound:
$$ \begin{aligned}[b] &\|u_{1}\|^{2}_{L^{2}}+\|u_{2} \|^{2}_{L^{2}}+\|\theta\|^{2}_{L^{2}}+ \mu_{2} \int^{t}_{0}\big\| \partial_{y}u_{1}( \tau)\big\| ^{2}_{L^{2}}\,d\tau+\kappa_{1} \int^{t}_{0}\big\| \partial_{x}\theta (\tau) \big\| ^{2}_{L^{2}}\,d\tau\\ &\quad \leq{C}\big\| \bigl(u^{1}_{0},u^{0}_{2}, \theta^{0}\bigr)\big\| ^{2}_{L^{2}}\end{aligned} $$
(3.27)
for any \(t\geq0\). Here C is a constant depending only on \(\mu_{2}\), \(\kappa_{1}\), and T.

Proposition 3.4

Assume that \((u^{0}_{1},u^{0}_{2},\theta^{0})\in{H}^{2}(\mathbb{R}^{2})\), \(\nabla\cdot {u^{0}}=0\). Let \((u_{1}, u_{2}, \theta)\) be the corresponding solution of (1.2)(1.3). Then, for any \(T>0\) and \(t\leq{T}\),
$$ \big\| \bigl(u_{1}(t), u_{2}(t), \theta(t)\bigr) \big\| _{H^{1}(\mathbb{R}^{2})}\leq{C_{1}}e^{C_{2}\int ^{t}_{0}\|\partial_{x}u\|^{2}_{L^{\infty}}\,d\tau}, $$
(3.28)
where \(C_{1}\) is a constant depending on T and initial data, and \(C_{2}\) is a pure constant.

Proof

Taking the inner product of the first three equations in (1.2) with \(-\Delta{u_{1}}\), \(-\Delta{u_{2}}\) and \(-\Delta\theta\), respectively, integrating with respect to space
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt} \bigl(\big\| \nabla{u_{1}}(t)\big\| ^{2}_{L^{2}}+\big\| \nabla {u_{2}}(t)\big\| ^{2}_{L^{2}}+\big\| \nabla\theta(t) \big\| ^{2}_{L^{2}} \bigr)+\mu_{2}\|\nabla \partial_{y}u_{1}\|^{2}_{L^{2}}+ \kappa_{1}\|\nabla\partial_{x}\theta\|^{2}_{L^{2}} \\ &\quad=\underbrace{- \iint_{\mathbb{R}^{2}}\theta\cdot\Delta{u_{2}}\,dx\, dy}_{K_{1}}+\underbrace{ \iint_{\mathbb{R}^{2}}\bigl[(u\cdot\nabla)\theta\bigr]\cdot \Delta\theta{ \,dx\,dy}}_{K_{2}}\underbrace{- \iint_{\mathbb{R}^{2}}u_{2}\cdot \Delta\theta{\,dx \,dy}}_{K_{3}}; \end{aligned} $$
(3.29)
then, for notational convenience, we set
$$ A(t)=\big\| \nabla{u_{1}}(\cdot,t)\big\| ^{2}_{L^{2}}+ \big\| \nabla{u_{2}}(\cdot,t)\big\| ^{2}_{L^{2}}+\big\| \nabla\theta( \cdot,t)\big\| ^{2}_{L^{2}}. $$
(3.30)
For \(K_{1}\) and \(K_{3}\), integrating by parts and applying the Hölder inequality gives
$$ K_{1}+K_{3}\leq\|\nabla{u} \|^{2}_{L^{2}}+\|\nabla\theta\|^{2}_{L^{2}}. $$
(3.31)
To estimate \(K_{2}\) we write component-wise
$$\begin{aligned}& \begin{aligned}[b] K_{2}={}&{-} \iint_{\mathbb{R}^{2}}\nabla\theta\cdot\nabla{u}\cdot\nabla\theta {\,dx \,dy} \\ ={}&\underbrace{ -\iint_{\mathbb{R}^{2}}\partial_{x}\theta\partial_{x}u_{1} \partial _{x}\theta{\,dx\,dy}}_{K_{21}}\underbrace{- \iint_{\mathbb{R}^{2}}\partial _{x}\theta\partial_{x}u_{2} \partial_{y}\theta{\,dx\,dy}}_{K_{22}}\underbrace {- \iint_{\mathbb{R}^{2}}\partial_{y}\theta\partial_{y}u_{1} \partial_{x}\theta{\, dx\,dy}}_{K_{23}} \\ &\underbrace{- \iint_{\mathbb{R}^{2}}\partial_{y}\theta\partial_{y}u_{2} \partial _{y}\theta{\,dx\,dy}}_{K_{24}}, \end{aligned} \end{aligned}$$
(3.32)
$$\begin{aligned}& \begin{aligned}[b] K_{21}={}&2 \iint_{\mathbb{R}^{2}}u_{1}\partial_{x}\theta \partial_{xx}\theta{\, dx\,dy} \\ \leq{}&C\|\partial_{xx}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{x}\theta \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xx}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{6}\|\nabla\partial_{x}\theta \|^{2}_{L^{2}}+C\|u_{1}\| ^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}\|\nabla \theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.33)
$$\begin{aligned}& K_{22}\leq{C}\|\partial_{x}u_{2} \|_{L^{\infty}}\|\nabla\theta\|^{2}_{L^{2}}, \end{aligned}$$
(3.34)
$$\begin{aligned}& \begin{aligned}[b] K_{23}\leq{}&C\| \partial_{y}u_{1}\|_{L^{2}}\|\partial_{y} \theta\|^{\frac {1}{2}}_{L^{2}}\|\partial_{xy}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy}\theta \|^{\frac {1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{6}\|\nabla\partial_{x}\theta \|^{2}_{L^{2}}+C\|\partial _{y}u_{1} \|^{2}_{L^{2}}\|\nabla\theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.35)
$$\begin{aligned}& \begin{aligned}[b] K_{24}={}&{-}2 \iint_{\mathbb{R}^{2}}u_{1}\partial_{y}\theta \partial_{xy}\theta{\, dx\,dy} \\ \leq{}&{C}\|\partial_{xy}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{y}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{6}\|\nabla\partial_{x}\theta \|^{2}_{L^{2}}+C\|u_{1}\| ^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}\|\nabla \theta\|^{2}_{L^{2}}. \end{aligned} \end{aligned}$$
(3.36)
After combining inequalities,
$$ \begin{aligned}[b]\frac{1}{2}\frac{d}{dt}A(t)+ \mu_{2}\|\nabla\partial_{y}u_{1} \|^{2}_{L^{2}}+\kappa _{1}\|\nabla\partial_{x} \theta\|^{2}_{L^{2}}\leq{}&\frac{\mu_{2}}{2}\|\nabla\partial _{y}u_{1}\|^{2}_{L^{2}}+\frac{\kappa_{1}}{2}\| \nabla\partial_{x}\theta\|^{2}_{L^{2}} \\ &+C\|\partial_{x}u_{2}\|^{2}_{L^{\infty}}A(t), \end{aligned} $$
(3.37)
after applying Gronwall’s lemma, we get the \(H^{1}\)-norm for \(u_{1}\), \(u_{2}\), and θ. This completes the proof of Proposition 3.4. □

\(H^{2}\) bound

To estimate the \(H^{2}\)-norm of \((u_{1}, u_{2}, \theta)\), we consider the equation of \(\omega=\nabla\times{u}\), θ
$$ \left\{\textstyle\begin{array}{l} \partial_{t}\omega+(u\cdot\nabla)\omega=-\mu_{2}\partial_{yyy}u_{1}+\partial _{x}\theta,\\ \partial_{t}\nabla\theta+\nabla[(u\cdot\nabla)\theta]=\kappa_{1}\nabla \partial_{xx}\theta+\nabla{u_{2}}. \end{array}\displaystyle \right. $$
(3.38)

Proposition 3.5

Assume that \((u^{0}_{1},u^{0}_{2},\theta^{0})\in{H}^{2}(\mathbb{R}^{2})\), \(\nabla\cdot {u^{0}}=0\) and let \((u_{1},u_{2},\theta)\) be the solution of (1.2)(1.3). Then \((\omega,\nabla\theta)\) satisfy
$$ \|\omega\|^{2}_{L^{2}}+\|\Delta\theta \|^{2}_{L^{2}}+\mu_{2} \int^{T}_{0}\|\Delta \partial_{y}u_{1} \|^{2}_{L^{2}}\,dt+\kappa_{1} \int^{T}_{0}\|\Delta\partial_{x}\theta\| ^{2}_{L^{2}}\,dt\leq{C} $$
(3.39)
if \(\partial_{x}u\in{L}^{2}([0,T];L^{\infty}(\mathbb{R}^{2}))\).

Proof

Taking the inner product of (3.38)1 with \(-\Delta\omega\) and (3.38)2 with \(-\Delta\nabla\theta\) in \(L^{2}(\mathbb{R}^{2})\), respectively, we find
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt} \bigl(\|\nabla\omega\|^{2}_{L^{2}}+\|\Delta\theta\| ^{2}_{L^{2}} \bigr)+\mu_{2}\|\Delta \partial_{y}u_{1}\|^{2}_{L^{2}}+ \kappa_{1}\|\Delta \partial_{x}\theta\|^{2}_{L^{2}} \\ &\quad= \underbrace{\iint_{\mathbb{R}^{2}}(u\cdot\nabla)\omega\cdot\Delta\omega{\,dx\, dy}}_{L_{1}}+ \underbrace{\iint_{\mathbb{R}^{2}}\nabla\partial_{x}\theta\cdot\nabla\omega{\,dx\, dy}}_{L_{2}} \\ &\qquad\underbrace{- \iint_{\mathbb{R}^{2}}\Delta\bigl[(u\cdot\nabla)\theta\bigr]\Delta\theta{ \,dx\, dy}}_{L_{3}}+ \underbrace{\iint_{\mathbb{R}^{2}}\Delta{u_{2}}\cdot\Delta\theta{\,dx\,dy}}_{L_{4}}. \end{aligned} $$
(3.40)
We now estimate \(L_{1}\) through \(L_{4}\). We firstly write the four terms in \(L_{1}\) explicitly,
$$ \begin{aligned}[b] L_{1}={}&{-} \iint_{\mathbb{R}^{2}}\nabla\omega\cdot\nabla{u}\cdot\nabla\omega {\,dx \,dy} \\ ={}&\underbrace{- \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\omega \partial_{x}\omega{\,dx \,dy}}_{L_{11}}\underbrace{- \iint_{\mathbb {R}^{2}}\partial_{x}u_{2} \partial_{x}\omega\partial_{y}\omega{\,dx\, dy}}_{L_{12}} \\ &\underbrace{- \iint_{\mathbb{R}^{2}}\partial_{y}u_{1}\partial _{x}\omega\partial_{y}\omega{\,dx\,dy}}_{L_{13}}\underbrace{- \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{y}\omega\partial _{y}\omega{\,dx \,dy}}_{L_{14}}. \end{aligned} $$
(3.41)
These terms can be bounded as follows:
$$\begin{aligned}& L_{11}\leq\|\partial_{x}u_{1} \|_{L^{\infty}}\|\partial_{x}\omega\|^{2}_{L^{2}} \leq\| \partial_{x}u_{1}\|_{L^{\infty}}\|\nabla\omega \|^{2}_{L^{2}}, \end{aligned}$$
(3.42)
$$\begin{aligned}& L_{12}\leq\|\partial_{x}u_{2} \|_{L^{\infty}}\|\partial_{x}\omega\|_{L^{2}}\| \partial_{y}\omega\|_{L^{2}}\leq\|\partial_{x}u_{2} \|_{L^{\infty}}\|\nabla\omega\| ^{2}_{L^{2}}, \end{aligned}$$
(3.43)
$$\begin{aligned}& \begin{aligned}[b] L_{13}\leq{}&C\| \partial_{x}\omega\|_{L^{2}}\|\partial_{y}u_{1} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{y}\omega \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\omega \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\mu_{2}}{6}\|\Delta\partial_{y}u_{1} \|^{2}_{L^{2}}+C\|\partial_{y}u_{1}\| ^{\frac{2}{3}}_{L^{2}}\|\nabla\partial_{y}u_{1} \|^{\frac{2}{3}}_{L^{2}}\|\nabla \omega\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.44)
$$\begin{aligned}& \begin{aligned}[b] L_{14}={}&2 \iint_{\mathbb{R}^{2}}u_{2}\partial_{y}\omega \partial_{yy}\omega{\, dx\,dy} \\ \leq{}&C\|\partial_{yy}\omega\|_{L^{2}}\|u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{y}\omega \|^{\frac{1}{2}}_{L^{2}}\| \partial_{yy}\omega \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\mu_{2}}{6}\|\Delta\partial_{y}u_{1} \|^{2}_{L^{2}}+C\|u_{2}\|^{2}_{L^{2}} \| \nabla{u}\|^{2}_{L^{2}}\|\nabla\omega\|^{2}_{L^{2}}. \end{aligned} \end{aligned}$$
(3.45)
The terms \(L_{2}\) and \(L_{4}\) can easily be bounded,
$$\begin{aligned}& L_{2}\leq\|\underbrace{\nabla\omega\|^{2}_{L^{2}}+\|\Delta \theta\|^{2}_{L^{2}}}_{\|\nabla\partial_{x}\theta\|_{L^{2}}\leq\|\Delta\theta\|_{L^{2}}}, \end{aligned}$$
(3.46)
$$\begin{aligned}& L_{4}\leq\underbrace{\|\nabla\omega\|^{2}_{L^{2}}+ \|\Delta\theta\| ^{2}_{L^{2}}}_{(\|\Delta{u_{2}}\|_{L^{2}}\leq\|\nabla\omega\|_{L^{2}})}. \end{aligned}$$
(3.47)
Finally, we deal with \(L_{3}\).
$$ L_{3}=- \iint_{\mathbb{R}^{2}}\Delta(u_{1}\partial_{x} \theta+u_{2}\partial _{y}\theta)\Delta\theta{\,dx\,dy} \triangleq{L}_{31}+L_{32}. $$
(3.48)
We first split \(L_{31}\) and \(L_{32}\) each into two terms,
$$\begin{aligned}& L_{31}=- \iint_{\mathbb{R}^{2}}\partial_{xx}(u_{1} \partial_{x}\theta +u_{2}\partial_{y}\theta)\Delta \theta{\,dx\,dy}\triangleq {L}_{311}+L_{312}, \end{aligned}$$
(3.49)
$$\begin{aligned}& \begin{aligned}[b] L_{32}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{yy}(u_{1} \partial_{x}\theta +u_{2}\partial_{y}\theta) \partial_{xx}\theta{\,dx\,dy}- \iint_{\mathbb {R}^{2}}\partial_{yy}(u_{1} \partial_{x}\theta+u_{2}\partial_{y}\theta) \partial _{yy}\theta{\,dx\,dy} \\ &\triangleq{L}_{321}+L_{322}.\end{aligned} \end{aligned}$$
(3.50)
These terms are bounded as follows:
$$\begin{aligned}& \begin{aligned}[b] L_{311}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{xx}(u_{1} \partial_{x}\theta)\Delta \theta{\,dx\,dy} \\ ={}& \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\Delta\partial _{x}\theta{\,dx\,dy}+ \iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta\Delta \partial_{x}\theta{\,dx\,dy} \\ \leq{}&C\|\Delta\partial_{x}\theta\|_{L^{2}}\| \partial_{x}u_{1}\|^{\frac {1}{2}}_{L^{2}}\| \partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\| \partial_{x}\theta \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xx}\theta\|^{\frac{1}{2}}_{L^{2}} \\ &+C\|\Delta\partial_{x}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xx}\theta \|^{\frac{1}{2}}_{L^{2}}\| \partial_{xxx}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{18}\| \Delta\partial_{x}\theta\|^{2}_{L^{2}}+C\|\nabla {u} \|^{2}_{L^{2}}\|\nabla\omega\|^{2}_{L^{2}}\\ &+C \bigl(\|\nabla\theta\|^{2}_{L^{2}}+\|u_{1}\| ^{2}_{L^{2}}\|\partial_{y}u_{1} \|^{2}_{L^{2}}\bigr)\|\Delta\theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.51)
$$\begin{aligned}& \begin{aligned}[b] L_{312}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{xx}(u_{2} \partial_{y}\theta)\Delta \theta{\,dx\,dy} \\ ={}& \iint_{\mathbb{R}^{2}}\partial_{x}u_{2} \partial_{y}\theta\Delta\partial _{x}\theta{\,dx\,dy}+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}\theta\Delta \partial_{x}\theta{\,dx\,dy} \\ \leq{}&\|\partial_{x}u_{2}\|_{L^{\infty}}\| \partial_{y}\theta\|_{L^{2}}\|\Delta \partial_{x} \theta\|_{L^{2}}\\&+C\|\Delta\partial_{x}\theta\|_{L^{2}} \|u_{2}\|^{\frac {1}{2}}_{L^{2}}\|\partial_{x}u_{2} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xy}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ \leq{}&\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}}+C\|\partial _{x}u_{2} \|^{2}_{L^{\infty}}\|\nabla\theta\|^{2}_{L^{2}}+C \|u_{2}\|^{2}_{L^{2}}\|\nabla {u}\|^{2}_{L^{2}} \|\nabla\partial_{x}\theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.52)
$$\begin{aligned}& \begin{aligned}[b] L_{321}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{yy}(u_{1} \partial_{x}\theta +u_{2}\partial_{y}\theta) \partial_{xx}\theta{\,dx\,dy} \\ ={}& \iint_{\mathbb{R}^{2}}\partial_{x}u_{1} \partial_{x}\theta\partial_{xyy}\theta {\,dx\,dy}+ \iint_{\mathbb{R}^{2}}u_{1}\partial_{xx}\theta\partial _{xyy}\theta{\,dx\,dy} \\ &+ \iint_{\mathbb{R}^{2}}\partial_{x}u_{2}\partial _{y}\theta\partial_{xyy}\theta{\,dx\,dy}+ \iint_{\mathbb{R}^{2}}u_{2}\partial_{xy}\theta \partial_{xyy}\theta{\,dx\, dy} \\ \leq{}&\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}}+C\|\partial _{x}u \|^{2}_{L^{\infty}}\|\nabla\theta\|^{2}_{L^{2}}+C\|u_{1}\|^{2}_{L^{2}}\|\nabla{u}\|^{2}_{L^{2}}\|\Delta\theta\|^{2}_{L^{2}}\\ &+C \|u_{2}\|^{2}_{L^{2}}\|\nabla {u}\|^{2}_{L^{2}} \|\nabla\partial_{x}\theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.53)
$$\begin{aligned}& \begin{aligned}[b] L_{322}={}&{-} \iint_{\mathbb{R}^{2}}\partial_{yy}(u_{1} \partial_{x}\theta +u_{2}\partial_{y}\theta) \partial_{yy}\theta{\,dx\,dy} \\ ={}&{-} \iint_{\mathbb{R}^{2}}\partial_{y}(\partial_{y}u_{1} \partial_{x}\theta +u_{1}\partial_{xy}\theta) \partial_{yy}\theta{\,dx\,dy}\\ &- \iint_{\mathbb {R}^{2}}\partial_{y}(\partial_{y}u_{2} \partial_{y}\theta+u_{2}\partial_{yy}\theta ) \partial_{yy}\theta{\,dx\,dy} \\ ={}&{-} \iint_{\mathbb{R}^{2}}(\partial_{yy}u_{1} \partial_{x}\theta+2\partial _{y}u_{1} \partial_{xy}\theta+u_{1}\partial_{xyy}\theta) \partial_{yy}\theta{\, dx\,dy} \\ &- \iint_{\mathbb{R}^{2}}(\partial_{yy}u_{2} \partial_{y}\theta+2\partial _{y}u_{2} \partial_{yy}\theta+u_{2}\partial_{yyy}\theta) \partial_{yy}\theta{\, dx\,dy}, \end{aligned} \end{aligned}$$
(3.54)
$$\begin{aligned}& \begin{aligned}[b] & \bigg|{-} \iint_{\mathbb{R}^{2}}\partial_{yy}u_{1} \partial_{x}\theta\partial _{yy}\theta{\,dx\,dy} \bigg| \\ &\quad\leq C\|\partial_{x}\theta\|_{L^{2}}\|\partial_{yy}u_{1} \|^{\frac {1}{2}}_{L^{2}}\|\partial_{yyy}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial _{yy}\theta \|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac {1}{2}}_{L^{2}} \\ &\quad\leq\frac{\mu_{2}}{6}\|\Delta\partial_{y}u_{1} \|^{2}_{L^{2}}+\frac{\kappa _{1}}{18}\|\Delta\partial_{x} \theta\|^{2}_{L^{2}}+C\|\partial_{x}\theta\| ^{2}_{L^{2}}\bigl(\|\nabla\omega\|^{2}_{L^{2}}+ \|\Delta\theta\|^{2}_{L^{2}}\bigr), \end{aligned} \end{aligned}$$
(3.55)
$$\begin{aligned}& \begin{aligned}[b] & \bigg|-2 \iint_{\mathbb{R}^{2}}\partial_{y}u_{1} \partial_{xy}\theta\partial _{yy}\theta{\,dx\,dy} \bigg| \\ &\quad\leq C\|\partial_{yy}\theta \|_{L^{2}}\|\partial_{y}u_{1}\|^{\frac {1}{2}}_{L^{2}} \|\partial_{xy}u_{1}\|^{\frac{1}{2}}_{L^{2}}\|\partial _{xy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy} \theta\|^{\frac {1}{2}}_{L^{2}} \\ &\quad\leq\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}}+C\|\partial _{y}u_{1} \|^{2}_{L^{2}}\|\nabla\omega\|^{2}_{L^{2}}+C\| \nabla\partial_{x}\theta\| _{L^{2}}\|\Delta\theta \|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.56)
$$\begin{aligned}& \begin{aligned}[b] &\bigg| -\iint_{\mathbb{R}^{2}}u_{1}\partial_{xyy}\theta \partial_{yy}\theta {\,dx\,dy}\bigg| \\ &\quad\leq C\|\partial_{xyy}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ &\quad\leq\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}}+C\|u_{1}\| ^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}\|\Delta \theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.57)
$$\begin{aligned}& \begin{aligned}[b] & \bigg|{-} \iint_{\mathbb{R}^{2}}\partial_{yy}u_{2} \partial_{y}\theta\partial _{yy}\theta{\,dx\,dy} \bigg| \\ &\quad\leq{C}\|\partial_{yy}u_{2}\|_{L^{2}}\| \partial_{y}\theta\|^{\frac {1}{2}}_{L^{2}}\| \partial_{yy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial _{yy}\theta\|^{\frac{1}{2}}_{L^{2}}\|\partial_{xyy} \theta\|^{\frac {1}{2}}_{L^{2}} \\ &\quad\leq\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}}+C\|\nabla \theta\|^{2}_{L^{2}}+C \|\nabla\partial_{y}u_{1}\|^{2}_{L^{2}}\| \Delta\theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.58)
$$\begin{aligned}& \begin{aligned}[b] & \bigg|{-}2 \iint_{\mathbb{R}^{2}}\partial_{y}u_{2} \partial_{yy}\theta\partial _{yy}\theta{\,dx\,dy} \bigg| \\&\quad= \bigg|-4 \iint_{\mathbb{R}^{2}}u_{1}\partial_{yy}\theta \partial_{xyy}\theta {\,dx\,dy} \bigg| \\ &\quad\leq C\|\partial_{xyy}\theta\|_{L^{2}}\|u_{1} \|^{\frac{1}{2}}_{L^{2}}\| \partial_{y}u_{1} \|^{\frac{1}{2}}_{L^{2}}\|\partial_{yy}\theta \|^{\frac {1}{2}}_{L^{2}}\|\partial_{xyy}\theta \|^{\frac{1}{2}}_{L^{2}} \\ &\quad\leq\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}}+C\|u_{1}\| ^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}\|\Delta \theta\|^{2}_{L^{2}}, \end{aligned} \end{aligned}$$
(3.59)
$$\begin{aligned}& \begin{aligned}[b] & \bigg|{-} \iint_{\mathbb{R}^{2}}u_{2}\partial_{yyy}\theta \partial_{yy}\theta {\,dx\,dy} \bigg| \\&\quad= \bigg| \iint_{\mathbb{R}^{2}}u_{1}\partial_{yy}\theta \partial_{xyy}\theta {\,dx\,dy} \bigg| \\ &\quad\leq\frac{\kappa_{1}}{18}\|\Delta\partial_{x}\theta \|^{2}_{L^{2}} +C\|u_{1}\|^{2}_{L^{2}} \|\partial_{y}u_{1}\|^{2}_{L^{2}}\|\Delta \theta\|^{2}_{L^{2}}. \end{aligned} \end{aligned}$$
(3.60)

After combining all inequalities, together with Gronwall’s inequality, we obtain the \(H^{1}\)-bound for ω, θ. Therefore, we obtain the global \(H^{2}\) bound for \((u,\theta)\) for the 2D Bénard system (1.2)–(1.3) with vertical dissipation in the horizontal velocity equation and horizontal dissipation in the temperature equation. We thus complete the proof of Theorem 3.2. □

4 Global existence and regularity criteria of weak solution for Bénard system with other partial dissipation

We devote this section to showing the global weak solution and regularity criteria for the 2D Bénard system with other cases for partial viscosity and thermal diffusivity. More precisely, we shall show the following cases and theorems.

Theorem 4.1

Let (i) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (ii) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa _{1}>0\), \(\kappa_{2}=0\). Suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{1}(\mathbb{R}^{2})}\) and \(\nabla\cdot {u^{0}}=0\). Then the problem (1.2)(1.3) with (i) and (ii) admits a global weak solution \((u_{1}, u_{2}, \theta)\), which obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial _{x}u_{2},\partial_{y}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr); \\ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial _{x}u_{2},\partial_{x}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr)\end{gathered} $$
for any \(T>0\), respectively. Moreover, suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{2}(\mathbb{R}^{2})}\) and \(\nabla\cdot{u^{0}}=0\). If one of the following two conditions holds:
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{yy}u_{1}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty ; \end{aligned}$$
(4.1)
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{y}u_{1}( \tau)\big\| _{L^{\infty}(\mathbb{R}^{2})}\,d\tau < \infty \end{aligned}$$
(4.2)
for any fixed \(T>0\), then the problem (1.2)(1.3) with (i) and (ii) admits a global classical solution \((u_{1}, u_{2}, \theta)\), which, respectively, obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\ \partial _{xxy}u_{2}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial_{x}u_{2}, \partial _{y}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr); \\ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\ \partial _{xxy}u_{2}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr), \qquad\partial_{x}u_{2}, \partial _{x}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr).\end{gathered} $$

Theorem 4.2

Let (iii) \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (iv) \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa _{1}=0\), \(\kappa_{2}>0\). Suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{1}(\mathbb{R}^{2})}\) and \(\nabla\cdot {u^{0}}=0\). Then the problem (1.2)(1.3) with (iii) and (iv) admits a global weak solution \((u_{1}, u_{2}, \theta)\), which obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial _{x}u_{1},\partial_{x}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr); \\ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial _{x}u_{1},\partial_{y}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr)\end{gathered} $$
for any \(T>0\), respectively. Furthermore, suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{2}(\mathbb{R}^{2})}\) and \(\nabla\cdot{u^{0}}=0\). If one of the following three conditions holds:
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{yy}u_{2}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty ; \end{aligned}$$
(4.3)
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{xy}u_{1}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty ; \end{aligned}$$
(4.4)
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{y}u_{2}( \tau)\big\| _{L^{\infty}(\mathbb{R}^{2})}\,d\tau < \infty \end{aligned}$$
(4.5)
for any fixed \(T>0\), then the problem (1.2)(1.3) with (iii) and (iv) admits a global classical solution \((u_{1}, u_{2}, \theta)\), which, respectively, obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\ \partial _{xxy}u_{1}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial_{x}u_{1}, \partial _{x}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr); \\ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\ \partial _{xxy}u_{1}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial_{x}u_{1}, \partial _{y}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr).\end{gathered} $$

Theorem 4.3

Let (v) \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\). Suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{1}(\mathbb {R}^{2})}\) and \(\nabla\cdot{u^{0}}=0\). Then the problem (1.2)(1.3) with (v) admits a global weak solution \((u_{1}, u_{2}, \theta)\), which obeys
$$ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr), \qquad\partial _{y}u_{1},\partial_{y}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr) $$
for any \(T>0\). Moreover, suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in {H^{2}(\mathbb{R}^{2})}\) and \(\nabla\cdot{u^{0}}=0\). If one of the following two conditions holds:
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{xx}u_{2}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty ; \end{aligned}$$
(4.6)
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{x}u_{2}( \tau)\big\| _{L^{\infty}(\mathbb{R}^{2})}\,d\tau < \infty \end{aligned}$$
(4.7)
for any fixed \(T>0\), then the problem (1.2)(1.3) with (v) admits a global classical solution \((u_{1}, u_{2}, \theta)\), which obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\ \partial _{xyy}u_{1}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial_{y}u_{1}, \partial _{y}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr).\end{gathered} $$

Theorem 4.4

Let (vi) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}>0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\) and (vii) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}>0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\). Suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{1}(\mathbb{R}^{2})}\) and \(\nabla\cdot {u^{0}}=0\). Then the problem (1.2)(1.3) with (vi) and (vii) admits a global weak solution \((u_{1}, u_{2}, \theta)\), which obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr),\quad\quad \partial _{y}u_{2},\partial_{x}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr); \\ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr), \qquad\partial _{y}u_{2},\partial_{y}\theta\in{L^{2}} \bigl([0,T);H^{1}\bigl(\mathbb{R}^{2}\bigr)\bigr)\end{gathered} $$
for any \(T>0\), respectively. Furthermore, suppose that \(u^{0}_{1}, u^{0}_{2}, \theta^{0}\in{H^{2}(\mathbb{R}^{2})}\) and \(\nabla\cdot{u^{0}}=0\). If one of the following three conditions holds:
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{xx}u_{1}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty ; \end{aligned}$$
(4.8)
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{xy}u_{2}( \tau)\big\| _{L^{4}(\mathbb{R}^{2})}\,d\tau< \infty ; \end{aligned}$$
(4.9)
$$\begin{aligned}& \int^{T}_{0}\big\| \partial_{x}u_{1}( \tau)\big\| _{L^{\infty}(\mathbb{R}^{2})}\,d\tau < \infty \end{aligned}$$
(4.10)
for any fixed \(T>0\), then the problem (1.2)(1.3) admits a global classical solution \((u_{1}, u_{2}, \theta)\), which, respectively, obeys
$$\begin{gathered} u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\\partial _{xyy}u_{2}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr), \qquad\partial_{y}u_{2}, \partial _{x}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr); \\ u_{1}, u_{2},\theta\in{L^{\infty}}\bigl([0,T );H^{2}\bigl(\mathbb{R}^{2}\bigr)\bigr),\\ \partial _{xyy}u_{2}\in{L^{2}}\bigl([0,T);L^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr),\qquad \partial_{y}u_{2}, \partial _{y}\theta\in{L^{2}}\bigl([0,T);H^{2} \bigl(\mathbb{R}^{2}\bigr)\bigr).\end{gathered} $$

Due to those theorems’ proofs being similar to the results of Sect. 3, we can leave the proofs of those theorems to the interested readers.

5 Conclusion

The Bénard fluid problem is a very classical problem in the fluid dynamics area. The global existence or non-existence of the classical solution to an inviscid Bénard system is an open and challenging problem, even in the two-dimensional case. Therefore, it is of interest to consider the Bénard system with partial viscosity. Inspired by recent work [13, 31, 32], we first consider the global weak solution for the 2D Bénard system with partial dissipation. Secondly, we establish some regularity criteria for the corresponding system.

Declarations

Acknowledgements

The author would like to thank professor Lili Du for many helpful discussions and suggestions. This paper is partially supported by The National Natural Science Foundation of China (No. 61563044, 71702119, 61762074); National Natural Science Foundation of Qinghai Province (2017-ZJ-902); Open Research Fund Program of State key Laboratory of Hydroscience and Engineering (No. sklhse-2017-A-05); The Science and Technology Major Project of Qinghai Province Natural Science Foundation (No. 2015-SF-A4-3).

Availability of data and materials

Not applicable.

Funding

Not applicable.

Authors’ contributions

LM conceived and designed the work. LM drafted the manuscript. LM revised the manuscript. LM and LZ read and approved the final version.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics and Computer, Panzhihua University, Panzhihua, P.R. China
(2)
College of Computer Science, Sichuan University, Chengdu, P.R. China
(3)
State Key Laboratory of Plateau Ecology and Agriculture, Department of Computer Technology and Applications, Qinghai University, Xining, P.R. China
(4)
Information Management Center, Sichuan University, Chengdu, P.R. China

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