Regularity of large solutions for the compressible magnetohydrodynamic equations

Abstract

In this paper, we consider the initial-boundary value problem of one-dimensional compressible magnetohydrodynamics flows. The existence and continuous dependence of global solutions in H¹ have been established in Chen and Wang (Z Angew Math Phys 54, 608-632, 2003). We will obtain the regularity of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.

1 Introduction

Magnetohydrodynamics (MHD) is concerned with the flow of electrically conducting fluids in the presence of magnetic fields, either externally applied or generated within the fluid by inductive action. The application of magnetohydrodynamics covers a very wide range of physical areas from liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. There is a complex interaction between the magnetic and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. For convenience, we consider the following plane magnetohydrodynamic equations in the Lagrangian coordinate system:

$$v_t-u_y=0,$$

(1.1)

$$u_t+{\left(p+\frac{1}{2}\mid \mathit{b}{\mid }^2\right)}_y={\left(\frac{\lambda u_y}{v}\right)}_y,$$

(1.2)

$${\mathit{w}}_t-{\mathit{b}}_y={\left(\frac{\mu {\mathit{w}}_y}{v}\right)}_y,$$

(1.3)

$${\left(v\mathit{b}\right)}_t-{\mathit{w}}_y={\left(\frac{\nu {\mathit{b}}_y}{v}\right)}_y,$$

(1.4)

$$E_t+{\left(u\left(p+\frac{1}{2}\mid \mathit{b}{\mid }^2\right)-\mathit{w}\cdot \mathit{b}\right)}_y={\left(\frac{\lambda u{u}_y+\mu \mathit{w}\cdot {\mathit{w}}_y+\nu \mathit{b}\cdot {\mathit{b}}_y+\kappa {\theta }_y}{v}\right)}_y.$$

(1.5)

Here, v, u, w , b , θ, and p are the specific volume, the longitudinal velocity, the transverse velocity, the transverse magnetic filed, the absolute temperature, and the pressure, respectively; λ, μ, ν, and κ are the bulk viscosity coefficient, the shear viscosity coefficient, the magnetic diffusivity, and the heat conductivity, respectively.

We consider problem (1.1)-(1.5) in the region {y ∈ Ω: = (0, 1), t ≥ 0} under the initial-boundary conditions

$$\left(v,u,\mathit{w},\mathit{b},\theta \right){|}_{t=0}=\left(v_0,u_0,{\mathit{w}}_0,{\mathit{b}}_0,\theta \right)\left(y\right),y\in \Omega ,$$

(1.6)

$$\left(u,\mathit{w},\mathit{b},{\theta }_y\right){|}_{\partial \Omega }=0.$$

(1.7)

In this paper, we focus on an initial-boundary problem for the magnetohydrodynamic flows of a perfect gas with following equations of state:

$$p=\frac{R\theta }{v},e=c_v\theta ,$$

where R is the gas constant and c _v is the heat capacity of the gas at constant volume. For concreteness, we assume that λ, μ, and ν are constants, and κ depends on the temperature θ with C₁ ≤ κ(θ)/(1 + θ^r ) ≤ C₂ for some positive constants C₁, C₂ and, r ≥ 2. The growth condition assumed on κ is motivated by the physical fact: κ ∝ θ^5/2 for important physical regimes (see [1, 2]). The total energy of the magnetohydrodynamics flows is

$$E=e+\frac{1}{2}\left(u^2+\mid \mathit{w}{\mid }^2\right)+\frac{1}{2}v\mid \mathit{b}{\mid }^2.$$

Before showing our main results, let us first recall the related results in the literature. For the one-dimensional ideal gas, i.e.,

$$e=c_v\theta ,\sigma =\frac{R\theta }{v}+\mu \frac{u_y}{v},Q=-\kappa \frac{{\theta }_y}{v},$$

(1.8)

with suitable positive constants c _v , R. Kazhikhov and Shelukhin [3–5], Kawashima and Nishida [6] established the existence of global smooth solutions. Zheng and Qin [7] proved the existence of maximal attractors in Hⁱ (i = 1, 2). However, under very high temperatures and densities, constitutive relations (1.8) become inadequate. Thus, a more realistic model would be a linearly viscous gas (or Newtonian fluid)

$$\sigma \left(v,\theta ,u_y\right)=-p\left(v,\theta \right)+\frac{\mu \left(v,\theta \right)}{v}{u}_y$$

(1.9)

satisfying Fourier's law of heat flux

$$Q\left(v,\theta ,{\theta }_y\right)=-\frac{\kappa \left(v,\theta \right)}{v}{\theta }_y$$

(1.10)

whose internal energy e and pressure p are coupled by the standard thermodynamical relation (1.8). In this case, Kawohl [8] obtained the existence of global solutions with the exponents r ∈ [0, 1], q ≥ 2r + 2. Jiang [9] also established the global existence with basically same constitutive relations as those in [8] but with the exponents r ∈ [0, 1], q ≥ r + 1. When the exponents q, r satisfy the more general constitutive relations than those in [8, 9], Qin [10] established the regularity and asymptotic behavior of global solutions with arbitrary initial data for a one-dimensional viscous heat-conductive real gas.

For the radiative and reactive gas, Ducomet [11] established the global existence and exponential decay in H¹ of smooth solutions, and Umehara and Tani [12] proved the global existence of smooth solutions for a self-gravitating radiative and reactive gas.

For the radiative magnetohydrodynamic equations with self-gravitation, Ducomet and Feireisl [13] proved the existence of global-in-time solutions of this problem with arbitrarily large initial data and conservative boundary conditions on a bounded spatial domain in ℝ³. Recently, under the technical condition that κ(ρ, θ) satisfies

$$k_1\left(1+{\theta }^q\right)\le \kappa \left(\rho ,\theta \right)\le k_2\left(1+{\theta }^q\right),k_1\left(1+{\theta }^q\right)\le \left|{\kappa }_{\rho }\left(\rho ,\theta \right)\right|\le k_2\left(1+{\theta }^q\right),$$

for some $q>\frac{5}{2}$, Zhang and Xie [14] investigated the existence of global smooth solutions.

For the non-radiative and non self-gravitation magnetohydrodynamic flows, there have been a number of studies under various conditions by several authors (see, e.g., [2, 15–22]). The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20]. Chen and Wang [15] investigated a free boundary problem with general large initial data with exponents r ∈ [0, 1], q ≥ 2r + 2. Under the technical condition that κ(ρ, θ) satisfies

$$C^{-1}\left(1+{\theta }^q\right)\le \kappa \left(\rho ,\theta \right)\le C\left(1+{\theta }^q\right)$$

for q ≥ 2, Chen and Wang [16] also proved the existence and continuous dependence of global strong solutions with large initial data. Wang [22] established large solutions to the initial-boundary value problem for planar magnetohydrodynamics. Under the technical condition upon

$$\kappa \left(\rho ,\theta \right)\equiv \kappa \left(\rho \right)>\frac{C}{\rho },$$

Fan et al. [18] investigated the uniqueness of the weak solutions of MHD with Lebesgue initial data. Fan et al. [19] also considered a one-dimensional plane compressible MHD flows and proved that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. The uniqueness and continuous dependence of weak solutions for the Cauchy problem have been proved by Hoff and Tsyganov [17].

As mentioned above, the global existence in $H_{+}^i\left(i=\mathsf{\text{ 2}},\mathsf{\text{4}}\right)$ of global solutions has never been studied for Equations (1.1)-(1.5) of the nonlinear one-dimensional compressible magnetohydrodynamics flows with initial-boundary conditions (1.6)-(1.7). The main aim of this paper is to prove the regularity of solutions in the subspace $H_{+}^i$ of (Hⁱ [0, 1])⁷(i = 2, 4) for systems (1.1)-(1.7). In order to obtain higher regularity of global solutions, there are many complicated estimates on higher derivations of solutions to be involved, this is our main difficulty. To overcome this difficulty, we should use some proper embedding theorems, the interpolation techniques as well as many delicate estimates. This is the novelty of the paper.

We define three spaces as follows:

$$\begin{array}{c}{H}_{+}^1=\left\{\left(v,u,\mathit{w},\mathit{b},\theta \right)\in {\left(H^1\left(\Omega \right)\right)}^7:v\left(x\right)>0,\theta \left(x\right)>0,x\in \Omega ,\right.\\ \left.u\left(0\right)=u\left(1\right)=0,\mathit{w}\left(0\right)=\mathit{w}\left(1\right)=\mathit{b}\left(0\right)=\mathit{b}\left(1\right)=\mathbf{0}\right\},\\ H_{+}^i=\left\{\left(v,u,\mathit{w},\mathit{b},\theta \right)\in {\left(H^i\left(\Omega \right)\right)}^7:v\left(x\right)>0,\theta \left(x\right)>0,x\in \Omega ,\right.\\ u\left(0\right)=u\left(1\right)=0,\mathit{w}\left(0\right)=\mathit{w}\left(1\right)=\mathit{b}\left(0\right)=\mathit{b}\left(1\right)=\mathbf{0},\\ \left.{\theta }^{\prime }\left(0\right)={\theta }^{\prime }\left(1\right)=0\right\},i=2,4.\end{array}$$

The notation in this paper will be stated as follows:

L^p , 1 ≤ p ≤ +∞, W^{m, p} , m ∈ N, H¹ = W^1,2, $H_0^1=W_0^{1,2}$ denote the usual (Sobolev) spaces on Ω. In addition, ||·|| _B denotes the norm in the space B, we also put $\parallel \cdot \parallel =\parallel \cdot {\parallel }_{L^2\left(\Omega \right)}$. Constants C _i (i = 1, 2, 3, 4) depend on the $H_{+}^i$ norm of the initial data (v₀, u₀, w ₀ , b₀, θ₀) and T > 0.

Now we are in a position to state our main results.

Theorem 1.1 Assume that the initial data$\left(v_0,u_0,{\mathit{w}}_0,{\mathit{b}}_0,{\theta }_0\right)\in H_{+}^2$and e, p, and κ are C³functions. Then, the problem (1.1) -(1.7) admits a unique global solution$\left(v\left(t\right),u\left(t\right),\mathit{w}\left(t\right),\mathit{b}\left(t\right),\theta \left(t\right)\right)\in H_{+}^2$such that for any T > 0,

$$\begin{array}{l}\parallel v(t)-\overline{v}{\parallel }_{H^2}^2+\parallel u(t){\parallel }_{H^2}^2+\parallel \mathit{w}(t){\parallel }_{H^2}^2+\parallel \mathit{b}(t){\parallel }_{H^2}^2+\parallel \theta (t)-\overline{\theta }{\parallel }_{H^2}^2+\parallel u_t(t){\parallel }^2+\parallel {\mathit{w}}_t(t){\parallel }^2\\ +\parallel {\mathit{b}}_t(t){\parallel }^2+\parallel {\theta }_t(t){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel v-\overline{v}{\parallel }_{H^2}^2}+\parallel u{\parallel }_{H^3}^2+\parallel \mathit{w}{\parallel }_{H^3}^2+\parallel \mathit{b}{\parallel }_{H^3}^2+\parallel \theta -\overline{\theta }{\parallel }_{H^3}^2\\ +\parallel u_{ty}{\parallel }^2+\parallel {\mathit{w}}_{ty}{\parallel }^2+\parallel {\mathit{b}}_{ty}{\parallel }^2+\parallel {\theta }_{ty}{\parallel }^2)(s)ds\le C_2,\forall t\in [0,T],\end{array}$$

(1.11)

where $\bar{v}={\int }_0^{1}v\mathsf{\text{d}}y={\int }_0^1{v}_0\mathsf{\text{d}}y$, constant $\bar{\theta }>0$ is determined by

$$e\left(\bar{v},\bar{\theta }\right)=\underset{0}{\overset{1}{\int }}\left(\frac{1}{2}\left(u_0^2+|{\mathbf{w}}_0{|}^2+v_0|{\mathbf{b}}_0{|}^2\right)+e\left(v_0,{\theta }_0\right)\right)\left(y\right)\mathsf{\text{d}}y.$$

Theorem 1.2 Assume that the initial data$\left(v_0,u_0,{\mathit{w}}_0,{\mathit{b}}_0,{\theta }_0\right)\in H_{+}^4$and e, p, and κ are C⁵functions on 0 < v < +∞ and 0 ≤ θ < +∞. Then, the problem (1.1)-(1.7) admits a unique global solution$\left(v\left(t\right),u\left(t\right),\mathit{w}\left(t\right),\mathit{b}\left(t\right),\theta \left(t\right)\right)\in H_{+}^4$such that for any T > 0,

$$\begin{array}{l}\parallel v(t)-\overline{v}{\parallel }_{H^4}^2+\parallel u(t){\parallel }_{H^4}^2+\parallel \mathit{w}(t){\parallel }_{H^4}^2+\parallel \mathit{b}(t){\parallel }_{H^4}^2+\parallel \theta (t)-\overline{\theta }{\parallel }_{H^4}^2+\parallel u_{tt}(t){\parallel }^2+\parallel {\mathit{w}}_{tt}(t){\parallel }^2\\ +\parallel {\mathit{b}}_{tt}(t){\parallel }^2+\parallel u_t(t){\parallel }_{H^2}^2+\parallel {\mathit{w}}_t(t){\parallel }_{H^2}^2+\parallel {\mathit{b}}_t(t){\parallel }_{H^2}^2+\parallel {\theta }_t(t){\parallel }_{H^2}^2+\parallel {\theta }_{tt}(t){\parallel }^2\\ +{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel v-\overline{v}{\parallel }_{H^4}^2}+\parallel u{\parallel }_{H^5}^2+\parallel \mathit{w}{\parallel }_{H^5}^2+\parallel \mathit{b}{\parallel }_{H^5}^2+\parallel \theta -\overline{\theta }{\parallel }_{H^5}^2+\parallel u_t{\parallel }_{H^3}^2+\parallel {\mathit{w}}_t{\parallel }_{H^3}^2+\parallel {\mathit{b}}_t{\parallel }_{H^3}^2\\ +\parallel {\theta }_t{\parallel }_{H^3}^2+\parallel u_{tt}{\parallel }_{H^1}^2+\parallel {\mathit{w}}_{tt}{\parallel }_{H^1}^2+\parallel {\mathit{b}}_{tt}{\parallel }_{H^1}^2+\parallel {\theta }_{tt}{\parallel }_{H^1}^2)(s)ds\le C_4,\forall t\in [0,T\text{]}\text{.}\end{array}$$

(1.12)

2 Proof of Theorem 1.1

In this section, we study the global existence of problem (1.1)-(1.7) in $H_{+}^2$ by establishing a series of priori estimates. Without loss of generality, we take c _v = R = 1. We begin with the following lemma.

Lemma 2.1 Assume that the initial data$\left(v_0,u_0,{\mathit{w}}_0,{\mathit{b}}_0,{\theta }_0\right)\in H_{+}^1$and e, p, and κ are C²functions on 0 < v < +∞ and 0 ≤ θ < +∞ and there exists a positive constant C₀such that

$$0<C_0^{-1}\le v_0\left(y\right)\le C_0,0<C_0^{-1}\le {\theta }_0\left(y\right)\le C_0.$$

Then, for the initial data$\left(v_0,u_0,{\mathit{w}}_0,{\mathit{b}}_0,{\theta }_0\right)\in H_{+}^1$, the problem (1.1) -(1.7) admits a unique global solution$\left(v\left(t\right),u\left(t\right),\mathit{w}\left(t\right),\mathit{b}\left(t\right),\theta \left(t\right)\right)\in H_{+}^1$such that for any T > 0

$$0<C_1^{-1}\le v\left(y,t\right)\le C_1,0<C_1^{-1}\le \theta \left(y,t\right)\le C_1,\forall \left(y,t\right)\in \left[0,1\right]\times \left[0,T\right]$$

(2.1)

and for any t ∈ [0, T],

$$\begin{array}{l}\parallel v(t)-\overline{v}{\parallel }_{H^1}^2+\parallel u(t){\parallel }_{H^1}^2+\parallel \mathit{w}(t){\parallel }_{H^1}^2+\parallel \mathit{b}(t){\parallel }_{H^1}^2+\parallel \theta (t)-\overline{\theta }{\parallel }_{H^1}^2+{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel v-\overline{v}{\parallel }_{H^1}^2}+\parallel u{\parallel }_{H^2}^2\\ +\parallel \mathit{w}{\parallel }_{H^2}^2+\parallel \mathit{b}{\parallel }_{H^2}^2+\parallel \theta -\overline{\theta }{\parallel }_{H^2}^2+\parallel u_t{\parallel }^2+\parallel {\mathit{w}}_t{\parallel }^2+\parallel {\mathit{b}}_t{\parallel }^2+\parallel {\theta }_t{\parallel }^2)(s)ds\le C_1.\end{array}$$

(2.2)

Proof. See, e.g., [16].

Lemma 2.2 Under the assumptions in Theorem 1.1, the following estimate holds:

$$\begin{array}{l}\parallel u_t(t){\parallel }^2+\parallel {\mathit{w}}_t(t){\parallel }^2+\parallel {\mathit{b}}_t(t){\parallel }^2+\parallel {\theta }_t(t){\parallel }^2\\ +{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel u_{ty}{\parallel }^2}+\parallel {\mathit{w}}_{ty}{\parallel }^2+\parallel {\mathit{b}}_{ty}{\parallel }^2+\parallel {\theta }_{ty}{\parallel }^2)(s)ds\le C_2,\forall t\in [0,T\text{]}\text{.}\end{array}$$

(2.3)

Proof. Differentiating (1.2) with respect to t, multiplying the resultant by u _t , and then integrating the resulting equation over Q _t : = Ω × [0, t], we infer

$$\begin{array}{c}\parallel u_t\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel u_{ty}\left(s\right){\parallel }^2\mathsf{\text{d}}s\\ \le \epsilon \underset{0}{\overset{t}{\int }}\parallel u_{ty}\left(s\right){\parallel }^2\mathsf{\text{d}}s+C_2\underset{0}{\overset{t}{\int }}\left(\parallel {\theta }_t\left(s\right){\parallel }^2\parallel +\parallel \mathbf{b}\cdot {\mathbf{b}}_t\left(s\right){\parallel }^2+\parallel u_y\left(s\right){\parallel }_{L^4}^4\right)\mathsf{\text{d}}s\\ \le \epsilon \underset{0}{\overset{t}{\int }}\parallel u_{ty}\left(s\right){\parallel }^2\mathsf{\text{d}}s+C_2\underset{0}{\overset{t}{\int }}\left(\parallel u_y\left(s\right){\parallel }_{L^{\infty }}^2+\parallel {\theta }_t\left(s\right){\parallel }^2+\parallel {\mathbf{b}}_t\left(s\right){\parallel }^2\right)\mathsf{\text{d}}s\\ \le C_2+\epsilon \underset{0}{\overset{t}{\int }}\parallel u_{ty}\left(s\right){\parallel }^2\mathsf{\text{d}}s,\end{array}$$

which implies

$$\parallel u_t\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel u_{ty}\left(s\right){\parallel }^2\mathsf{\text{d}}s\le C_2.$$

(2.4)

Analogously, we have

$$\parallel {\mathit{w}}_t\left(t\right){\parallel }^2+{\mathit{b}}_t\left(t\right){\parallel }^2+\parallel {\theta }_t\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{ty}{\parallel }^2+\parallel {\mathit{b}}_{ty}{\parallel }^2+\parallel {\theta }_{ty}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s\le C_2.$$

(2.5)

Thus, (2.3) follows from (2.4)-(2.5).

Lemma 2.3 Under the assumptions in Theorem 1.1, the following estimate holds:

$$\begin{array}{c}\parallel u_{yy}(t){\parallel }^2+\parallel {\mathit{w}}_{yy}(t){\parallel }^2+\parallel {\mathit{b}}_{yy}(t){\parallel }^2+\parallel {\theta }_{yy}(t){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel u_{yyy}{\parallel }^2}\\ +\parallel {\mathit{w}}_{yyy}{\parallel }^2+\parallel {\mathit{b}}_{yyy}{\parallel }^2+\parallel {\theta }_{yyy}{\parallel }^2)(s)ds\le C_2,\forall t\in [0,T\text{]}\text{.}\end{array}$$

(2.6)

Proof. Equation (1.2) can be rewritten as

$$u_t=-\frac{\lambda {\theta }_y}{v}+\frac{\lambda \theta v_y}{v}-\frac{\lambda u_y{v}_y}{v^2}+\frac{\lambda u_{yy}}{v}-\mathbf{b}\cdot {\mathbf{b}}_y.$$

(2.7)

Using equation (2.7), Lemmas 2.1-2.2, Sobolev's embedding theorem and Young's inequality, we have

$$\begin{array}{c}\parallel u_{yy}\left(t\right)\parallel \le C_2\left(\left|\right|u_t\left(t\right)\left|\right|+\left|\right|{\theta }_y\left(t\right)\left|\right|+\left|\right|\mathbf{b}\cdot {\mathbf{b}}_y\left(t\right)\left|\right|+\left|\right|\theta v_y\left(t\right)\left|\right|+\left|\right|v_y{u}_y\left(t\right)\left|\right|\right)\\ \le C_2\left(\left|\right|u_t\left(t\right)\left|\right|+\left|\right|{\theta }_y\left(t\right)\left|\right|+\parallel \theta \left(t\right){\parallel }_{L^{\infty }}\left|\right|v_y\left(t\right)\left|\right|+\left|\right|{\mathbf{b}}_y\left(t\right)|{|}^2+|\left|u_y\left(t\right)\right|{|}_{L^{\infty }}\left|\right|v_y\left(t\right)\left|\right|\right)\\ \le \epsilon \left|\right|u_{yy}\left(t\right)\left|\right|+C_2\left(\left|\right|u_t\left(t\right)\left|\right|+1\right),\end{array}$$

which leads to

$$\parallel u_{yy}(t)\parallel \le C_2,{\displaystyle \underset{0}{\overset{t}{\int }}\parallel u_{yyy}(s{)\parallel }^2\text{d}s}\le C_2{\displaystyle \underset{0}{\overset{t}{\int }}\parallel u_{ty}(s{)\parallel }^2\text{d}s}\le C_2.$$

(2.8)

Similarly, we derive

$$\parallel {\mathit{w}}_{yy}\left(t\right)\parallel +\parallel {\mathit{b}}_{yy}\left(t\right)\parallel +\parallel {\theta }_{yy}\left(t\right)\parallel \le C_2\left(\parallel {\mathit{w}}_t\left(t\right)\parallel +\parallel {\mathit{b}}_t\left(t\right)\parallel +\parallel {\theta }_t\left(t\right)\parallel +1\right)\le C_2,$$

(2.9)

$$\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{yyy}{\parallel }^2+\parallel {\mathit{b}}_{yyy}{\parallel }^2+\parallel {\theta }_{yyy}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s\le C_2.$$

(2.10)

Thus, (2.6) follows from (2.8)-(2.10).

Lemma 2.4 Under the assumptions in Theorem 1.1, the following estimate holds:

$$\parallel v_{yy}(t){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}\parallel v_{yy}(s{)\parallel }^2}ds\le C_2,\forall t\in [0,T].$$

(2.11)

Proof. Differentiating (1.2) with respect to y, we obtain

$$\lambda \frac{d}{\mathsf{\text{d}}t}\left(\frac{v_{yy}}{v}\right)+\frac{\theta }{v^2}{v}_{yy}=u_{ty}+E\left(y,t\right),$$

(2.12)

where

$$E\left(y,t\right)=\frac{{\theta }_{yy}}{v}+\frac{2v_y\left(\lambda u_{yy}-{\theta }_y\right)}{v^2}+\frac{2v_y^2\left(\theta -\lambda u_y\right)}{v^3}+\mathit{b}\cdot {\mathit{b}}_{yy}+\mid {\mathit{b}}_y{\mid }^2.$$

Multiplying (2.12) by $\frac{v_{yy}}{v}$, integrating the resulting equation over Q _t , and then using the Young inequality and interpolation theorem, we can conclude

$$\begin{array}{l}{\Vert \frac{v_{yy}}{v}(t)\Vert }^2+C_1^{-1}{\displaystyle \underset{0}{\overset{t}{\int }}{\Vert \frac{v_{yy}}{v}(s)\Vert }^2}\text{d}s\le \frac{1}{4C_1}{\displaystyle \underset{0}{\overset{t}{\int }}{\Vert \frac{v_{yy}}{v}(s)\Vert }^2}\text{d}s+C_2{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel u_{ty}{\parallel }^2+\parallel {\theta }_{yy}{\parallel }^2}\\ +\parallel v_y{u}_{yy}{\parallel }^2+\parallel v_y{\parallel }_{L^4}^4+\parallel u_y{v}_y^2{\parallel }^2+\parallel {\mathit{b}}_y{\parallel }_{L^4}^4+\parallel \mathit{b}{\parallel }_{L^{\infty }}^2\parallel {\mathit{b}}_{yy}{\parallel }^2)(s)\text{d}s\\ \le \frac{1}{2C_1}{\displaystyle \underset{0}{\overset{t}{\int }}{\Vert \frac{v_{yy}}{v}(s)\Vert }^2}\text{d}s+C_2{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel v_y{\parallel }^2}+\parallel u_{ty}{\parallel }^2+\parallel u_{yy}{\parallel }_{L^{\infty }}^2+\parallel {\theta }_{yy}{\parallel }^2+\parallel {\mathit{b}}_{yy}{\parallel }^2)(s)\text{d}s,\end{array}$$

which, together with Lemmas 2.1-2.3, yields (2.11).

Proof of Theorem 1.1. By Lemmas 2.1-2.4, we complete the proof of Theorem 1.1.

3 Proof of Theorem 1.2

In this section, we study the global existence of problem (1.1)-(1.7) in $H_{+}^4$ by establishing a series of priori estimates. We begin with the following lemmas.

Lemma 3.1 Under the assumptions in Theorem 1.2, the following estimates hold:

$$\parallel u_{ty}\left(y,0\right)\parallel +\parallel {\mathit{w}}_{ty}\left(y,0\right)\parallel +\parallel {\mathit{w}}_{ty}\left(y,0\right)\parallel +\parallel {\theta }_{ty}\left(y,0\right)\parallel \le C_3,$$

(3.1)

$$\begin{array}{c}\parallel u_{tt}\left(y,0\right)\parallel +\parallel {\mathit{w}}_{tt}\left(y,0\right)\parallel +\parallel {\mathit{b}}_{tt}\left(y,0\right)\parallel +\parallel {\theta }_{tt}\left(y,0\right)\parallel \\ +\parallel u_{tyy}\left(y,0\right)\parallel +\parallel {\mathit{w}}_{tyy}\left(y,0\right)\parallel +\parallel {\mathit{b}}_{tyy}\left(y,0\right)\parallel +\parallel {\theta }_{tyy}\left(y,0\right)\parallel \le C_3.\end{array}$$

(3.2)

Proof. We easily infer from (1.2), Lemma 2.1 and Theorems 1.1 that

$$\begin{array}{ll}\hfill \parallel u_t\left(t\right)\parallel & \le C_3\left(\parallel v_y\left(t\right)\parallel +\parallel {\theta }_y\left(t\right)\parallel +\parallel u_{yy}\left(t\right)\parallel +\parallel u_y\left(t\right){\parallel }_{L^{\infty }}\parallel v_y\left(t\right)\parallel +\parallel \mathit{b}\left(t\right){\parallel }_{L^{\infty }}\parallel {\mathit{b}}_y\left(t\right)\parallel \right)\\ \le C_3\left(\parallel v_y\left(t\right)\parallel +\parallel {\theta }_y\left(t\right)\parallel +\parallel u_{yy}\left(t\right)\parallel +\parallel {\mathit{b}}_y\left(t\right)\parallel \right).\end{array}$$

Differentiating (1.2) with respect to y, and using Theorem 1.1, we get

$$\parallel u_{ty}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^1}+\parallel {\theta }_y\left(t\right){\parallel }_{H^1}+\parallel u_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^1}\right),$$

(3.3)

or

$$\parallel u_{yyy}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^1}+\parallel {\theta }_y\left(t\right){\parallel }_{H^1}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^1}+\parallel u_{ty}\left(t\right)\parallel \right).$$

(3.4)

Differentiating (1.2) with respect to y twice, using the embedding theorem and Theorem 1.1, we conclude

$$\parallel u_{tyy}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel {\theta }_y\left(t\right){\parallel }_{H^2}+\parallel u_y\left(t\right){\parallel }_{H^3}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}\right),$$

(3.5)

or

$$\parallel u_{yyyy}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel {\theta }_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel u_{tyy}\left(t\right)\parallel \right).$$

(3.6)

Similarly, we have

$$\begin{array}{c}\parallel {\mathit{w}}_t\left(t\right)\parallel \le C_3\left(\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^1}+\parallel {\mathit{b}}_y\left(t\right)\parallel +\parallel v_y\left(t\right)\parallel \right),\\ \parallel {\mathit{w}}_{ty}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^1}+\parallel v_y\left(t\right){\parallel }_{H^1}\right),\end{array}$$

(3.7)

or

$$\parallel {\mathit{w}}_{yyy}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^1}+\parallel v_y\left(t\right){\parallel }_{H^1}+\parallel {\mathit{b}}_{ty}\left(t\right)\parallel \right),$$

(3.8)

$$\parallel {\mathit{w}}_{tyy}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^3}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel v_y\left(t\right){\parallel }_{H^2}\right),$$

(3.9)

or

$$\parallel {\mathit{w}}_{yyyy}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{w}}_{tyy}\left(t\right)\parallel \right),$$

(3.10)

$$\begin{array}{c}\parallel {\mathit{b}}_t\parallel \le C_3\left(\parallel {\mathit{b}}_y{\parallel }_{H^1}+\parallel {\mathit{w}}_y\parallel +\parallel v_y\parallel \right),\\ \parallel {\mathit{b}}_{ty}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^1}+\parallel v_y\left(t\right){\parallel }_{H^1}\right),\end{array}$$

(3.11)

or

$$\parallel {\mathit{b}}_{yyy}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^1}+\parallel v_y\left(t\right){\parallel }_{H^1}+\parallel {\mathit{b}}_{ty}\left(t\right)\parallel \right),$$

(3.12)

$$\parallel {\mathit{b}}_{tyy}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^3}+\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^2}+\parallel v_y\left(t\right){\parallel }_{H^2}\right),$$

(3.13)

or

$$\parallel {\mathit{b}}_{yyyy}\left(t\right)\parallel \le C_3\left(\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^2}+\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_{tyy}\left(t\right)\parallel \right),$$

(3.14)

$$\begin{array}{c}\parallel {\theta }_t(t)\parallel \le C_3(\parallel u_y(t)\parallel +\parallel v_y(t)\parallel +\parallel {\theta }_{yy}(t)\parallel +\parallel u_y(t){\parallel }_{L^{\infty }}\parallel u_y(t)\parallel \\ +\parallel {\mathit{w}}_y(t){\parallel }_{L^{\infty }}\parallel {\mathit{w}}_y(t)\parallel +\parallel {\mathit{b}}_y(t){\parallel }_{L^{\infty }}\parallel {\mathit{b}}_y(t)\parallel +\parallel {\theta }_y(t){\parallel }_{L^{\infty }}\parallel {\theta }_y(t)\parallel ),\\ \le C_3(\parallel {\theta }_{yy}(t)\parallel +\parallel u_{yy}(t)\parallel +\parallel {\mathit{w}}_{yy}(t)\parallel +\parallel {\mathit{b}}_{yy}(t)\parallel ),\end{array}$$

(3.15)

$$\begin{array}{l}\parallel {\theta }_{ty}(t)\parallel \le C_3(\parallel {\theta }_t(t)\parallel +\parallel {\theta }_y(t){\parallel }_{H^2}+\parallel v_y(t){\parallel }_{H^1}+\parallel u_y(t){\parallel }_{H^1}+\parallel {\mathit{w}}_y(t){\parallel }_{H^1}\\ +\parallel {\mathit{b}}_y(t){\parallel }_{H^1}),\end{array}$$

(3.16)

or

$$\begin{array}{l}\parallel {\theta }_{yyy}(t)\parallel \le C_3(\parallel v_y(t){\parallel }_{H^1}+\parallel u_y(t){\parallel }_{H^1}+\parallel {\mathit{w}}_y(t){\parallel }_{H^1}+\parallel {\mathit{b}}_y(t){\parallel }_{H^1}\\ +\parallel {\theta }_{ty}(t)\parallel ),\end{array}$$

(3.17)

$$\parallel {\theta }_{tyy}\left(t\right)\parallel \le C_3\left(\parallel {\theta }_y\left(t\right){\parallel }_{H^3}+\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel u_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}\right),$$

(3.18)

or

$$\begin{array}{l}\parallel {\theta }_{yyyy}(t)\parallel \le C_3(\parallel v_y(t){\parallel }_{H^2}+\parallel u_y(t){\parallel }_{H^2}+\parallel {\mathit{w}}_y(t){\parallel }_{H^2}+\parallel {\mathit{b}}_y(t){\parallel }_{H^2}\\ +\parallel {\theta }_{tyy}(t)\parallel ).\end{array}$$

(3.19)

Differentiating (1.2) with respect to t, and using Theorem 1.1, (3.3), (3.5), (3.11)-(3.12) and (3.16), we derive

$$\parallel u_{tt}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel u_y\left(t\right){\parallel }_{H^3}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel {\theta }_y\left(t\right){\parallel }_{H^2}\right).$$

(3.20)

Similarly, we can conclude

$$\parallel {\mathit{w}}_{tt}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^3}\right),$$

(3.21)

$$\parallel {\mathit{b}}_{tt}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^3}+\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^2}\right),$$

(3.22)

$$\parallel {\theta }_{tt}\left(t\right)\parallel \le C_3\left(\parallel v_y\left(t\right){\parallel }_{H^2}+\parallel u_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{b}}_y\left(t\right){\parallel }_{H^2}+\parallel {\mathit{w}}_y\left(t\right){\parallel }_{H^2}+\parallel {\theta }_y\left(t\right){\parallel }_{H^3}\right).$$

(3.23)

Thus, (3.1) follows from (3.3), (3.7), (3.11) and (3.16), and (3.2) from (3.5), (3.9), (3.13), (3.18) and (3.20)-(3.23).

Lemma 3.2 Under the assumptions in Theorem 1.2, the following estimates hold, for any t ∈ [0, T],

$$\parallel u_{tt}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel u_{tty}\left(s\right){\parallel }^{2}ds\le C_3+C_3\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\theta }_{tyy}{\parallel }^2\right)\left(s\right)ds,$$

(3.24)

$$\begin{array}{c}\parallel {\mathit{w}}_{tt}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{tt}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{tty}\left(s\right){\parallel }^2+\parallel {\mathit{b}}_{tty}\left(s\right){\parallel }^2\right)ds\\ \le C_3+C_3\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{b}}_{tyy}\left(s\right){\parallel }^2+\parallel {\mathit{w}}_{tyy}\left(s\right){\parallel }^2\right)ds,\end{array}$$

(3.25)

$$\begin{array}{c}\parallel {\theta }_{tt}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel {\theta }_{tty}\left(s\right){\parallel }^{2}ds\le C_3+C_2{\epsilon }^{-1}\underset{0}{\overset{t}{\int }}\parallel {\theta }_{tyy}\left(s\right){\parallel }^{2}ds\\ +C_1\epsilon \underset{0}{\overset{t}{\int }}\left(\parallel u_{tyy}{\parallel }^2+\parallel u_{tty}{\parallel }^2+\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\mathit{w}}_{tty}{\parallel }^2+\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tty}{\parallel }^2\right)\left(s\right)ds.\end{array}$$

(3.26)

Proof. Differentiating (1.2) with respect to t twice, multiplying the resulting equation by u _tt , performing an integration by parts, and using Lemma 2.1, we have

$$\begin{array}{l}\frac{1}{2}\frac{d}{\text{d}t}{\displaystyle \underset{0}{\overset{1}{\int }}{u}_{tt}^2}(y,t)\text{d}y\le -\lambda \parallel u_{tty}(t){\parallel }^2+C_4(\parallel {\theta }_{tt}(t)\parallel +\parallel u_{yy}(t)\parallel +\parallel {\theta }_t{u}_y(t)\parallel \\ +\parallel \mathit{b}\cdot {\mathit{b}}_{tt}(t)\parallel +\parallel {\mathit{b}}_t^2(t){\parallel }^2+\parallel u_{ty}(t)\parallel )\parallel u_{tty}(t)\parallel \\ \le -C_1^{-1}\parallel u_{tty}(t){\parallel }^2+C_4(\parallel {\theta }_{tt}(t){\parallel }^2+\parallel {\mathit{b}}_{tt}(t){\parallel }^2+\parallel u_{ty}(t){\parallel }^2+\parallel u_y(t){\parallel }^2+\parallel {\theta }_t(t){\parallel }^2)\end{array}$$

(3.27)

Thus, using Theorem 1.1 and Lemma 3.1, we get

$$\parallel u_{tt}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel u_{tty}\left(s\right){\parallel }^2\mathsf{\text{d}}s\le C_3+C_3\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\theta }_{tyy}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s.$$

Analogously, we obtain

$$\begin{array}{c}\parallel {\mathit{w}}_{tt}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{tt}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{tty}{\parallel }^2+\parallel {\mathit{b}}_{tty}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s\\ \le C_3+C_3\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tyy}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s.\end{array}$$

Equation (1.5) can be rewritten as

$${\left(c_v\theta \right)}_t+p{u}_y={\left(\frac{\kappa {\theta }_y}{v}\right)}_y+\frac{\lambda u_y^2+\mu |{\mathit{w}}_y{|}^2+\nu |{\mathbf{b}}_y{|}^2}{v}.$$

(3.28)

Differentiating (3.28) with respect to t twice, multiplying the resulting equation by θ _tt in L² 0[1] and integrating by parts, we arrive at

$$\begin{array}{c}\frac{1}{2}\frac{d}{\mathsf{\text{d}}t}\underset{0}{\overset{1}{\int }}{c}_v{\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y=-\underset{0}{\overset{1}{\int }}{\left(\frac{\kappa {\theta }_y}{v}\right)}_{tt}{\theta }_{tty}\left(y,t\right)\mathsf{\text{d}}y\\ -\underset{0}{\overset{1}{\int }}\left(p-\frac{\lambda u_y}{v}\right)u_{tty}{\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y+\underset{0}{\overset{1}{\int }}\left(\frac{\mu {\mathit{w}}_y}{v}{\mathit{w}}_{tty}+\frac{\nu {\mathit{b}}_y}{v}{\mathit{b}}_{tty}\right){\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y\\ -2\underset{0}{\overset{1}{\int }}\left(p_t-{\left(\frac{\lambda u_y}{v}\right)}_t\right)u_{ty}{\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y+2\underset{0}{\overset{1}{\int }}\left[{\left(\frac{\mu {\mathit{w}}_y}{v}\right)}_t{\mathit{w}}_{ty}+{\left(\frac{\nu {\mathit{b}}_y}{v}\right)}_t{\mathit{b}}_{ty}\right]{\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y\\ +\underset{0}{\overset{1}{\int }}{\left(-p+\frac{\lambda u_y}{v}\right)}_{tt}{u}_y{\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y+\underset{0}{\overset{1}{\int }}\left[{\left(\frac{\mu {\mathit{w}}_y}{v}\right)}_{tt}{\mathit{w}}_y+{\left(\frac{\nu {\mathit{b}}_y}{v}\right)}_{tt}{\mathit{b}}_y\right]{\theta }_{tt}\left(y,t\right)\mathsf{\text{d}}y\\ =B_1+B_2+B_3+B_4+B_5+B_6+B_7.\end{array}$$

By virtue of Theorem 1.1 and Lemmas 3.1-3.2, using the embedding theorem, we deduce for any ε ∈ (0, 1),

$$\begin{array}{l}{B}_1\le -C^{-1}\parallel {\theta }_{tty}(t){\parallel }^2+C_2\parallel {\theta }_{ty}(t){\parallel }_{L^{\infty }}(\parallel u_y(t){\parallel }^2+\parallel {\theta }_t(t){\parallel }^2)\parallel {\theta }_{tty}(t)\parallel \\ +C_2\parallel (\frac{\kappa }{v}{)}_{tt}(t){\parallel }^2\parallel {\theta }_y(t){\parallel }_{L^{\infty }}\parallel {\theta }_{tty}(t)\parallel \\ \le -2C^{-1}\parallel {\theta }_{tty}(t){\parallel }^2+C_2(\parallel u_y(t){\parallel }^2+\parallel {\theta }_t(t){\parallel }^2+\parallel u_{ty}(t){\parallel }^2\\ +\parallel {\theta }_{ty}(t){\parallel }^2+\parallel {\theta }_{tt}(t){\parallel }^2+\parallel {\theta }_{tyy}(t){\parallel }^2),\\ B_2\le \epsilon \parallel u_{tty}(t){\parallel }^2+C_2{\epsilon }^{-1}\parallel {\theta }_{tt}(t){\parallel }^2,\\ B_3\le \epsilon (\parallel {\mathit{w}}_{tty}(t){\parallel }^2+\parallel {\mathit{b}}_{tty}(t){\parallel }^2)+C_2{\epsilon }^{-1}\parallel {\theta }_{tt}(t){\parallel }^2\end{array}$$

and

$$\begin{array}{ll}\hfill B_4& \le C_1\underset{0}{\overset{1}{\int }}\left(\left|{\theta }_t\right|+\left|u_y\right|+\left|u_{ty}\right|+|u_y{|}^2\right)|{\theta }_{tt}\parallel u_{ty}|\left(y,t\right)\mathsf{\text{d}}y\\ \le C_2\parallel u_{ty}\left(t\right)|{|}^{\frac{1}{2}}\parallel u_{tyy}\left(t\right)|{|}^{\frac{1}{2}}\left(\parallel {\theta }_t\left(t\right)\parallel +\parallel u_y\left(t\right)\parallel +\parallel u_{ty}\left(t\right)\parallel \right)\parallel {\theta }_{tt}\left(t\right)\parallel \end{array}$$

which implies

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}{B}_4\mathsf{\text{d}}s\le C_2\underset{0\le s\le t}{sup}\parallel {\theta }_{tt}\left(s\right)\parallel {\left(\underset{0}{\overset{t}{\int }}\parallel u_{ty}\left(s\right)|{|}^2\mathsf{\text{d}}s\right)}^{\frac{1}{4}}{\left(\underset{0}{\overset{t}{\int }}\parallel u_{tyy}\left(s\right)|{|}^2\mathsf{\text{d}}s\right)}^{\frac{1}{4}}\\ \times {\left(\underset{0}{\overset{t}{\int }}\left(\parallel u_{ty}|{|}^2+\parallel {\theta }_t|{|}^2+\parallel u_y|{|}^2\right)\left(s\right)\mathsf{\text{d}}s\right)}^{\frac{1}{2}}\\ \le \epsilon \left(\underset{0\le s\le t}{sup}\parallel {\theta }_{tt}\left(s\right)|{|}^2+\underset{0}{\overset{t}{\int }}\parallel u_{tyy}\left(s\right)|{|}^2\mathsf{\text{d}}s\right)+C_3{\epsilon }^{-3}.\end{array}$$

$$\begin{array}{ll}\hfill B_5& \le C_1\underset{0}{\overset{1}{\int }}\left[\left(|{\mathit{w}}_y{|}^2+|{\mathit{w}}_{ty}|\right)\left|{\mathit{w}}_{ty}\right|+\left(|{\mathit{w}}_y{|}^2+|{\mathit{w}}_{ty}|\right)\left|{\mathit{w}}_{ty}\right|\right]\left|{\theta }_{tt}\right|\left(y,t\right)\mathsf{\text{d}}y\\ \le C_2\parallel {\mathit{w}}_{ty}\left(t\right)|{|}_{L^{\infty }}\left(\parallel {\mathit{w}}_y\left(t\right)|{|}^2+\parallel {\mathit{w}}_{ty}\left(t\right)\parallel \right)\parallel {\theta }_{tt}\left(t\right)\parallel +C_2\parallel {\mathit{w}}_{ty}\left(t\right)|{|}_{L^{\infty }}\left(\parallel {\mathit{b}}_y\left(t\right)|{|}^2+\parallel {\mathit{b}}_{ty}\left(t\right)\parallel \right)\parallel {\theta }_{tt}\left(t\right)\parallel \end{array}$$

which implies

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{t}{\int }}{B}_5}\text{d}s\le \epsilon \left(\underset{0\le s\le t}{{\displaystyle \sup }}\parallel {\theta }_{tt}(s){\parallel }^2+{\displaystyle {\int }_0^t}(\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\mathit{w}}_{tyy}{\parallel }^2(s)\text{d}s\right)+C_3{\epsilon }^{-3}\mathrm{,}\\ B_6\le C_2\parallel u_y(t)|{|}_{L^{\infty }}\parallel {\theta }_{tt}(t)\parallel [(\parallel {\theta }_t(t)|{|}_{L^{\infty }}+\parallel u_y(t)|{|}_{L^{\infty }})(\parallel {\theta }_t(t)\parallel +\parallel u_y(t)\parallel )+\parallel {\theta }_{tt}(t)\parallel \\ +\parallel u_{ty}(t)\parallel +\parallel u_y(t)\parallel +\parallel u_{tty}(t)\parallel ]\\ \le C_2\parallel {\theta }_{tt}(t)\parallel (\parallel {\theta }_t(t)\parallel +\parallel u_y(t)|{|}_{H^1}+\parallel {\theta }_{ty}(t)\parallel +\parallel {\theta }_{tt}(t)\parallel +\parallel u_{ty}(t)\parallel +\parallel u_{tty}(t)\parallel )\\ \le \epsilon \parallel u_{tty}(t){\parallel }^2+C_2{\epsilon }^{-1}(\parallel {\theta }_t(t){\parallel }^2+\parallel u_y(t)|{|}_{H^1}^2+\parallel {\theta }_{ty}(t){\parallel }^2+\parallel {\theta }_{tt}(t){\parallel }^2+\parallel u_{ty}(t){\parallel }^2),\\ B_7\le C_2\parallel {\mathit{w}}_y(t)|{|}_{L^{\infty }}\parallel {\theta }_{tt}(t)\parallel (\parallel {\mathit{w}}_{tty}(t)\parallel +\parallel {\mathit{w}}_{ty}(t)\parallel +\parallel {\mathit{w}}_y(t)\parallel )+C_2\parallel {\mathit{b}}_y(t)|{|}_{L^{\infty }}\parallel {\theta }_{tt}(t)\parallel \\ \times (\parallel {\mathit{b}}_{tty}(t)\parallel +\parallel {\mathit{b}}_{ty}(t)\parallel +\parallel {\mathit{b}}_y(t)\parallel )\\ \le C_2\parallel {\theta }_{tt}(t)\parallel (\parallel {\mathit{w}}_{ty}(t)\parallel +\parallel {\mathit{w}}_y(t)|{|}_{H^1}+\parallel {\mathit{w}}_{tty}(t)\parallel +\parallel {\mathit{b}}_{ty}(t)\parallel +\parallel {\mathit{b}}_y(t)|{|}_{H^1}+\parallel {\mathit{b}}_{tty}(t)\parallel )\\ \le \epsilon (\parallel {\mathit{w}}_{tty}(t){\parallel }^2+\parallel {\mathit{b}}_{tty}(t){\parallel }^2)+C_2{\epsilon }^{-1}(\parallel {\theta }_{tt}(t){\parallel }^2+\parallel {\mathit{w}}_y(t)|{|}_{H^1}^2\\ +\parallel {\mathit{w}}_{ty}(t){\parallel }^2+\parallel {\mathit{b}}_y(t)|{|}_{H^1}^2+\parallel {\mathit{b}}_{ty}(t){\parallel }^2).\end{array}$$

Thus, for ε ∈ (0, 1) small enough, we derive from above estimates

$$\begin{array}{l}\parallel {\theta }_{tt}(t){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}\parallel {\theta }_{tty}(s{)\parallel }^2}\text{d}s\le C_2{\epsilon }^{-1}{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel {\theta }_{tyy}(s{)\parallel }^2+\parallel {\theta }_{tt}(s{)\parallel }^2)}\text{d}s+C_3{\epsilon }^{-3}\\ +C_1\epsilon [\underset{0\le s\le t}{{\displaystyle \sup }}\parallel {\theta }_{tt}(s){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel u_{tyy}{\parallel }^2+\parallel u_{tty}{\parallel }^2+\parallel {\mathit{w}}_{tyy}{\parallel }^2}\\ +\parallel {\mathit{w}}_{tty}{\parallel }^2+\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tty}{\parallel }^2)(s)\text{d}s].\end{array}$$

(3.29)

Thus, taking supremum in t on the left-hand side of (3.29), picking ε ∈ (0, 1) small enough, and using (3.23), we can derive estimate (3.26).

Lemma 3.3 Under the assumptions in Theorem 1.2, the following estimates hold, for any t ∈ [0,T],

$$\begin{array}{c}\parallel u_{ty}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel u_{tyy}\left(s\right){\parallel }^{2}ds\\ \le C_3{\epsilon }^{-6}+C_2{\epsilon }^2\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\theta }_{tyy}{\parallel }^2+\parallel u_{tty}{\parallel }^2\right)\left(s\right)ds,\end{array}$$

(3.30)

$$\begin{array}{c}\parallel {\mathit{w}}_{ty}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{ty}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tyy}{\parallel }^2\right)\left(s\right)ds\\ \le C_3{\epsilon }^{-6}+C_2{\epsilon }^2\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tty}{\parallel }^2\right)\left(s\right)ds,\end{array}$$

(3.31)

$$\begin{array}{c}\parallel {\theta }_{ty}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel {\theta }_{tyy}\left(s\right){\parallel }^{2}ds\le C_3{\epsilon }^{-6}\\ +C_2{\epsilon }^2\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel u_{tyy}{\parallel }^2+\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\theta }_{tty}{\parallel }^2+\parallel {\theta }_{yyy}{\parallel }^2\parallel {\theta }_{ty}{\parallel }^2\right)\left(s\right)ds.\end{array}$$

(3.32)

Proof. Differentiating (1.2) with respect to y and t, multiplying the resulting equation by u _ty , and integrating by parts, we arrive at

$$\frac{1}{2}\frac{d}{\mathsf{\text{d}}t}\parallel u_{ty}\left(t\right){\parallel }^2=D_0\left(y,t\right)+D_1\left(t\right),$$

(3.33)

where

$$D_0\left(y,t\right)={\sigma }_{ty}{u}_{ty}{|}_{y=0}^{y=1},D_1\left(t\right)=-\underset{0}{\overset{1}{\int }}{\sigma }_{ty}{u}_{tyy}\mathsf{\text{d}}y,\sigma =-\left(p+\frac{1}{2}|\mathit{b}{|}^2-\frac{\lambda u_y}{v}\right).$$

We use Theorem 1.1, Lemma 2.1, the interpolation inequality and Poincaré's inequality to obtain

$$\begin{array}{c}{D}_0\le C_1[(\parallel u_y(t{)\parallel }_{L^{\infty }}+\parallel {\theta }_t(t{)\parallel }_{L^{\infty }})(\parallel v_y(t{)\parallel }_{L^{\infty }}+\parallel {\theta }_y(t{)\parallel }_{L^{\infty }})\\ +\parallel {\mathit{b}}_t(t){\parallel }_{L^{\infty }}\parallel {\mathit{b}}_y(t){\parallel }_{L^{\infty }}+\parallel {\mathit{b}}_{ty}(t){\parallel }_{L^{\infty }}\parallel \mathit{b}(t){\parallel }_{L^{\infty }}+\parallel {\theta }_{ty}(t){\parallel }_{L^{\infty }}+\parallel u_y(t){\parallel }_{L^{\infty }}^2\\ +\parallel u_y(t{)\parallel }_{L^{\infty }}\parallel u_{yy}(t{)\parallel }_{L^{\infty }}+\parallel u_{ty}(t{)\parallel }_{L^{\infty }}\parallel v_y(t{)\parallel }_{L^{\infty }}+\parallel u_{tyy}(t{)\parallel }_{L^{\infty }}]\parallel u_{ty}(t){\parallel }_{L^{\infty }}\\ \le C_3(D_{01}+D_{01})\parallel u_{ty}(t){\parallel }^{\frac{1}{2}}\parallel u_{tyy}(t){\parallel }^{\frac{1}{2}},\end{array}$$

(3.34)

where

$$\begin{array}{ll}\hfill D_{01}& =\parallel u_y\left(t\right){\parallel }_{H^2}+\parallel {\theta }_t\left(t\right)\parallel +\parallel {\theta }_{ty}\left(t\right)\parallel +\parallel {\mathit{b}}_t\left(t\right)\parallel +\parallel {\mathit{b}}_{ty}\left(t\right)\parallel ,\\ \hfill D_{02}& =\parallel {\theta }_{ty}\left(t\right){\parallel }^{\frac{1}{2}}\parallel {\theta }_{tyy}\left(t\right){\parallel }^{\frac{1}{2}}+\parallel u_{tyy}\left(t\right){\parallel }^{\frac{1}{2}}\parallel u_{tyyy}\left(t\right){\parallel }^{\frac{1}{2}}+\parallel u_{tyy}\left(t\right)\parallel \\ +\parallel u_{ty}\left(t\right){\parallel }^{\frac{1}{2}}\parallel u_{tyy}\left(t\right){\parallel }^{\frac{1}{2}}+\parallel {\mathit{b}}_{ty}\left(t\right){\parallel }^{\frac{1}{2}}\parallel {\mathit{b}}_{tyy}\left(t\right){\parallel }^{\frac{1}{2}}.\end{array}$$

Using the Young inequality several times, we derive

$$\begin{array}{c}{C}_3{D}_{01}\parallel u_{ty}\left(t\right){\parallel }^{\frac{1}{2}}\parallel u_{tyy}\left(t\right){\parallel }^{\frac{1}{2}}\le \frac{{\epsilon }^2}{2}\parallel u_{tyy}\left(t\right){\parallel }^2\\ +C_3{\epsilon }^{-\frac{2}{3}}\left(\parallel u_{ty}\left(t\right){\parallel }^2+\parallel u_y\left(t\right){\parallel }_{H^2}^2+\parallel {\theta }_t\left(t\right){\parallel }^2+\parallel {\theta }_{ty}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_t\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{ty}\left(t\right){\parallel }^2\right)\end{array}$$

(3.35)

and

$$\begin{array}{ll}\hfill C_3{D}_{02}\parallel u_{ty}\left(t\right){\parallel }^{\frac{1}{2}}\parallel u_{tyy}\left(t\right){\parallel }^{\frac{1}{2}}& \le \frac{{\epsilon }^2}{2}\parallel u_{tyy}\left(t\right){\parallel }^2+{\epsilon }^2\left(\parallel u_{tyyy}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{tyy}\left(t\right){\parallel }^2+\parallel {\theta }_{tyy}\left(t\right){\parallel }^2\right)\\ +C_3{\epsilon }^{-6}\left(\parallel u_{ty}\left(t\right){\parallel }^2+\parallel {\theta }_{ty}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{ty}\left(t\right){\parallel }^2\right).\end{array}$$

(3.36)

Thus, we infer from (3.34)-(3.36) that

$$\begin{array}{ll}\hfill D_0& \le {\epsilon }^2\left(\parallel u_{tyyy}\left(t\right){\parallel }^2+\parallel u_{tyy}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{tyy}\left(t\right){\parallel }^2+\parallel {\theta }_{tyy}\left(t\right){\parallel }^2\right)\\ +C_3{\epsilon }^{-6}\left(\parallel u_{ty}\left(t\right){\parallel }^2+\parallel {\theta }_{ty}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{ty}\left(t\right){\parallel }^2+\parallel {\theta }_t\left(t\right){\parallel }^2+\parallel u_y\left(t\right){\parallel }_{H^2}^2+\parallel {\mathit{b}}_t\left(t\right){\parallel }^2\right),\end{array}$$

which, together with Theorem 1.1, Lemma 2.1, and Lemmas 3.1-3.2, yields

$$\underset{0}{\overset{t}{\int }}{D}_0\mathsf{\text{d}}s\le {\epsilon }^2\underset{0}{\overset{t}{\int }}\left(\parallel u_{tyyy}{\parallel }^2+\parallel u_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\theta }_{tyy}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s+C_3{\epsilon }^{-6}.$$

(3.37)

Similarly, by Theorem 1.1, Lemma 2.1, and Lemmas 3.1-3.2 and the embedding theorem, we have

$$D_1\le {\left(2C_3\right)}^{-1}\parallel u_{tyy}\left(t\right){\parallel }^2+C_3\left(\parallel u_{ty}\left(t\right){\parallel }^2+\parallel {\mathit{b}}_{ty}\left(t\right){\parallel }^2+\parallel {\theta }_t\left(t\right){\parallel }_{H^1}^2+\parallel u_y\left(t\right){\parallel }_{H^1}^2\right),$$

(3.38)

which, combined with (3.33), (3.37)-(3.38), Theorem 1.1, Lemma 2.1, and Lemmas 3.1-3.2, gives that for ε ∈ (0, 1) small enough,

$$\parallel u_{ty}\left(t\right){\parallel }^2+\underset{0}{\overset{t}{\int }}\parallel u_{tyy}\left(s\right){\parallel }^2\mathsf{\text{d}}s\le C_3{\epsilon }^{-6}+C_2{\epsilon }^2\underset{0}{\overset{t}{\int }}\left(\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\theta }_{tyy}{\parallel }^2+\parallel u_{tyyy}{\parallel }^2\right)\left(s\right)\mathsf{\text{d}}s.$$

(3.39)

On the other hand, differentiating (1.2) with respect to x and t, using Theorem 1.1 and Lemmas 3.1-3.2, we have

$$\begin{array}{c}\parallel u_{tyyy}(t)\parallel \le C_1\parallel u_{tty}(t)\parallel +C_2(\parallel u_{yy}(t){\parallel }_{H^2}^2+\parallel {\theta }_y(t){\parallel }_{H^1}^2+\parallel v_y(t){\parallel }_{H^1}^2\\ +\parallel {\mathit{b}}_y(t){\parallel }_{H^1}^2+\parallel {\theta }_t(t){\parallel }_{H^2}^2+\parallel {\mathit{b}}_t(t){\parallel }_{H^2}^2).\end{array}$$

(3.40)

Thus, inserting (3.40) into (3.39) implies estimate (3.30).

Analogously, we can obtain estimates (3.31)-(3.32). □

Lemma 3.4 Under the assumptions in Theorem 1.2, the following estimates hold for any t ∈ [0,T],

$$\begin{array}{l}\parallel u_{tt}(t){\parallel }^2+\parallel u_{ty}(t){\parallel }^2+\parallel {\mathit{w}}_{tt}(t){\parallel }^2+\parallel {\mathit{w}}_{ty}(t){\parallel }^2+\parallel {\mathit{b}}_{tt}(t){\parallel }^2+\parallel {\mathit{b}}_{ty}(t){\parallel }^2+\parallel {\theta }_{tt}(t){\parallel }^2\\ +\parallel {\theta }_{ty}(t){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel u_{tty}{\parallel }^2}+\parallel u_{tyy}{\parallel }^2+\parallel {\mathit{w}}_{tty}{\parallel }^2+\parallel {\mathit{w}}_{tyy}{\parallel }^2+\parallel {\mathit{b}}_{tty}{\parallel }^2\\ +\parallel {\mathit{b}}_{tyy}{\parallel }^2+\parallel {\theta }_{tty}{\parallel }^2+\parallel {\theta }_{tyy}{\parallel }^2)(s)ds\le C_4,\end{array}$$

(3.41)

$$\parallel v_{yyy}\left(t\right){\parallel }_{H^1}^2+\parallel v_{yy}\left(t\right){\parallel }_{{W^1}^{,\infty }}^2+\underset{0}{\overset{t}{\int }}\left(\parallel v_{yyy}{\parallel }_{H^1}^2+\parallel v_{yy}{\parallel }_{W^{1,\infty }}^2\right)\left(s\right)ds\le C_4,$$

(3.42)

$$\begin{array}{l}\parallel u_{yyy}(t){\parallel }_{H^1}^2+\parallel u_{yy}(t){\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel {\mathit{w}}_{yyy}(t){\parallel }_{H^1}^2+\parallel {\mathit{w}}_{yy}(t){\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel {\mathit{b}}_{yyy}(t){\parallel }_{H^1}^2+\parallel {\mathit{b}}_{yy}(t){\parallel }_{W^{\mathrm{1,}\infty }}^2\\ +\parallel {\theta }_{yyy}(t){\parallel }_{H^1}^2+\parallel {\theta }_{yy}(t){\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel v_{tyyy}(t){\parallel }^2+\parallel u_{tyy}(t){\parallel }^2+\parallel {\mathit{w}}_{tyy}(t){\parallel }^2+\parallel {\mathit{b}}_{tyy}(t){\parallel }^2\\ +\parallel {\theta }_{tyy}(t){\parallel }^2+{\displaystyle \underset{0}{\overset{t}{\int }}(\parallel u_{tt}{\parallel }^2}+\parallel {\mathit{w}}_{tt}{\parallel }^2+\parallel {\mathit{b}}_{tt}{\parallel }^2+\parallel {\theta }_{tt}{\parallel }^2+\parallel u_{yy}{\parallel }_{W^{2,\infty }}^2+\parallel {\mathit{w}}_{yy}{\parallel }_{W^{2,\infty }}^2\\ +\parallel {\mathit{b}}_{yy}{\parallel }_{W^{2,\infty }}^2+\parallel {\theta }_{yy}{\parallel }_{W^{2,\infty }}^2+\parallel {\theta }_{tyy}{\parallel }_{H^1}^2+\parallel u_{tyy}{\parallel }_{H^1}^2+\parallel {\mathit{w}}_{tyy}{\parallel }_{H^1}^2+\parallel {\mathit{b}}_{tyy}{\parallel }_{H^1}^2\\ +\parallel {\theta }_{ty}{\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel u_{ty}{\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel {\mathit{w}}_{ty}{\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel {\mathit{b}}_{ty}{\parallel }_{W^{\mathrm{1,}\infty }}^2+\parallel v_{tyyy}{\parallel }_{H^1}^2)(s)ds\le C_4,\end{array}$$

(3.43)

$$\underset{0}{\overset{t}{\int }}\left(\parallel u_{yyyy}{\parallel }_{H^1}^2+\parallel {\mathit{w}}_{yyyy}{\parallel }_{H^1}^2+\parallel {\mathit{b}}_{yyyy}{\parallel }_{H^1}^2+\parallel {\theta }_{yyyy}{\parallel }_{H^1}^2\right)\left(s\right)ds\le C_3.$$

(3.44)

Proof. Adding up (3.30)-(3.32), picking ε ∈ (0, 1) enough small, by Lemmas 3.1-3.3, and Gronwall's inequality, we get

$$\begin{array}{l}\parallel u_{ty}(t){}^{}\end{array}$$