Global existence and convergence rates for the smooth solutions to the compressible magnetohydrodynamic equations in the half space

Abstract

Background

With the characteristics of low pollution and low energy consumption, the magnetohydrodynamics has made widely attention. This paper provides the standard energy method to solve the magnetohydrodynamic equations (MHD) in the half space ${ℝ}_{+}^3$. It proves the global existence for the compressible (MHD) by combining the careful a priori estimates and the local existence result. This study also considers the large time behaviors of the solutions.

Results

The interactions between the viscous, compressible fluid motion and the magnetic field are modeled by the magnetohydrodynamic system which describes the coupling between the compressible Navier-Stokes equations and the magnetic equations. This study has applied the analytical method to obtain the solutions to (MHD) in ${ℝ}_{+}^3$. It proves that under the assumption that the initial data are close to the constant state, the global existence of smooth solutions can be established. Moreover, the various decay rates of such solutions in L^p-norm with 2≤p≤+∞ and their derivatives in L²-norm can also be derived from combining the decay estimates of the linearized system and the energy method.

Conclusions

This study demonstrates that the global existence and the decay rates for the compressible (MHD) can be established under the similar initial assumptions as for the compressible Navier-Stokes equations. Especially, the results suggest that if the initial velocity is small, the velocity decays at a certain rate. This implies that only under the initial assumption that the data are large, it may reach the requirements of (MHD) power generation, which can be used to achieve the value of industrial application and environmental protection.

Background

Magnetohydrodynamics, which combines the environmental fluid mechanics and electrodynamics theories to study the interaction discipline between the conduction fluid and electromagnetic, is the theory of the macroscopic, and it has spanned a very large range of applications (Gerebeau et al. [2006]). Due to the lower environmental pollution, especially in energy industry, magnetohydrodynamic power generation is used to conserve energy and mitigate pollution in order to protect the environment. In virtue of the industrial importance and theoretical challenges, the study on (MHD) has attracted many scientists. In the present paper, we are interested in the well-posedness theory of (MHD). Many results concerning the existence and uniqueness of (weak, strong or smooth) solutions in one dimension can be found in (Chen and Wang [2002], [2003]; Kawashima and Okada [1982]) and the references cited therein. In multi-dimensional case the global existence of weak solutions for the bounded domains has been established recently in (Ducomet and Feireisl [2006]; Tan and Wang [2009]). The local unique strong solution has been obtained in (Fan and Yu [2009]). In (Chen and Tan [2010], [2012]) we established the global existence and decay rates of the smooth solutions for the Cauchy problem. However, many fundamental problems for the compressible (MHD) in the half space are still open. In this paper, we will extend our results (Chen and Tan [2010]) to the initial boundary problem in the half space.

Results

In this paper, we will consider the initial boundary value problem for the compressible magnetohydrodynamic equations (MHD) in the half space ${ℝ}_{+}^3=\left\{x=(x^{\prime },x_3):x^{\prime }\in {ℝ}^2,x_3>0\right\}$ (cf. (Gerebeau et al. [2006])):

$$\left\{\begin{array}{c}{\rho }_t+\text{div}\left(\rho \mathbf{\text{u}}\right)=0,\\ {\left(\rho \mathbf{\text{u}}\right)}_t+\text{div}(\rho \mathbf{\text{u}}\otimes \mathbf{\text{u}}-\mathbf{\text{P}})={\mu }_0\text{curl}\mathbf{\text{H}}\times \mathbf{\text{H}},\\ {\mathbf{\text{H}}}_t-\text{curl}(\mathbf{\text{u}}\times \mathbf{\text{H}})+\frac{1}{\sigma {\mu }_0}\text{curlcurl}\mathbf{\text{H}}=0,\text{div}\mathbf{\text{H}}=0.\end{array}\right.$$

(1)

Here ρ,u=(u¹,u²,u³),H=(H¹,H²,H³) represent the density, velocity of the fluid and the magnetic field respectively. μ₀>0 stands for permeability of free space, and σ>0 is the electric conductivity. The stress tensor P is given by

$$\mathbf{\text{P}}=-p\mathbf{\text{I}}+\mu \left(\nabla \mathbf{\text{u}}+\nabla {\mathbf{\text{u}}}^T\right)+\lambda \text{div}\mathbf{\text{u}}\mathbf{\text{I}},$$

(2)

where p=p(ρ) is the pressure and the viscosity coefficients λ, μ satisfy

$$2\mu +3\lambda >0\text{and}\mu >0.$$

(3)

For convenience, we reformulate the system (1) as

$$\left\{\begin{array}{c}{\rho }_t+\text{div}\left(\rho \mathbf{\text{u}}\right)=0,\\ \rho {\mathbf{\text{u}}}_t+\rho \mathbf{\text{u}}·\nabla \mathbf{\text{u}}+\nabla p=\Delta \mathbf{\text{u}}+\nabla \text{div}\mathbf{\text{u}}+\text{curl}\mathbf{\text{H}}\times \mathbf{\text{H}},\\ {\mathbf{\text{H}}}_t+\text{curl}(\mathbf{\text{u}}\times \mathbf{\text{H}})-\Delta \mathbf{\text{H}}=0,\text{div}\mathbf{\text{H}}=0,\end{array}\right.$$

(4)

in $(0,\infty )\times {ℝ}_{+}^3$. Notice that we have normalized some physical constants to be unit but without reducing any essential difficulties for our analysis. We complement (2) the initial condition

$$(\rho ,\mathbf{\text{u}},\mathbf{\text{H}})(0,x)=\left({\rho }_0\right(x),{\mathbf{\text{u}}}_0(x),{\mathbf{\text{H}}}_0(x\left)\right),x\in {ℝ}_{+}^3,$$

(5)

and the following boundary conditions

$$\mathbf{\text{u}}{|}_{\{x_3=0\}}=\mathbf{\text{0}},\mathbf{\text{H}}{|}_{\{x_3=0\}}=\mathbf{\text{0}},$$

(6)

or

$$\mathbf{\text{u}}{|}_{\{x_3=0\}}=\mathbf{\text{0}},\mathbf{\text{H}}·\mathbf{\text{n}}{|}_{\{x_3=0\}}=0,\text{curl}\mathbf{\text{H}}\times \mathbf{\text{n}}{|}_{\{x_3=0\}}=\mathbf{\text{0}},$$

(7)

where n=(0,0,−1) is the normal vector of ${ℝ}_{+}^3$. We assume that throughout the paper the initial boundary data satisfy certain compatibility conditions as usual in (Matsumura and Nishida [1983]).

Before stating out our results, we shall introduce some standard notations.

Notations. We denote by L^p, W^{m, p} the usual Lebesgue and Sobolev spaces on ${ℝ}_{+}^3$ and H^m=W^m,2, with norms $|·{|}_{L^p},|·{|}_{W^{m,p}},|·{|}_{H^m}$ respectively. For the sake of conciseness, we do not precise in functional space names when they are concerned with scalar-valued or vector-valued functions. We denote ∇=∂ _x =(∂₁,∂₂,∂₃)^t, where ${\partial }_i={\partial }_{x_i}$, and put ${\partial }_x^{l}f={\nabla }^{l}f=\nabla \left({\nabla }^{l-1}f\right)$ for l=1,2,3,⋯. We assume that C be a positive generic constant throughout this paper that may vary at different places and the integration domain ${ℝ}_{+}^3$ will be always omitted without any ambiguity. Now our main results can be formulated as the following theorems. Firstly we state the results on the global existence and uniqueness of smooth solutions as:

Theorem 1.

Assume that the initial data are close enough to the constant state $(\bar{\rho },\mathbf{\text{0}},\mathbf{\text{0}})$ with $\bar{\rho }>0$, i.e., there exists a constant δ₀ such that

$$|\left({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0\right){|}_{H^3}\le {\delta }_0.$$

(8)

Then there exists a unique globally smooth solution (ρ,u,H) of the initial boundary problem (2)–(4) or (2), (3) and (5) such that for any t∈[ 0,∞), it holds

$$\begin{array}{c}\left|\right(\rho -\bar{\rho },\mathbf{\text{u}},\mathbf{\text{H}}\left)\right(·,t){|}_{H^3}^2+\underset{0}{\overset{t}{\int }}|{\partial }_x\rho (·,s){|}_{H^2}^2+\left|\right({\partial }_x\mathbf{\text{u}},{\partial }_x\mathbf{\text{H}}\left)\right(·,s){|}_{H^3}^2\mathit{\text{ds}}\\ \le C\left|\right({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0){|}_{H^3}^2.\end{array}$$

(9)

Remark 1.

We will only prove Theorem 1 under the boundary condition (4). Due to the special geometry of the boundary of the half space, a simple calculation shows that the boundary conditions on H in (5) are equivalent to the following Dirichlet-Neumann boundary conditions:

$${\partial }_3{H}^1{|}_{\{x_3=0\}}={\partial }_3{H}^2{|}_{\{x_3=0\}}=0,H^3{|}_{\{x_3=0\}}=0.$$

(10)

Hence to treat H¹,H² as in (Matsumura and Nishida [1983]), we can also prove Theorem 1 under the boundary conditions (5) in the similar way as we will proceed.

By imposing some additional conditions on the initial data we will establish the following various decay rates of the solutions obtained in Theorem 1:

Theorem 2.

Let (ρ,u, H) be the solution obtained in Theorem 1 and assume in addition that the initial data $({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0)\in L^1\left({ℝ}_{+}^3\right)$ and there exists δ₁>0 such that

$$\left|\right({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0){|}_{L^1}+|({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0){|}_{H^3}<{\delta }_1,$$

(11)

then for all t≥0, it holds that

$$\left|\right(\rho -\bar{\rho },\mathbf{\text{u}},\mathbf{\text{H}}\left)\right(t){|}_{L^p}=O\left({(1+t)}^{-\frac{3}{2}\left(1-\frac{1}{p}\right)}\right),\forall p\in [2,\infty ],$$

(12)

$$\left|{\partial }_x\right(\rho -\bar{\rho },\mathbf{\text{u}},\mathbf{\text{H}}\left)\right(t){|}_{L^2}=O\left({(1+t)}^{-\frac{5}{4}}\right),$$

(13)

and

$$\left|\right(\rho -\bar{\rho },\mathbf{\text{u}},\mathbf{\text{H}}\left)\right(t){|}_{H^3}\le C({\delta }_1)\left({(1+t)}^{-\left(\frac{3}{4}-{\epsilon }_1\right)}\right),\forall 0<{\epsilon }_1<\tilde{\epsilon },$$

(14)

where $\tilde{\epsilon }$ is some positive number.

Moreover, if the data satisfy $({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0)\in H^4\left({ℝ}_{+}^3\right)$ and there exists δ₂>0 such that

$$\left|\right({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0){|}_{L^1}+|({\rho }_0-\bar{\rho },{\mathbf{\text{u}}}_0,{\mathbf{\text{H}}}_0){|}_{H^4}<{\delta }_2,$$

(15)

then

$$\left|{\partial }_x\right(\rho -\bar{\rho },\mathbf{\text{u}},\mathbf{\text{H}}\left)\right(t){|}_{H^3}\le C({\delta }_2){(1+t)}^{-\left(\frac{5}{4}-{\epsilon }_2\right)},\forall 0<{\epsilon }_2<\hat{\epsilon },$$

(16)

for all t≥0, where $\hat{\epsilon }<\tilde{\epsilon }$ is a positive number. In fact, it holds

$$\left|\left({\partial }_x^2\rho ,{\partial }_x^2\mathbf{\text{u}},{\partial }_x^2\mathbf{\text{H}},{\partial }_x^3\mathbf{\text{H}}\right)\right(t){|}_{L^2}\le C({\delta }_2){(1+t)}^{-\frac{5}{4}},$$

(17)

for all t≥0.

We will prove the global existence of smooth solutions by the standard energy method in spirit of (Matsumura and Nishida [1983], [1979], [1980]). And we remark that we can also obtain the global existence of strong solutions for the small initial data in the H²-framework, which can be proved in the similar way. On the other hand, the L²- L^∞ decay rates for the smooth solutions and the L² decay rates for the derivatives of first order are optimal since (9)–(10) concerning $(\rho -\bar{\rho },\mathbf{\text{u}})$ and H are the same as the optimal decay rates for the compressible Navier-Stokes equations (Kagei and Kobayashi [2005]) and the heat equation respectively. Related convergence rates of solutions for the Navier-Stokes equations on the unbounded domain can be found in (Kobayashi [2002]; Kobayashi and Shibata [1999]; Matsumura and Nishida [1979]) and the references cited therein. Although our proofs are in spirit of those for the Navier-Stokes equations, (Kagei and Kobayashi [2005]; Kobayashi [2002]; Kobayashi and Shibata [1999]; Matsumura and Nishida [1979], [1980]), we should derive the new estimates arising from the presence of the magnetic field and overcome the strong coupling between the mass, momentum equations and magnetic equation. However, it is easy to obtain the optimal decay rates of H and its first derivatives by the properties of heat kernel. Indeed, we can rewrite the equation (19) ₃ analogously to the form of (19) ₁–(19) ₂, i.e.,

$$\left\{\begin{array}{c}{0}_t+\text{div}\mathbf{\text{H}}=0,\\ {\mathbf{\text{H}}}_t+\nabla 0-\Delta \mathbf{\text{H}}-\nabla \text{div}\mathbf{\text{H}}=S_3.\end{array}\right.$$

(18)

Thus, we can get the estimates of (0, H) which are similar to (ϱ,v) in the system (19) ₁–(19) ₂. Moreover, we will get the better decay rate of the magnetic field by the elliptic system.

Discussion

As well known, the heavy emissions of Greenhouse gases, such as C O₂, C H₄, N₂O, S F₆ cause global warming, and also result in a great deal of harm to the environment. It has been a hot topic and widespread concern to study on how to strictly control the greenhouse gases emissions. In order to profoundly reduce the environment pollution, we must focus on energy structure adjustment. Without any course of mechanical motion, Magnetohydrodynamics (MHD) power generation technology, also called plasma power generation technology, transforms thermal energy and kinetic energy directly into electricity. Thus by applying (MHD) power generation technology, we can realize the desulphuriz and reduce the production of N O _x effectively, so as to achieve the effect of high efficiency and low pollution.

To complete the (MHD) generation process, which is of high industrial application value, a conductive gas (plasma) will be directed through a magnetic field with a large velocity, under a high temperature condition. In this situation, how to control the initial velocity of the conductive gas has to be considered. From the results in Section Results, we can conclude that if we assume that the initial data are close enough to the constant state, then there exists a unique globally solution to the (MHD) system and the solution decays at some rates. This indicates that if the initial velocity is sufficiently small, although the solution to the (MHD) system exists globally, then the velocity will decays and never be large, which implies that it may never reach the requirements of (MHD) power generation. However, the problem of the global existence of the solutions with the large initial data is still open.

Conclusions

In this paper, we demonstrate that the global existence and the decay rates for the compressible (MHD) in ${ℝ}_{+}^3$ can be established under the similar initial assumptions as for the compressible Navier-Stokes equations which can be seen in (Matsumura and Nishida [1983]). It implies that the magnetic field does not affect the decay rates of the velocity. Indeed, the results (9)–(13) in Theorem 2 suggest that the decay rates for the derivatives of the magnetic field are the same as the velocity’s. And in (14), we cannot get the estimates for ${\partial }_x^3\mathbf{\text{u}}$ but ${\partial }_x^3\mathbf{\text{H}}$. Furthermore, the results suggest that if the initial velocity is small, the velocity decays at the optimal rate. This implies that it may never reach the requirements of (MHD) power generation unless giving the gas an large initial velocity.

Methods

Proof of theorem 1

In this section, we will prove the existence part of Theorem 1 and the uniqueness is standard so it will be omitted.

Some elementary inequalities

The first bright idea to reduce many complicated computations lies in that we just need to do the lowest-order and highest-order energy estimates for the solutions. This is motivated by the following observation:

$$|f{|}_{H^k}^2\le C|\left(f,{\partial }_x^{k}f\right){|}_{L^2}^2,\forall f\in H^k\left({ℝ}_{+}^3\right).$$

(19)

The inequality (15) can be easily proved by combing Young’s inequality and Gagliardo-Nirenberg’s inequality

$$|{\partial }_x^{i}f{|}_{L^p}\le C(p\left)\right|f{|}_{L^q}^{\alpha }|{\partial }_x^{k}f{|}_{L^r}^{(1-\alpha )},\forall f\in H^k\left({ℝ}_{+}^3\right)$$

(20)

where $\frac{1}{p}-\frac{i}{3}=\frac{1}{q}\alpha +\left(\frac{1}{r}-\frac{k}{3}\right)(1-\alpha )$ with i≤k. Indeed (16) can be proved by the extension technique together with Gagliardo-Nirenberg’s inequality in the whole space. We can obtain the following useful inequality by Hölder inequality and (16):

$$\begin{array}{cc}\left|{\partial }_x^k\right(\mathit{\text{fg}}){|}_{L^2}& \le C\left(\left|f{|}_{L^{\infty }}\right|{\partial }_x^{k}g{|}_{L^2}+\left|{\partial }_x^{k}f{|}_{L^2}\right|g{|}_{L^{\infty }}\right),\\ \forall f,g\in H^k\left({ℝ}_{+}^3\right),k\ge 2\end{array}$$

(21)

and the general form of (3) can be deduced directly in the following:

$$\begin{array}{c}|f_1{f}_2\cdots f_s{|}_k\le C{\sum }_{j=1}^s|f_1{|}_{L^{\infty }}\cdots \left|f_{j-1}{|}_{L^{\infty }}\right|f_j{|}_k|f_{j+1}{|}_{L^{\infty }}\cdots |f_s{|}_{L^{\infty }},\\ \forall f_j\in H^k\left({ℝ}_{+}^3\right),k\ge 2,\mathit{\text{for}}j=1,2,\cdots ,\mathrm{s.}\end{array}$$

(22)

The linearized system

We will linearize the problem (2)–(4) as follows. Setting $\gamma =\sqrt{P^{\prime }\left(\bar{\rho }\right)},\mu =1/\bar{\rho }$ and introducing new variables by

$$\begin{array}{c}\varrho =\rho -\bar{\rho },\mathbf{\text{v}}=\frac{1}{\mathrm{\mu \gamma }}\mathbf{\text{u}},\mathbf{\text{H}}=\mathbf{\text{H}}.\end{array}$$

(23)

Hence the initial boundary value problem (2)–(4) can be reformulated as

$$\left\{\begin{array}{c}{\varrho }_t+\gamma \text{div}\mathbf{\text{v}}=S_1,\\ {\mathbf{\text{v}}}_t+\gamma \nabla \varrho -\mathrm{\mu \Delta }\mathbf{\text{v}}-\mu \nabla \text{div}\mathbf{\text{v}}=S_2,\\ {\mathbf{\text{H}}}_t-\Delta \mathbf{\text{H}}=S_3,\text{div}\mathbf{\text{H}}=0,\\ (\varrho ,\mathbf{\text{v}},\mathbf{\text{H}})(x,0)=({\varrho }_0,{\mathbf{\text{v}}}_0,{\mathbf{\text{H}}}_0)\left(x\right)=({\rho }_0-\bar{\rho },{\mathbf{\text{v}}}_0,{\mathbf{\text{H}}}_0),\\ \mathbf{\text{v}}{|}_{\{x_3=0\}}=\mathbf{\text{H}}{|}_{\{x_3=0\}}=\mathbf{\text{0}},\end{array}\right.$$

(24)

where

$$\begin{array}{c}{S}_1=-\mathrm{\mu \gamma }\text{div}\left(\varrho \mathbf{\text{v}}\right),\end{array}$$

(25)

$$\begin{array}{cc}{S}_2=& \frac{1}{\mathrm{\mu \gamma \rho }}\text{curl}\mathbf{\text{H}}\times \mathbf{\text{H}}+\left(\frac{1}{\rho }-\frac{1}{\bar{\rho }}\right)\Delta \mathbf{\text{v}}+\left(\frac{1}{\rho }-\frac{1}{\bar{\rho }}\right)\nabla \text{div}\mathbf{\text{v}}\\ -\mathrm{\mu \gamma }\mathbf{\text{v}}·\nabla \mathbf{\text{v}}-\frac{1}{\mathrm{\mu \gamma }}\left[\frac{P^{\prime }\left(\rho \right)}{\rho }-\frac{P^{\prime }\left(\bar{\rho }\right)}{\bar{\rho }}\right]\nabla \varrho ,\end{array}$$

(26)

and

$$\begin{array}{c}{S}_3=-\mathrm{\mu \gamma }\text{curl}(\mathbf{\text{v}}\times \mathbf{\text{H}}).\end{array}$$

(27)

In order to state our results more concisely, we define an energy functional as:

$$\begin{array}{cc}N(t_1,t_2)& =\left\{\underset{t_1\le t\le t_2}{sup}\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}\left)\right(t)\underset{3}{\overset{2}{|}}+\underset{t_1}{\overset{t_2}{\int }}|{\partial }_x\varrho \left(s\right)\underset{2}{\overset{2}{|}}\right.\\ +{\left.\left|\right({\partial }_x\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}\left)\right(s){|}_3^2\mathit{\text{dx}}\right\}}^{\frac{1}{2}},\end{array}$$

(28)

and change the condition of the initial data (6) as

$$\left|\right({\varrho }_0,{\mathbf{\text{v}}}_0,{\mathbf{\text{H}}}_0){|}_{H^3}\le {\delta }_0^{\prime }=max\left(\frac{1}{2\mathrm{\mu \gamma }},1\right){\delta }_0.$$

(29)

Local and global existence

We will finish the proof of Theorem 1 in this subsection. First we state out the local existence without proof, since it can be proved in a standard way (Matsumura and Nishida [1980]) or can be found in (Ströhmer [1990], Vol’pert and Hudjaev [1972]):

Theorem 3.

(local existence) Under the assumption (23), there exists a positive constant T such that the initial boundary value problem (19) has a unique solution (ϱ,v,H) which is continuous in $[0,T]\times {ℝ}^3$ together with its derivatives of first order in t and of second order in x. Moreover, there exists a constant C₁>1 such that it holds N(0,t)≤C₁N(0,0), for any t∈[ 0,T].

We will prove in this subsection the following a priori estimate:

Theorem 4.

(a priori estimate) There exists a constant δ≪1 such that if N(0,T)≤δ, then there exists a constant C₂>1 such that N(0,T)≤C₂N(0,0).

The global existence of smooth solutions will be proved via a continued argument by combining the local existence theorem and the a priori estimate theorem. We shall state the global existence of smooth solutions to the linearized problem (19) as follows.

Proposition 3.1.

(global existence) Under the assumptions of Theorem 1, the initial boundary value problem (19) has a unique global solution such that for t∈[ 0,∞), it holds N(0,t)≤C N(0,0). Thus (ρ,u,H) which satisfies (4) uniquely solves the initial boundary value problem (2)–(4) for all time.

Proof.

See in (Chen and Tan [2010]). □

A priori estimates

We observe that the a priori assumption in Theorem 4 and the embedding inequality together with the continuity equation (19) ₁ imply

$$\begin{array}{c}\underset{0\le t\le T}{sup}\left|\right(\varrho ,{\varrho }_t,{\partial }_x\varrho ,\mathbf{\text{v}},{\partial }_x\mathbf{\text{v}},\mathbf{\text{H}},{\partial }_x\mathbf{\text{H}}\left)\right(t\left)\right|\\ \le C\underset{0\le t\le T}{sup}\left|\right(\rho -\bar{\rho },\mathbf{\text{v}},\mathbf{\text{H}}\left)\right(·,t){|}_{H^3}\le \mathrm{C\delta .}\end{array}$$

(30)

In particular

$$\begin{array}{c}\frac{\bar{\rho }}{2}\le \rho =\varrho +\bar{\rho }\le 2\bar{\rho }.\end{array}$$

(31)

In the sequel, we will always use the smallness assumption of δ and (24)–(25).

Next we shall do some preparatory work from Lemma 3.1 to Lemma 3.7. Firstly, we regard the equations (19) ₂–(19) ₃ as the elliptic system with respect to x variables, i.e.,

$$\left\{\begin{array}{c}\Delta \mathbf{\text{v}}+\nabla \text{div}\mathbf{\text{v}}=\frac{1}{\mu }{\mathbf{\text{v}}}_t+\frac{\gamma }{\mu }\nabla \rho -\frac{1}{\mu }{S}_2,\\ \Delta \mathbf{\text{H}}={\mathbf{\text{H}}}_t-S_3,\\ \mathbf{\text{v}}{|}_{\{x_3=0\}}=\mathbf{\text{0}},\mathbf{\text{H}}{|}_{\{x_3=0\}}=\mathbf{\text{0}}.\end{array}\right.$$

(32)

Thus we have the following estimates which we can found in (Cho et al. [2004]):

Lemma 3.1.

Under the assumptions of Theorem 4, we have for k=2,3,4 that

$$\begin{array}{c}|{\partial }_x^k\mathbf{\text{v}}{|}_{L^2}\le C\left\{|{\mathbf{\text{v}}}_t{|}_{k-2}+|{\partial }_x\varrho {|}_{k-2}+|S_2{|}_{k-2}+|\mathbf{\text{v}}{|}_{L^2}\right\},\end{array}$$

(33)

and

$$\begin{array}{c}|{\partial }_x^k\mathbf{\text{H}}{|}_{L^2}\le C\left\{|{\mathbf{\text{H}}}_t{|}_{k-2}+|S_3{|}_{k-2}+|{\partial }_x\mathbf{\text{H}}{|}_{L^2}\right\}.\end{array}$$

(34)

Next we derive the following stokes equation from the equations (19) ₁–(19) ₂:

$$\left\{\begin{array}{c}\gamma \text{div}\mathbf{\text{v}}=g=-\frac{\mathrm{d\rho }}{\mathit{\text{dt}}}-\mathrm{\mu \gamma \varrho }\text{div}\mathbf{\text{v}},\\ -\mathrm{\mu \Delta }\mathbf{\text{v}}+\gamma \nabla \rho =\mathbf{\text{h}}=-{\mathbf{\text{v}}}_t-\frac{\mu }{\gamma }\nabla g+S_2,\\ \mathbf{\text{v}}{|}_{\{x_3=0\}}=0.\end{array}\right.$$

(35)

We have the following estimates which can be found in (Galdi et al. [1994]):

Lemma 3.2.

Under the assumptions of Theorem 4, we have for k=2,3,4 that

$$\begin{array}{c}|{\partial }_x^k\mathbf{\text{v}}{|}_{L^2}^2+|{\partial }_x^{k-1}\rho {|}_{L^2}^2\le C\left\{|g{|}_{k-1}^2+|h{|}_{k-2}^2\right\}.\end{array}$$

(36)

Next we shall do the estimates for the terms contained in N(0,t).

Lemma 3.3.

Under the assumptions of Theorem 4, we have that

$$\begin{array}{c}\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }_x(\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2,\end{array}$$

(37)

and

$$\begin{array}{c}\left|\right({\varrho }_t,{\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t){|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }_x({\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t){|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(38)

Proof.

By multiplying the equations (19) ₁–(19) ₃ by ϱ, v and H respectively, integrating over ${ℝ}_{+}^3$ and adding the resulting equations, we have

$$\begin{array}{c}\frac{1}{2}\frac{d}{\mathit{\text{dt}}}\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2+\mu |(\text{div},{\partial }_x\mathbf{\text{v}}){|}_{L^2}^2+|{\partial }_x\mathbf{\text{H}}{|}_{L^2}\\ =〈S_1,\varrho 〉+〈S_2,\mathbf{\text{v}}〉+〈S_3,\mathbf{\text{H}}〉.\end{array}$$

(39)

We shall estimate the terms in the right-hand side term by term as:

$$\begin{array}{ll}〈S_1,\varrho 〉& =\int \mathrm{\gamma \mu }\left(\varrho \mathbf{\text{v}}\right)·{\partial }_x\mathrm{\varrho dx}\le C\left|\varrho {|}_{L^3}\right|\mathbf{\text{v}}{|}_{L^6}|{\partial }_x\varrho {|}_{L^2}\\ \le C\left|\varrho {|}_{H^1}\right|({\partial }_x\mathbf{\text{v}},{\partial }_x\varrho ){|}_{L^2}^2\le \mathrm{C\delta }\left|\right({\partial }_x\mathbf{\text{v}},{\partial }_x\varrho ){|}_{L^2}^2,\\ 〈S_2,\mathbf{\text{v}}〉& \le C\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}\left){|}_{L^3}\right|({\partial }_x\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^2}|\mathbf{\text{v}}{|}_{L^6}\\ \le C\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}\left){|}_{H^1}\right|({\partial }_x\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^2}^2\\ \le \mathrm{C\delta }\left|\right({\partial }_x\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^2}^2,\\ 〈S_3,\mathbf{\text{H}}〉& \le C\left|\right(\mathbf{\text{v}},\mathbf{\text{H}}\left){|}_{L^3}\right|({\partial }_x\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^6}|\mathbf{\text{H}}{|}_{L^2}\\ \le C\left|\right(\mathbf{\text{v}},\mathbf{\text{H}}\left){|}_{H^1}\right|({\partial }_x\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^6}|\mathbf{\text{H}}{|}_{L^2}\\ \le \mathrm{C\delta }\left|\right({\partial }_x\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^2}^2,\end{array}$$

(40)

where Hölder’s inequality and Sobolev’s inequality are used. Thus by integrating the above inequality in time and the definition of N(0,t), we have got (30).

Similarly, by taking ∂ _t to (19) ₁–(19) ₃, multiplying by ϱ _t , v _t and H _t respectively and integrating the resulting inequalities, we obtain

$$\begin{array}{c}\left|\right({\varrho }_t,{\mathit{v}}_t,{\mathit{H}}_t){|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }_x({\mathit{v}}_t,{\mathit{H}}_t){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+C\left|\underset{0}{\overset{t}{\int }}〈{\partial }_t{S}_1,{\varrho }_t〉+〈{\partial }_t{S}_2,v_t〉+〈{\partial }_t{S}_3,{\mathit{H}}_t〉\mathit{\text{ds}}\right|.\end{array}$$

(41)

Thus we have to estimate the terms in the right-hand side of the above inequality.

$$\begin{array}{cc}〈{\partial }_x{S}_1,{\varrho }_t〉& =\int \mathrm{\gamma \mu }{\partial }_t(\varrho \text{div}\mathbf{\text{v}}+\nabla \varrho ·\mathbf{\text{v}}){\partial }_t\mathrm{\varrho dx}\\ \le C\left|\right(\varrho ,{\partial }_x\varrho ,\mathbf{\text{v}},{\partial }_x\mathbf{\text{v}}\left){|}_{L^3}\right|({\varrho }_t,{\partial }_x{\varrho }_t,{\mathbf{\text{v}}}_t,{\partial }_x{\mathbf{\text{v}}}_t){|}_{L^2}|{\varrho }_t{|}_{L^6}\\ \le C\left|\right(\varrho ,{\partial }_x\varrho ,\mathbf{\text{v}},{\partial }_x\mathbf{\text{v}}\left){|}_{H^1}\right|({\varrho }_t,{\partial }_x{\varrho }_t,{\mathbf{\text{v}}}_t,{\partial }_x{\mathbf{\text{v}}}_t){|}_{L^2}^2\\ \le \mathrm{C\delta }\left|\right({\partial }_x\varrho ,{\partial }_x^2\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x^3\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}},{\partial }_x^2\mathbf{\text{H}}){|}_{L^2}^2\\ \le \mathrm{C\delta N}{(0,t)}^2,\end{array}$$

(42)

where we can get the L²-norm estimates of ϱ _t , ∂ _x ϱ _t , v _t , and ∂ _x v _t with the aid of the equation (19) ₁–(19) ₂ and (24). Similarly we have that

$$\begin{array}{cc}〈{\partial }_t{S}_2,{\mathbf{\text{v}}}_t〉& \le C|〈{\partial }_t(\varrho \text{curl}\mathbf{\text{H}}\times \mathbf{\text{H}},\mathrm{\varrho \Delta }\mathbf{\text{v}},\varrho \nabla \text{div}\mathbf{\text{v}},\mathbf{\text{v}}·\nabla \mathbf{\text{v}},\varrho ·\nabla \varrho ),{\mathbf{\text{v}}}_t〉|\\ \le C\left|\right(\varrho ,{\varrho }_t,\mathbf{\text{v}},{\partial }_x\mathbf{\text{v}},\mathbf{\text{H}},{\partial }_x\mathbf{\text{H}}){|}_{L^3}\\ \times \left|\right({\partial }_x\varrho ,{\partial }_x{\varrho }_t,{\mathbf{\text{v}}}_t,{\partial }_x^2\mathbf{\text{v}},{\partial }_x{\mathbf{\text{v}}}_t,{\partial }_x^2{\mathbf{\text{v}}}_t,{\partial }_x\mathbf{\text{H}},{\mathbf{\text{H}}}_t,{\partial }_x{\mathbf{\text{H}}}_t\left){|}_{L^2}\right|{\mathbf{\text{v}}}_t{|}_{L^6}\\ \le C\left|\right(\varrho ,{\partial }_x\varrho ,\mathbf{\text{v}},{\partial }_x\mathbf{\text{v}},\mathbf{\text{H}},{\partial }_x\mathbf{\text{H}}){|}_{H^1}\\ \times \left|\right({\partial }_x\varrho ,{\partial }_x^2\varrho ,{\partial }_x^3\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x^3\mathbf{\text{v}},{\partial }_x^4\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}},{\partial }_x^2\mathbf{\text{H}},{\partial }_x^3\mathbf{\text{H}}){|}_{L^2}^2\\ \le \mathrm{C\delta }\left|\right({\partial }_x\varrho ,{\partial }_x^2\varrho ,{\partial }_x^3\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x^3\mathbf{\text{v}},{\partial }_x^4\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}},{\partial }_x^2\mathbf{\text{H}},{\partial }_x^3\mathbf{\text{H}}){|}_{L^2}^2,\end{array}$$

(43)

and

$$\begin{array}{cc}〈{\partial }_t{S}_3,{\mathbf{\text{H}}}_t〉& \le C|〈{\partial }_t\text{curl}(\mathbf{\text{v}}\times \mathbf{\text{H}}),{\mathbf{\text{H}}}_t〉|\\ \le C\left|\right(\mathbf{\text{v}},{\partial }_x\mathbf{\text{v}},\mathbf{\text{H}},{\partial }_x\mathbf{\text{H}}\left){|}_{L^3}\right|({\mathbf{\text{v}}}_t,{\partial }_x{\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t,{\partial }_x{\mathbf{\text{H}}}_t){|}_{L^2}|{\mathbf{\text{H}}}_t{|}_{L^6}\\ \le C\left|\right(\mathbf{\text{v}},\mathbf{\text{H}}\left){|}_{H^2}\right|({\partial }_x\varrho ,{\partial }_x^2\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x^3\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}},{\partial }_x^2\mathbf{\text{H}},{\partial }_x^3\mathbf{\text{H}}){|}_{L^2}^2\\ \le \mathrm{C\delta }\left|\right({\partial }_x\varrho ,{\partial }_x^2\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x^3\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}},{\partial }_x^2\mathbf{\text{H}},{\partial }_x^3\mathbf{\text{H}}){|}_{L^2}^2.\end{array}$$

(44)

Together with these inequalities, we can deduce the inequality (31). Hence the proof of Lemma 3.3 is complete. □

Next we estimate the L²-norm of the first derivatives of v and H.

Lemma 3.4.

Under the assumptions of Theorem 4, we have

$$\begin{array}{cc}\left|{\partial }_x\right(\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2& +\underset{0}{\overset{t}{\int }}\left|\right({\varrho }_t,{\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t\left)\right(s){|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2\\ +\mathrm{C\delta N}{(0,t)}^2,\end{array}$$

(45)

and

$$\begin{array}{cc}\left|{\partial }_x\right({\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t){|}_{L^2}^2& +\underset{0}{\overset{t}{\int }}\left|\right({\varrho }_{\mathit{\text{tt}}},{\mathbf{\text{v}}}_{\mathit{\text{tt}}},{\mathbf{\text{H}}}_{\mathit{\text{tt}}}\left)\right(s){|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2\\ +\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(46)

Proof.

By multiplying ϱ _t , v _t and H _t to the equations (19) ₁–(19) ₃ respectively, integrating over ${ℝ}_{+}^3$ and adding the resulting equations, we have

$$\begin{array}{c}\frac{d}{\mathit{\text{dt}}}\int \left(\frac{\mu }{2}|{\partial }_x\mathbf{\text{v}}{|}^2+\frac{\mu }{2}|\text{div}\mathbf{\text{v}}{|}^2+\frac{1}{2}|{\partial }_x\mathbf{\text{H}}{|}^2-\mathrm{\gamma \varrho }\text{div}\mathbf{\text{v}}\right)\mathit{\text{dx}}\\ +\int \left(\underset{t}{\overset{2}{\varrho }}+|{\mathbf{\text{v}}}_t{|}^2+|{\mathbf{\text{H}}}_t{|}^2+2\gamma {\varrho }_t\text{div}\mathbf{\text{v}}\right)\mathit{\text{dx}}\\ =\int 〈S_1,{\varrho }_t〉+〈S_2,{\mathbf{\text{v}}}_t〉+〈S_3,{\mathbf{\text{H}}}_t〉\mathit{\text{dx}}\\ \le \epsilon \left(|{\varrho }_t{|}_{L^2}^2+|{\mathbf{\text{v}}}_t{|}_{L^2}^2+|{\mathbf{\text{H}}}_t{|}_{L^2}^2\right)+C\left(\epsilon \right)\left(|S_1{|}_{L^2}^2+|S_2{|}_{L^2}^2+|S_3{|}_{L^2}^2\right).\end{array}$$

(47)

Since

$$\begin{array}{cc}|S_1{|}_{L^2}^2& \le C\left|\right(\varrho ,\mathbf{\text{v}}\left){|}_{L^{\infty }}\right|{\partial }_x(\varrho ,\mathbf{\text{v}}){|}_{L^2}^2\le \mathrm{C\delta }\left|{\partial }_x\right(\varrho ,\mathbf{\text{v}}){|}_{L^2}^2,\end{array}$$

(48)

$$\begin{array}{cc}|S_2{|}_{L^2}^2& \le C\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}\left){|}_{L^{\infty }}\right|({\partial }_x\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^2}^2\\ \le \mathrm{C\delta }\left|\right({\partial }_x\varrho ,{\partial }_x\mathbf{\text{v}},{\partial }_x^2\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}){|}_{L^2}^2,\end{array}$$

(49)

and

$$\begin{array}{c}|S_3{|}_{L^2}^2\le C|(\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^{\infty }}\left|{\partial }_x\right(\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2\le \mathrm{C\delta }|{\partial }_x(\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2,\end{array}$$

(50)

thus by integrating (34) in time and with the aid of the smallness of ε and Cauchy’s inequality, we obtain

$$\begin{array}{c}\left|{\partial }_x\right(\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|({\varrho }_t,{\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t)\left(s\right){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\left(|\varrho {|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }_x\mathbf{\text{v}}\left(s\right){|}_{L^2}^2\mathit{\text{ds}}\right),\end{array}$$

(51)

which together with (30) yields (32). And we can also get (33) in the similar way. The proof of Lemma 3.4 is complete. □

Next we estimate the L²-Norms of the first derivatives of ϱ. We shall divide the estimates into two parts. Firstly we denote the tangential derivatives by ∂=(∂₁,∂₂). And it is easy to see that the tangential derivatives of the solution of (19) satisfy the same boundary conditions in (19).

By taking ∂ to the equations (19) ₁–(19) ₂ and multiplying them with ∂ ϱ and ∂ v respectively, we can deduce that

$$\begin{array}{c}\frac{1}{2}\frac{d}{\mathit{\text{dt}}}\int |\mathrm{\partial \varrho }{|}^2+|\partial \mathbf{\text{v}}{|}^2\mathit{\text{dx}}+\int \mu |\partial {\partial }_x\mathbf{\text{v}}{|}^2+\mu |\partial \text{div}\mathbf{\text{v}}{|}^2\mathit{\text{dx}}\\ =〈\partial S_1,\mathrm{\partial \varrho }〉+〈\partial S_2,\partial \mathbf{\text{v}}〉.\end{array}$$

(52)

Define the material derivative $\frac{\mathrm{d\rho }}{\mathit{\text{dt}}}={\rho }_t+\nabla \rho ·\mathbf{\text{u}}$. Then by the continuity equation (19) ₁ and the formula of variable substitution (18), we have

$$\begin{array}{c}\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}=-\gamma \text{div}\mathbf{\text{v}}-\mathrm{\mu \gamma \varrho }\text{div}\mathbf{\text{v}},\end{array}$$

(53)

where $\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}=\frac{\mathrm{d\rho }}{\mathit{\text{dt}}}={\varrho }_t+\mathrm{\mu \gamma }\nabla \varrho ·\mathbf{\text{v}}$. By taking ∂ to the equation (36), we can obtain the following inequality:

$${\left|\partial \left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right|}_{L^2}^2\le C\left(\partial {\partial }_x\mathbf{\text{v}}{|}_{L^2}^2+\left|\partial \right(\varrho \text{div}\mathbf{\text{v}}){|}_{L^2}^2\right),$$

(54)

then by multiplying a small enough constant α to it, together with (35), and integrating in time, we have

$$\begin{array}{c}\left|\partial \right(\varrho ,\mathbf{\text{v}}){|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\partial \left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}},{\partial }_x\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+C\left|\underset{0}{\overset{t}{\int }}〈\partial S_1,\mathrm{\partial \varrho }〉+〈\partial S_2,\partial \mathbf{\text{v}}〉\mathit{\text{ds}}\right|\\ +C\underset{0}{\overset{t}{\int }}\left|\partial \right(\varrho \text{div}\mathbf{\text{v}}){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathit{\text{CN}}{(0,t)}^2+C\left|\underset{0}{\overset{t}{\int }}〈\partial S_1,\mathrm{\partial \varrho }〉+〈\partial S_2,\partial \mathbf{\text{v}}〉\mathit{\text{ds}}\right|.\end{array}$$

(55)

Similarly, we can obtain the following lemma and we omit the proof of it.

Lemma 3.5.

Under the assumptions of Theorem 4, we have for k = 1, 2, 3 that

$$\begin{array}{c}\left|{\partial }^k\right(\varrho ,\mathbf{\text{v}}){|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|{\partial }^k\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}},{\partial }_x\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathit{\text{CN}}{(0,t)}^2+C\left|\underset{0}{\overset{t}{\int }}〈{\partial }^k{S}_1,{\partial }^k\varrho 〉\right.\\ \left.+〈{\partial }^k{S}_2,{\partial }^k\mathbf{\text{v}}〉\mathit{\text{ds}}\right|.\end{array}$$

(56)

Next we have to obtain the estimates for the normal derivatives of solution. We can derive the following equations from (19) ₁–(19) ₂.

$$\begin{array}{c}{\partial }_3{\varrho }_t+\gamma {\partial }_3\text{div}\mathbf{\text{v}}=-\mathrm{\mu \gamma }{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right)-\mathrm{\mu \gamma }{\partial }_3(\nabla \varrho ·\mathbf{\text{v}}),\\ {\mathbf{\text{v}}}_t^3+\gamma {\partial }_3\varrho -\mathrm{\mu \Delta }{\mathbf{\text{v}}}^3-\mu {\partial }_3\text{div}\mathbf{\text{v}}=S_2^3,\end{array}$$

(57)

where $S_2={\left(S_2^1,S_2^2,S_2^3\right)}^T$. To eliminate the term ∂₃₃v³, we have the following equality from the above equalities

$$\begin{array}{cc}\frac{2\mu }{\gamma }{\partial }_3{\varrho }_t+\gamma {\partial }_3\varrho & =-{\mathbf{\text{v}}}_t^3+\mu \left({\partial }_{11}{\mathbf{\text{v}}}^3+{\partial }_{22}{\mathbf{\text{v}}}^3+{\partial }_{13}{\mathbf{\text{v}}}^1+{\partial }_{23}{\mathbf{\text{v}}}^2\right)\\ -2{\mu }^2{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right)-2{\mu }^2{\partial }_3(\nabla \varrho ·\mathbf{\text{v}})+S_2^3.\end{array}$$

(58)

And we can also derive the following equality from (36)

$$\begin{array}{cc}\frac{2\mu }{\gamma }{\partial }_3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)+\gamma {\partial }_3\varrho & =-{\mathbf{\text{v}}}_t^3+\mu \left({\partial }_{11}{\mathbf{\text{v}}}^3+{\partial }_{22}{\mathbf{\text{v}}}^3+{\partial }_{13}{\mathbf{\text{v}}}^1+{\partial }_{23}{\mathbf{\text{v}}}^2\right)\\ -2{\mu }^2{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right)+S_2^3.\end{array}$$

(59)

By multiplying (38) with ∂₃ϱ and integrating on ${ℝ}_{+}^3$, we have

$$\begin{array}{c}\frac{\mu }{\gamma }\frac{d}{\mathit{\text{dt}}}|{\partial }_3\varrho {|}_{L^2}^2+\gamma |{\partial }_3\varrho {|}_{L^2}^2\\ =\int \left\{-\underset{t}{\overset{3}{\mathbf{\text{v}}}}+\mu \left({\partial }_{11}{\mathbf{\text{v}}}^3+{\partial }_{22}{\mathbf{\text{v}}}^3+{\partial }_{13}{\mathbf{\text{v}}}^1+{\partial }_{23}{\mathbf{\text{v}}}^2\right)\right.\\ \left.-2{\mu }^2{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right)-2{\mu }^2{\partial }_3(\nabla \varrho ·\mathbf{\text{v}})+S_2^3\right\}{\partial }_3\mathrm{\varrho dx}\\ \le \frac{\gamma }{2}|{\partial }_3\varrho {|}_{L^2}^2+C(\gamma \left)\right|\left({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}},{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right),S_2\right){|}_{L^2}^2\\ +C\left|\int {\partial }_3(\nabla \varrho ·\mathbf{\text{v}}){\partial }_3\mathrm{\varrho dx}\right|,\end{array}$$

(60)

then integrating this in time, it implies

$$\begin{array}{c}|{\partial }_3\varrho {|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }_3\varrho \left(s\right){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+C\underset{0}{\overset{t}{\int }}\left|\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}},{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right),S_2){|}_{L^2}^2\\ +|\int {\partial }_3(\nabla \varrho ·\mathbf{\text{v}}\left){\partial }_3\mathrm{\varrho dx}\right|\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+C\underset{0}{\overset{t}{\int }}\left|\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}}){|}_{L^2}^2\mathit{\text{ds}}+\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(61)

Similarly, by multiplying ${\partial }_3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)$ to (39) and integrating it, we have

$$\begin{array}{c}|{\partial }_3\varrho {|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }_3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\left(s\right){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+C\underset{0}{\overset{t}{\int }}\left|\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}},{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right),S_2){|}_{L^2}^2\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}\left|\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}}){|}_{L^2}^2.\end{array}$$

(62)

This together with (41) yields

$$\begin{array}{cc}|& {\partial }_3\varrho {|}_{L^2}^2+\int \left|{\partial }_3\varrho \right(s){|}_{L^2}^2+|{\partial }_3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\left(s\right){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}\left|\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}}){|}_{L^2}^2\mathrm{ds.}\end{array}$$

(63)

Similarly, by taking ${\partial }^k{\partial }_3^l$ to (38) and (39), multiplying with ${\partial }^k{\partial }_3^{1+l}\varrho$ and ${\partial }^k{\partial }_3^{1+l}\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)$ respectively, we will get the general form of (42) as follows:

Lemma 3.6.

Under the assumptions of Theorem 4, we have for k+l=0,1,2 that

$$\begin{array}{c}|{\partial }^k{\partial }_3^{1+l}\varrho {|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }^k{\partial }_3^{1+l}\varrho {|}_{L^2}^2+\left|{\partial }^k{\partial }_3^{1+l}\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right(s){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}\left|{\partial }^k{\partial }_3^l\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}}){|}_{L^2}^2\mathrm{ds.}\end{array}$$

(64)

Proof.

Here we only to prove the case when k+l=2. As in (42), we can obtain

$$\begin{array}{c}|{\partial }^k{\partial }_3^{1+l}\varrho {|}_{L^2}^2+\underset{0}{\overset{t}{\int }}|{\partial }^k{\partial }_3^{1+l}\varrho {|}_{L^2}^2+\left|{\partial }^k{\partial }_3^{1+l}\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right(s){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+C\underset{0}{\overset{t}{\int }}\left|{\partial }^k{\partial }_3^l\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}},{\partial }_3\left(\varrho \text{div}\mathbf{\text{v}}\right),S_2){|}_{L^2}^2\\ +|\int {\partial }^k{\partial }_3^{1+l}(\nabla \varrho ·\mathbf{\text{v}})·{\partial }^k{\partial }_3^{1+l}\mathrm{\varrho dx}|\mathrm{ds.}\end{array}$$

(65)

By (17), we have

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}\left|{\partial }_x^3\right(\varrho \text{div}\mathbf{\text{v}}){|}_{L^2}^2\mathit{\text{ds}}& \le C\underset{0}{\overset{t}{\int }}|\varrho ,\text{div}\mathbf{\text{v}}{|}_{L^{\infty }}^2|{\partial }_x^3\varrho ,{\partial }_x^3\text{div}\mathbf{\text{v}}{|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(66)

Similarly,

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}|{\partial }_x^2{S}_2{|}_{L^2}^2\mathit{\text{ds}}& \le C\underset{0}{\overset{t}{\int }}|\varrho ,\mathbf{\text{v, H}}{|}_{L^{\infty }}^2|{\partial }_x^3\varrho ,{\partial }_x^3\mathbf{\text{v, H}}{|}_{L^2}^2\mathit{\text{ds}}\\ +C\underset{0}{\overset{t}{\int }}|\varrho ,{\partial }_x\varrho {|}_{L^{\infty }}^2|{\partial }_x^2\mathbf{\text{v}}{|}_1^2+\left|{\partial }_x^2\varrho {|}_{L^4}^2\right|{\partial }_x^2\mathbf{\text{v}}{|}_{L^4}^2\mathit{\text{ds}}\\ \le \mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}\left|{\partial }_x^2\varrho {|}_1^2\right|{\partial }_x^2\mathbf{\text{v}}{|}_1^2\mathit{\text{ds}}\\ \le \mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(67)

Moreover, by using the identity

$$({\partial }^k{\partial }_3^{1+l}\nabla \varrho ·\mathbf{\text{v}})·{\partial }^k{\partial }_3^{1+l}\varrho =\nabla \frac{|{\partial }^k{\partial }_3^{1+l}\varrho {|}^2}{2}·\mathbf{\text{v}}$$

(68)

and integration by parts, we shall obtain

$$\underset{0}{\overset{t}{\int }}\int {\partial }^k{\partial }_3^{1+l}(\nabla \varrho ·\mathbf{\text{v}})·{\partial }^k{\partial }_3^{1+l}\mathrm{\varrho dxds}\le \mathrm{C\delta N}{(0,t)}^2.$$

(69)

Substituting these inequalities into (44), we get (43). □

Last by taking ${\partial }_x^l{\partial }^k$ to (28), and by Lemma 3.2, we have

Lemma 3.7.

Under the assumptions of Theorem 4, we have for k+l=0,1,2 that

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}|{\partial }_x^{2+l}{\partial }^k\mathbf{\text{v}}{|}_{L^2}& +|{\partial }_x^{1+l}{\partial }^k\varrho {|}_{L^2}\mathit{\text{ds}}\le C\underset{0}{\overset{t}{\int }}|{\partial }^k{\mathbf{\text{v}}}_t{|}_l\\ +|{\partial }^k\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right){|}_{1+l}\mathit{\text{ds}}+\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(70)

Now we will finish the proof of Theorem 4 by doing the estimates for the lowest-order and highest-order derivatives. In the sequel, we divide the a priori estimates into three parts.

Part 1: estimates for the lowest derivatives of ϱ,v,H

Proposition 3.2.

Under the assumptions of Theorem 4, we have

$$\begin{array}{cc}\left|\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2& +\underset{0}{\overset{t}{\int }}\left|{\partial }_x\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}\left)\right(s){|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2\\ +\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(71)

Proof.

Due to (30), we have only to estimate $\underset{0}{\overset{t}{\int }}\left|{\partial }_x\varrho \right(s){|}_{L^2}^2\mathit{\text{ds}}$. First by Lemma 3.5, k=0,1, we have

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\left|\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right(s){|}_{L^2}^2+|\partial \left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\left(s\right){|}_{L^2}^2+|\partial {\partial }_x\mathbf{\text{v}}{|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+\mathrm{C\delta }\underset{0}{\overset{t}{\int }}\left|\right(S_1,S_2,S_3\left)\right(s){|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(72)

And by (42), we have

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}|{\partial }_3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right){|}_{L^2}^2\mathit{\text{ds}}& \le \mathit{\text{CN}}{(0,0)}^2+C\underset{0}{\overset{t}{\int }}\left|\right({\mathbf{\text{v}}}_t,\partial {\partial }_x\mathbf{\text{v}}){|}_{L^2}^2\mathit{\text{ds}}\\ +\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(73)

Thus (47)–(48) together with (32) implies

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\left|\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right(s){|}_1^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(74)

By Lemma 3.4, k=l=0, and integrating in time, we obtain

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}|{\partial }_x\varrho {|}_{L^2}^2\mathit{\text{ds}}\le C\underset{0}{\overset{t}{\int }}|{\mathbf{\text{v}}}_t{|}_{L^2}^2+|\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}{|}_1^2\right)\mathit{\text{ds}}+\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(75)

This combining with (49) and (32), yields the estimate for $\underset{0}{\overset{t}{\int }}\left|{\partial }_x\varrho \right(s){|}_{L^2}^2\mathit{\text{ds}}$. □

Part 2: estimates for the highest derivatives of ϱ,v,H

Proposition 3.3.

Under the assumptions of Theorem 4, we have

$$\begin{array}{cc}\left|{\partial }_x^3\right(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}}){|}_{L^2}^2& +\underset{0}{\overset{t}{\int }}\left|\left({\partial }_x^3\varrho ,{\partial }_x^4\mathbf{\text{v}},{\partial }_x^4\mathbf{\text{H}}\right)\right(s){|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2\\ +\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(76)

Proof.

We divide the proof into three steps as follows.

i) By Lemma 2.8, k=3 and Lemma 2.9, k=2,l=0, we have

$$\begin{array}{c}{\left|{\partial }^2{\partial }_x\varrho \right|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\left({\partial }^2{\partial }_3\varrho ,{\partial }^2{\partial }_x\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right),{\partial }^3{\partial }_x\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\left|\underset{0}{\overset{t}{\int }}〈{\partial }^k{S}_1,{\partial }^k\varrho 〉\right.\\ +\left.〈{\partial }^k{S}_2,{\partial }^k\mathbf{\text{v}}〉\mathit{\text{ds}}\right|+C\underset{0}{\overset{t}{\int }}|{\partial }^2{\mathbf{\text{v}}}_t{|}_{L^2}^2\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}|{\partial }^2{\mathbf{\text{v}}}_t{|}_{L^2}^2\mathit{\text{ds}},\end{array}$$

(77)

where we get the last inequality as in the proof of Lemma 3.6. By Lemma 3.7, k=2,l=0, we have

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}& {\left|\left({\partial }_x^2{\partial }^2\mathbf{\text{v}},{\partial }_x{\partial }^2\varrho \right)\right|}_{L^2}^2\mathit{\text{ds}}\le C\underset{0}{\overset{t}{\int }}{\left|\left({\partial }^2{\mathbf{\text{v}}}_t,\right.{\partial }^2{\partial }_x\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ +\mathrm{C\delta N}{(0,t)}^2.\end{array}$$

(78)

This together with (51) yields

$$\begin{array}{c}{\left|{\partial }^2{\partial }_x\varrho \right|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\left({\partial }^2{\partial }_x\varrho ,{\partial }^2{\partial }_x\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right),{\partial }^2{\partial }_x^2\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }^2{\mathbf{\text{v}}}_t\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(79)

By Lemma 3.6, k=l=1, and (52), we have

$$\begin{array}{c}{\left|\partial {\partial }_x^2\varrho \right|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\left(\partial {\partial }_x^2\varrho ,\partial {\partial }_x^2\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right),{\partial }^2{\partial }_x^2\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{v}}}_t\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(80)

Then by Lemma 3.7, k=l=1, and with the help of (15), (31) and (47), we have

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x^3\partial \mathbf{\text{v}},{\partial }_x^2\mathrm{\partial \varrho }\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le C\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x{\mathbf{\text{v}}}_t,{\partial }_x^2{\mathbf{\text{v}}}_t,\partial \left(\frac{\mathrm{D\varrho }}{\mathit{\text{dt}}}\right),\right.\partial {\partial }_x\left(\frac{\mathrm{D\varrho }}{\mathit{\text{dt}}}\right),\partial {\partial }_x^2\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ +\mathrm{C\delta N}{(0,t)}^2\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{v}}}_t,\partial {\partial }_x^2\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right|}_{L^2}^2\mathit{\text{ds}},\end{array}$$

(81)

which together with (53) yields

$$\begin{array}{c}{\left|\partial {\partial }_x^2\varrho \right|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\left(\partial {\partial }_x^2\varrho ,\partial {\partial }_x^2\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right),\partial {\partial }_x^3\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{v}}}_t\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(82)

By Lemma 3.6, k=0,l=2, and by (54), we have

$$\begin{array}{c}{\left|{\partial }_x^3\varrho \right|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x^3\varrho ,{\partial }_x^3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right),\partial {\partial }_x^3\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{v}}}_t\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(83)

Thus by Lemma 3.7, k=0,l=2, we have

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x^4\mathbf{\text{v}},{\partial }_x^3\varrho \right)\right|}_{L^2}^2\mathit{\text{ds}}& \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2\\ +C\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x^2{\mathbf{\text{v}}}_t,{\partial }_x^3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right)\right|\right.}_{L^2}^2\mathit{\text{ds}},\end{array}$$

(84)

together it with (55), then we have got

$$\begin{array}{cc}{\left|{\partial }_x^3\varrho \right|}_{L^2}^2& +\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x^3\varrho ,{\partial }_x^3\left(\frac{\mathrm{D\varrho }}{\mathit{\text{Dt}}}\right),{\partial }_x^4\mathbf{\text{v}}\right)\right|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2\\ +\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{v}}}_t\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(85)

i i) By Lemma 3.1, k=3, we have

$$\begin{array}{cc}{\left|{\partial }_x^3(\mathbf{\text{v}},\mathbf{\text{H}})\right|}_{L^2}^2& \le C\left({\left|\left({\partial }_x\varrho ,{\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t\right)\right|}_1^2+{\left|(S_2,S_3)\right|}_1^2+{\left|\left(\mathbf{\text{v}},{\partial }_x\mathbf{\text{H}}\right)\right|}_{L^2}^2\right)\\ \le \mathrm{C\delta N}{(0,t)}^2+C{\left|\varrho ,{\partial }_x^3\varrho ,\mathbf{\text{v}},{\mathbf{\text{v}}}_t,{\partial }_x{\mathbf{\text{v}}}_t,{\partial }_x\mathbf{\text{H}},{\mathbf{\text{H}}}_t,{\partial }_x{\mathbf{\text{H}}}_t\right|}_{L^2}^2,\end{array}$$

(86)

which together with (30)–(33), (46) and (56) implies

$$\begin{array}{c}{\left|{\partial }_x^3\left(\mathbf{\text{v}},\mathbf{\text{H}}\right)\right|}_{L^2}^2\le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{v}}}_t\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(87)

i i i) By Lemma 3.1, k=4, we have

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^4\mathbf{\text{H}}\right|}_{L^2}^2\mathit{\text{ds}}\le C\underset{0}{\overset{t}{\int }}{\left|{\mathbf{\text{H}}}_t\right|}_2^2+|S_3{|}_2^2+|{\partial }_x\mathbf{\text{H}}{|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2{\mathbf{\text{H}}}_t\right|}_{L^2}^2\mathit{\text{ds}},\end{array}$$

(88)

this together with (56) and (58), then it implies

$$\begin{array}{c}{\left|{\partial }_x^3(\varrho ,\mathbf{\text{v}},\mathbf{\text{H}})\right|}_{L^2}^2+\underset{0}{\overset{t}{\int }}{\left|\left({\partial }_x^3\varrho ,{\partial }_x^4\mathbf{\text{v}},{\partial }_x^4\mathbf{\text{H}}\right)\left(s\right)\right|}_{L^2}^2\mathit{\text{ds}}\\ \le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2+C\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2\left({\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t\right)\right|}_{L^2}^2\mathrm{ds.}\end{array}$$

(89)

We can derive from Lemma 3.1 that

$$\begin{array}{c}{\left|{\partial }_x^2({\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t)\right|}_{L^2}^2\le C({\left|\left({\partial }_x{\varrho }_t,{\mathbf{\text{v}}}_t,{\mathbf{\text{v}}}_{\mathit{\text{tt}}},{\partial }_x{\mathbf{\text{H}}}_t,{\mathbf{\text{H}}}_{\mathit{\text{tt}}},{\partial }_t{S}_2,{\partial }_t{S}_3\right)\right|}_{L^2}^2.\end{array}$$

(90)

Here with the help of (19) ₁, we have

$$\begin{array}{cc}\underset{0}{\overset{t}{\int }}{\left|{\partial }_x{\varrho }_t\right|}_{L^2}^2\mathit{\text{ds}}& \le C\underset{0}{\overset{t}{\int }}|{\partial }_x^2\varrho {|}_{L^2}^2+{\left|{\partial }_x^2\mathbf{\text{v}}\right|}_{L^2}^2\mathit{\text{ds}}\\ \le C\left(\epsilon \right)\underset{0}{\overset{t}{\int }}{\left|{\partial }_x\varrho ,{\partial }_x\mathbf{\text{v}}\right|}_{L^2}^2\mathit{\text{ds}}+\epsilon \underset{0}{\overset{t}{\int }}|{\partial }_x^3\varrho ,{\partial }_x^4\mathbf{\text{v}}{|}_{L^2}^2\mathit{\text{ds}},\end{array}$$

(91)

where the terms in the right-hand side can be absorbed by (46) and (59). Thus by (30)–(33), we obtain

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}{\left|{\partial }_x^2({\mathbf{\text{v}}}_t,{\mathbf{\text{H}}}_t)\right|}_{L^2}^2\mathit{\text{ds}}\le \mathit{\text{CN}}{(0,0)}^2+\mathrm{C\delta N}{(0,t)}^2,\end{array}$$

(92)

which together with (59) implies (50). □

Part 3: ConclusionCombining the inequalities in Proposition 3.2 and Proposition 3.3, it yields

$$\begin{array}{c}{\left|\left(\varrho ,{\partial }_x^3\varrho ,\mathbf{\text{v}},{\partial }_x^3\mathbf{\text{v}},\mathbf{\text{H}},{\partial }_x^3\mathbf{\text{H}}\right)\right|}_{L^2}^2\end{array}$$