Global existence and convergence rates for the smooth solutions to the compressible magnetohydrodynamic equations in the half space
© Chen and Tan; licensee Springer 2014
Received: 3 March 2014
Accepted: 6 June 2014
Published: 11 July 2014
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 . 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.
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 . 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.
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.
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. ). 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 , ; Kawashima and Okada ) 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 ; Tan and Wang ). The local unique strong solution has been obtained in (Fan and Yu ). In (Chen and Tan , ) 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 ) to the initial boundary problem in the half space.
where n=(0,0,−1) is the normal vector of . We assume that throughout the paper the initial boundary data satisfy certain compatibility conditions as usual in (Matsumura and Nishida ).
Before stating out our results, we shall introduce some standard notations.
Notations. We denote by L p , Wm, p the usual Lebesgue and Sobolev spaces on and H m =Wm,2, with norms 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 , and put 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 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:
By imposing some additional conditions on the initial data we will establish the following various decay rates of the solutions obtained in Theorem 1:
where is some positive number.
for all t≥0.
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.
In this paper, we demonstrate that the global existence and the decay rates for the compressible (MHD) in can be established under the similar initial assumptions as for the compressible Navier-Stokes equations which can be seen in (Matsumura and Nishida ). 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 but . 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.
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 linearized system
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 ) or can be found in (Ströhmer , Vol’pert and Hudjaev ):
(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 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:
(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.
(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.
See in (Chen and Tan ). □
A priori estimates
Thus we have the following estimates which we can found in (Cho et al. ):
We have the following estimates which can be found in (Galdi et al. ):
Next we shall do the estimates for the terms contained in N(0,t).
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).
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.
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).
Similarly, we can obtain the following lemma and we omit the proof of it.
Last by taking to (28), and by Lemma 3.2, we have
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
Part 2: estimates for the highest derivatives of ϱ,v,H
We divide the proof into three steps as follows.
Proof of Theorem 2
In this section we shall prove the decay rates of the solution obtained in Theorem 1 to finish the proof of Theorem 2.
Some elementary decay-in-time estimates
for some . Here |·|X∩Y=|·| X +|·| Y .
for any t≥1, 2≤q≤+∞.