Time-decay solutions of the initial-boundary value problem of rotating magnetohydrodynamic fluids

R E S E A R C H

Open Access

Time-decay solutions

of the initial-boundary value problem

of rotating magnetohydrodynamic fluids

Weiwei Wang

_{and Youyi Zhao}

*_{Correspondence:} [email protected] College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

Abstract

We have investigated an initial-boundary problem for the perturbation equations of rotating, incompressible, and viscous magnetohydrodynamic (MHD) fluids with zero resistivity in a horizontally periodic domain. The velocity of the fluid in the domain is non-slip on both upper and lower flat boundaries. We switch the analysis of the initial-boundary problem from Euler coordinates to Lagrangian coordinates under proper initial data, and get a so-called transformed MHD problem. Then, we exploit the two-tiers energy method. We deduce the time-decay estimates for the

transformed MHD problem which, together with a local well-posedness result, implies that there exists a unique time-decay solution to the transformed MHD problem. By an inverse transformation of coordinates, we also obtain the existence of a unique time-decay solution to the original initial-boundary problem with proper initial data.

Keywords: magnetohydrodynamic fluid; equilibrium state; magnetic field; decay estimates; rotation

1 Introduction

The three-dimensional (D) rotating, incompressible and viscous magnetohydrodynamic (MHD) equations with zero resistivity in a domain⊂R_{read as follows:}

⎧ ⎪ ⎨ ⎪ ⎩

ρvt+ρv· ∇v+∇(p+λ|M|/) + ρ(ω×v) =μv+λM· ∇M,

Mt=M· ∇v–v· ∇M,

divv=divM= .

(.)

Here the unknownsv=v(x,t),M:=M(x,t) andp=p(x,t) denote the velocity, the magnetic field, and the pressure of the incompressible MHD fluid respectively;μ> ,ρandλstand

for the coefficients of the shear viscosity, the density constant, and the permeability of vacuum, respectively. ρ(ω×v) represents the Coriolis force, andω= (, ,ω) denotes the constant angular velocity in the vertical direction. In system (.), equation (.)describes

the balance law of momentum, while (.) is called the induction equation. As for the

constraintdivM= , it can be seen just as a restriction on the initial value ofMsince (divM)t=  due to (.).

LetM¯ := (m,m,m) be a constant vector withm= , and (,M¯,p¯) be a rest state of

the system (.). We denote the perturbation to an equilibrium state (,M¯) by

v=v– , N=M–M¯, q˜=p–p¯. Then, (v,N,q) satisfies the perturbation equations

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ρvt+ρv· ∇v+∇(q˜+λ|N+M¯|/)

=μv+λ(N+M¯)· ∇(N+M¯) + ρω(ve–ve),

Nt= (N+M¯)· ∇v–v· ∇(N+M¯),

divv=divN= ,

(.)

where we have used the relationω×v=ω(ve–ve). For system (.), we impose the

initial and the boundary conditions:

(v,N)|t== (v,N) in, (.)

v(·,t)|∂=  for anyt> , (.)

wherevandNshould satisfy the compatibility conditionsdivv=divN= . We call the

initial-boundary value problem (.)-(.) the MHD problem (with rotation) for simplicity. In this article, we always assume that the domain is horizontally periodic with finite height,

i.e.,

:=x:=x,x

∈R_{|x∈T,  <x}

<h withh> ,

whereT := (πLT)×(πLT),T=R/Z, and πL, πL>  are the periodicity lengths.

The effects of magnetic fields and rotation on the motion of pure fluids were widely in-vestigated; see [–] and the references cited therein. In particular, Tan and Wang [] showed that the well-posedness problem of the initial-boundary problem (.)-(.) for

ω=  (i.e., without the effect of rotation). In this article, we further considerω= , and show that there also exists a unique time-decay solution to the initial-boundary problem (.)-(.) in Lagrangian coordinates (see Theorem .), which, together with the inverse transformation of coordinates, implies the existence of a time-decay solution to the origi-nal initial-boundary problem (.)-(.) with proper initial data inH(). Our result also holds for the caseω= , thus improves Tan and Wang’s result in [], in which the suffi-ciently small initial data at least belong toH_().

In the next section we introduce the form of the initial-boundary problem (.)-(.) in Lagrangian coordinates, and the details of our result.

2 Main results

2.1 Reformulation

In general, it is difficult to directly show the existence of a unique global-in-time solution to (.)-(.). Instead, we switch our analysis to Lagrangian coordinates as in [, ]. To this end, we assume that there is an invertible mappingζ:=ζ(y) :→, such that

whereζ_denotes the third component ofζ. We define the flow mapζ as the solution to

ζt(y,t) =v(ζ(y,t),t), ζ(y, ) =ζ.

(.)

We denote the Eulerian coordinates by (x,t) withx=ζ(y,t), where (y,t)∈×R+_stand

for the Lagrangian coordinates. In order to switch back and forth from Lagrangian to Eulerian coordinates, we assume thatζ(·,t) is invertible and=ζ(,t). In other words, the Eulerian domain of the fluid is the image ofunder mappingζ. In view of the non-slip boundary conditionv|∂= , we have∂=ζ(∂,t). In addition, sincedet∇ζ= , we

have

det(∇ζ) =  (.)

due todivv= ; see [], Proposition ..

Now, we further define the Lagrangian unknowns by

(u,p˜,B)(y,t) =v,p+λ|M|/,Mζ(y,t),t for (y,t)∈×R+. (.)

Thus in Lagrangian coordinates the evolution equations foru,p˜andBread as

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ζt=u,

ρut–μAu+∇Ap˜=λB· ∇AB+ ρω(ue–ue),

Bt–B· ∇Au= ,

div_Au= ,

(.)

with initial and boundary conditions

(u,ζ–y)|∂=  and (ζ,u,B)|t== (ζ,u,B).

Moreover,div_AB=  if the initial dataζandBsatisfy

div_AB= . (.)

Here A denotes the initial value of A, the matrix A:= (Aij)× via AT = (∇ζ)–:=

(∂jζi)–×, and the differential operators ∇A, divA and A are defined by ∇Af :=

(Ak∂kf,Ak∂kf,Ak∂kf)T,divA(X,X,X)T:=Alk∂kXl, andAf :=divA∇Af for

appro-priatefandX. It should be noted that we have used the Einstein convention of summation over repeated indices, and∂k=∂yk. In addition, in view of the definition ofAand (.), we

can see thatA= (A∗_ij)×, whereA∗ijis the algebraic complement minor of the (i,j)th entry ∂jζi. Since∂kA∗ik= , we can get an important relation

div_Au=∂l(Akluk) = , (.)

Our next goal is to eliminateBby expressing it in terms ofζ. This can be achieved in the same manner as in [, ]. For the reader’s convenience, we give the derivation here. In view of the definition ofA, one has

∂iζkAkj=Aik∂kζj=δij,

whereδij=  fori=j, andδij=  fori=j. Thus, applyingAjlto (.), we obtain

Ajl∂tBj=BiAik∂kujAjl=BiAik∂t(∂kζj)Ajl= –BiAik∂kζj∂tAjl= –Bj∂tAjl,

which implies that∂t(AjlBj) =  (i.e., (ATB)t= ). Hence,

AjlBj=AjlBj, (.)

which yieldsBi=∂lζiAjlBj,i.e.,

B=∇ζAT_B. (.)

Here and in what follows, the notationf_{also denotes the initial data of the functionf. To}

obtain the asymptotic stability in time, we naturally expect

(ζ,B) converges to (y,M¯) ast→ ∞. (.)

Thus (.) formally implies

AT

B=M¯, i.e., B=M¯ · ∇ζ. (.)

Putting the above expression ofBinto (.), we get

B=M¯ · ∇ζ. (.)

Moreover, in view of (.), (.) and (.), the Lorentz force term can be represented by

B· ∇AB=BlAlk∂kB=AlkBl∂k(M¯ · ∇ζ) =M¯ · ∇(M¯ · ∇ζ). (.)

Summing up the above analyses, we can see that, under the initial conditions (.) and (.), one can use the relation (.) to change (.) into the following Navier-Stokes sys-tem:

⎧ ⎪ ⎨ ⎪ ⎩

ζt=u,

ρut–μAu+∇Ap˜=λM¯ · ∇(M¯ · ∇ζ) + ρω(ue–ue),

div_Au= ,

andBis defined by (.). Now, we introduce the shift functions

Then the evolution equations for the shift functions (η,q) anduread as ⎧ ⎪ ⎨ ⎪ ⎩

ηt=u,

ρut–μAu+∇Aq–λM¯ · ∇(M¯ · ∇ζ) = ρω(ue–ue),

div_Au= ,

(.)

whereA= (I+∇η)–_{,I= (δ}

ij)×. The associated initial and boundary conditions read as

follows:

(η,u)|t== (η,u), (η,u)|∂= . (.)

It should be noted that the shift functionqis the sum of the perturbed fluid and the mag-netic pressures in Lagrangian coordinates. Hence we still callqthe perturbation pressure for the sake of simplicity. In this article, we call the initial-boundary value problem (.)-(.) the transformed MHD problem.

2.2 Main results

Before stating our first main result on the transformed MHD problem in detail, we intro-duce some simplified notations that shall be used throughout this paper:

R+ := [,∞), :=

, Lp:=Lp() :=W,p() for  <p≤ ∞,

H_:=W_,(), Hk:=Wk,(), · k:= · _Hk₍₎ fork≥, ∂_hkdenotes∂α

 ∂

α

 for anyα+α=k, · m,k:=

α+α=m ∂α

 ∂

α  · k,

abmeans thata≤cb,

where, and in what follows, the letterc denotes a generic constant which may depend on the domainand some physical parameters, such asλ,M¯,g,μandρin the MHD

equations (.). It should be noted that a product space (X)n_{of vector functions is still}

denoted byX, for example, a vector functionu∈(H)is denoted byu∈H with norm

uH := (_k=uk

H)/. Finally, we define some functionals:

EL_:=∇η

,+(η,u) 

+(ut,∇q)  ,

DL_:=(η,M¯ _{· ∇η,∇u)}

,+(η,u)  +



k=

∂_tku_–k+∇q_+∇qt_,

EH_:=∇η

,+η+  k= ∂k tu  –k+  k=

_∇∂_tkq_–k,

DH_:=(η,M¯ _{· ∇η,∇u)}

,+(η,u)  +  k= ∂k tu  –k+  k=

_∇∂_tkq_–k+∇q_,

G(t) = sup ≤τ<t

η(τ)_, G(t) = t



(η,u)(τ) 

( +τ)/ dτ,

G(t) = sup ≤τ<tE

H_{(τ) +} t



DH_{(τ) dτ, G}

(t) = sup ≤τ<t

Next, we introduce our main result.

Theorem . Letbe a horizontally periodic domain with finite height,ωbe an arbitrary

real number,and m= .Then there is a sufficiently small constantδ> ,such that for any

(η,u)∈H×Hsatisfying the following conditions:

() η_+u_≤δ; () ζ:=y+ηsatisfies(.);

() (η,u)satisfies necessary compatibility conditions(i.e.,∂tju(x, )|∂= forj= 

and),

there exists a unique global solution(η,u)∈C(R+_,H×H)to the transformed MHD problem(.)-(.)with an associated perturbation pressure q.Moreover, (η,u,q)enjoys the following stability estimate:

G(∞) :=



k=

Gk(∞)≤c

η_+u_. (.)

Here the positive constantsδ and c depend on the domainand other known physical parametersλ,M¯,μandρ.

Remark . Exploiting the inverse transformation ofζ, we can easily deduce from

Theo-rem . the global well-posedness of the original MHD problem (.)-(.). More precisely, there is a sufficiently small constantδ> , such that, for any (v,N)∈Hsatisfying the

following conditions:

() there exists an invertible mappingζ:=ζ(x) :→, such that (.) holds, where

AT

= (∇ζ)–;

() (M¯ +N)(ζ) =M¯ · ∇ζ;

() ζ–x+v≤δ;

() the initial data,v,Nsatisfy necessary compatibility conditions (i.e., ∂_tjv(x, )|∂= forj= and),

there exists a unique global solution (v,N)∈C_(R+

,H) to the original MHD problem

(.)-(.) with an associated perturbation pressureq˜. Moreover, (v,N,q˜) enjoys the fol-lowing stability estimate:

sup

≤t<∞

N_+



k=

∂_tkv(t)_–k+



k=

_∇∂_tkq˜(t)_–k

+ sup

≤t<∞

( +t)_(v,N)

+(vt,∇ ˜q)  

≤cζ–x+v

. (.)

Now we briefly describe the basic idea in the proof of Theorem .. By the standard energy method, there are two functionalsE˜L _{andQof (η,u) satisfying the lower-order}

energy inequality (see Proposition .)

d dtE˜

L_+DL_≤QDL_{, (.)}

that the structure of the energy inequality above is very similar to that of the surface wave problem [], for which Guo and Tice developed a two-tier energy method to overcome this difficulty. In the spirit of the two-tier energy method, we look after a higher-order energy inequality to match the lower-order energy inequality (.). SinceE˜Lcontains

η, we find that the higher-order energy at least includesη. Thus, similar to (.), we establish the higher-order energy inequality (see Proposition .)

d dtE˜

H_+DH_≤√_EL_(η,u)

, (.)

where the functionalE˜H_{is equivalent toE}H_{. Moreover, the highest-order norm(η,u)} 

enjoys the highest-order energy inequality

d dtη



,∗+(η,u)E

H_+DH_{, (.)}

where the normη,∗is equivalent toη. In the derivation of thea prioriestimates, we haveQEH_{, and thus (.) implies (see Proposition .)}

d dtE˜

L_+DL_{≤. (.)}

Consequently, by the two-tier energy method, we can deduce the global-in-time stability estimate (.) based on (.)-(.).

The rest of the sections are mainly devoted to the proof of Theorem .. In Section , we first derive the lower-order energy inequality (.) for the transformed MHD problem. Then in Section  we derive the higher-order energy inequality (.) and the highest-order energy inequality (.). Based on these three energy inequalities, we prove Theo-rem . by adapting the two-tier energy method in Section .

3 Lower-order energy inequality

In this section, we start to derive the lower-order energy inequality in thea prioriestimates for the transformed MHD problem. To this end, let (η,u) be a solution of the transformed MHD problem with perturbed pressureq, such that

G(T) + sup ≤τ≤T

EH_{(τ)≤δ∈(, ) for someT> , (.)}

whereδis sufficiently small. It should be noted that the smallness depends on the known physical parameters in (.), and will be repeatedly used in what follows. Moreover, we assume that the solution (η,u,q) possesses proper regularity, so that the formal calcu-lation makes sense. We remind the reader that in the calcucalcu-lations, we shall repeatedly use Cauchy-Schwarz’s inequality, Hölder’s inequality, the embedding inequalities (see [], . Theorem)

fLpf for ≤p≤ and fL∞f, (.)

and the interpolation inequalities (see [], . Theorem)

fjf–

j i  f

j i

for any ≤j<iand any constant> , where the constantCdepends on the domain and. In addition, we shall also frequently use the following two estimates:

fgjfjgκ(j) forj≥ (.)

and

f∂f forf ∈H, (.)

whereκ(j) =jforj≥ andκ(j) =  forj≤. We also introduce the following inequality, see (.) in []:

f≤h ¯M· ∇f/π. (.)

Before deriving the lower-order energy inequality defined on (,T], we first give some preliminary estimates, temporal derivative estimates, horizontal spatial estimates and Stokes estimates in sequence.

3.1 Preliminary estimates

In this subsection, we derive some preliminary estimates forA. To begin with, we give an expression ofA. Using (.), we have

∂tdet(I+∇η) =

≤i,j≤

∂t∂jηiA∗ij=

≤i,j≤

A∗_ij∂jui, (.)

where A∗_ijis the algebraic complement minor of the (i,j)th entry in the matrix I+∇η. Recalling the definition ofA, we see that

A=A∗_ij×/det(I+∇η).

Inserting this relation into (.), we get

∂tdet(I+∇η) =det(I+∇η)

≤i,j,≤

Aij∂jui= ,

which, together with initial conditiondet(I+∇η) = , implies

det(I+∇η) = . Thus we obtain

A=A∗_ij×. (.)

Now, exploiting (.), (.), (.) and (.), we easily see that

Aj +ηj+ +ηj+ for ≤j≤, (.)

Similarly, we further deduce that

∂_tiA_j i–

k=

∂_tk∇u_j for any ≤i≤ and ≤j≤ – i.

LettingA˜:=A–I, we next boundA. To this end, we assume that˜ δis so small that the following expansion holds:

AT_{=I–∇η+ (∇η)}

∞

i=

(–∇η)i=I–∇η+ (∇η)AT,

whence

˜

AT_{= (∇η)}_AT_–∇η.

Using (.), (.) and (.), we find that

˜Aj∇ηj for ≤j≤.

3.2 Temporal derivative estimates

In this subsection, we try to control temporal derivatives. For this purpose, we apply∂_tjto (.) to get

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

∂_tj+η=∂_tju,

ρ∂_tj+u–μ_A∂_tju+∇_A∂_tjq

=λ∂tjM¯ · ∇(M¯ · ∇ζ) + ρω∂ j

t(ue–ue) +μNut,j+Nqt,j,

div_A∂_tju=divDt,ju,

(.)

where

N_ut,j:=

≤m<j,≤n≤j

∂_tj–m–nAil∂l

∂_tnAik∂tm∂ku

,

N_qt,j:= –

≤l<j

∂_tj–lAik∂tl∂kq

×,

Dt,j_u :=

–

≤l<j

C_jj–l∂_tj–lAki∂tluk

×

, (.)

C_jj–ldenotes the number of (j–l)-combinations from a given setSofjelements, and we have used relation (.) in (.). Then from (.) we show the following estimates:

Lemma . It holds that for j= and,

d dt

√

ρ∂_tju_+λM¯ · ∇∂tjη  

+μ∇A∂_tju_√EHDL_{, (.)}

d

dt∇Aut

 +



μ √

Proof () We only prove (.) forj= , since the derivation of the casej=  is similar. Multiplying (.)withj=  byut, integrating (by parts) the resulting equality over,

and using (.), we get

 

d dt

√ρut_+λ ¯M·ηt

+μ∇_Aut_

= ρω

∂t(ue–ue)·utdy+

qtdivAutdy

+μ

N_ut,·utdy+

N_qt,·utdy

:=



k=

I_kL. (.)

The last three integralsI_L, . . . ,I_Lcan be estimated as follows:

I_L:= –

∇qt·Dt,u dy∇qtAtLuL∇qtA_tu

√EHDL_{, (.)}

I_LAAtuut√EHDL_{, (.)}

I_LAt∇qL∞utAt∇qut √

EH_DL_{, (.)}

where we have used (.) in (.). Consequently, the desired estimate (.) follows

from (.)-(.).

() Now we turn to the proof of (.). Multiplying (.)withj=  byutt, integrating

(by parts) the resulting equality over, and using (.), we conclude

μ

 d

dt∇Aut

 +

√ ρutt_

= ρω∂t(ue–ue)·utt+λM¯ · ∇(M¯ · ∇u)·utt

dy

+

qtdivAuttdy+μ

N_ut,·uttdy+

N_qt,·uttdy

+μ

∇Aut:∇Atutdy

:=



k=

J_kH. (.)

On the other hand, the five integralsJH

 , . . . ,JHcan be bounded as follows:

J_Hut+uutt, (.)

J_H= –

∇qt·Dt,u dx∇qt

Attu+Atut

√EHDL_{, (.)}

J_HAAtuutt √

J_HAt∇qutt√EHDL_{, (.)}

J_HAAtut √

EHDL_{. (.)}

Thus, substituting (.)-(.) into (.) and using Cauchy-Schwarz’s inequality, we

im-mediately get (.).

3.3 Horizontal spatial estimates

In this subsection, we establish the estimates of horizontal spatial derivatives. For this purpose, we rewrite (.) as the following non-homogeneous linear form:

⎧ ⎪ ⎨ ⎪ ⎩

ηt=u,

ρut–μu+∇q–λM¯ · ∇(M¯ · ∇η) = ρω(ue–ue) +μNuh+Nqh,

divu=Dh u,

(.)

where

N_uh:=∂l

(A˜jlA˜jk+A˜lk+A˜kl)∂kui

×,

Nh

q := –(A˜ik∂kq)× and Dhu:=A˜lk∂kul.

Then we have the following estimate on horizontal spatial derivatives ofη.

Lemma . It holds that

d dt

ρ∂_hjη·∂_hjudy+μ ∇∂

j hη

 

+λM¯ · ∇∂_hjη_ √EH_DL_+u

j,, ≤j≤.

Proof We only show the casej= ; the remaining three cases can be verified similarly. Applying∂_hto (.), multiplying the resulting equation by∂hη, and then using (.),

we get

ρ∂t

∂_hη·∂_hu–μ∂_hηt+λM¯ · ∇ _¯

M· ∇∂_hη·∂_hη

=ρω∂_h(ue–ue) +μ∂hNuh+∂hNqh–∇∂hq

·∂_hη+ρ∂_hu. If we integrate (by parts) the above identity over, we obtain

d dt

ρ∂_hη·∂_hudy+μ ∇∂

 hη

 

+λM¯ · ∇∂hη  

= ρω

∂_h(ue–ue)·∂hηdy+ μ∂hNuh+∂hNqh

·∂_hηdy

+

∂_hqdiv∂_hηdy+√ρ∂_hu_



k=

K_kL+u_,, (.)

We have the boundedness

K_L≤∂_hu_∂_hη_. (.)

Noting that

μ∂

hNuh+∂hNqh ˜Au+ ˜Au+ ˜A∇q+ ˜A∇q √EHDL_,

we find that

K_Lμ∂_hN_uh+∂_hN_qh_∂_hη_√EH_DL_{. (.)}

Next we estimate the third integralKL. To start with, we analyze the property ofdivη.

Since

det(I+∇η) = ,

we have by Sarrus’ rule

divη=∂η∂η+∂η∂η+∂η∂η–∂η∂η–∂η∂η–∂η∂η

+∂η(∂η∂η–∂η∂η) +∂η(∂η∂η–∂η∂η)

+∂η(∂η∂η–∂η∂η).

Multiplying the above identity by a smooth test functionφ, and then integrating (by parts) the resulting identity over, we derive that

φdivηdy= –

∇φ·ψdy,

where

ψ:=

⎛ ⎜ ⎝

η(∂η+∂η) +η(∂η∂η–∂η∂η) η∂η–η∂η+η(∂η∂η–∂η∂η)

–η∂η–η∂η+η(∂η∂η–∂η∂η) ⎞ ⎟ ⎠.

This means that

divη=divψ.

Thus, it follows immediately that

K_L= –

∂_h∇q·∂_hψdy= –

∂h∇q·∂_hψdy

∇q∂_hψ

η∇qη √

Now, substituting (.), (.) and (.) into (.), we immediately obtain the desired

estimate for the casej= .

Similarly, we also establish the following estimates of horizontal spatial derivatives ofu:

Lemma . We have

d dt

√

ρ∂_hju_+λM¯ · ∇∂hjη  

+μ∇∂_hju_√EHDL_{, j= , , .}

Proof We only prove the casej= ; the remaining two cases can be shown similarly. Ap-plying∂_hto (.), and multiplying the resulting equality by∂hu, we make use of (.)

to get

ρ∂_hut·∂hu–μ∂hu·∂hu–λM¯ · ∇ _¯

M· ∇∂_hη·∂_hηt

=ρω∂_h(ue–ue) +μ∂hNuh+∂hNqh–∇∂hq

·∂_hu.

Integrating (by parts) the above identity over, we have

 

d dt

ρ∂_hudy+λM¯ · ∇∂hη  

+μ

∇∂

 hu

 

= μ∂_hN_uh+∂_hN_qh·∂_hudy+

∂_hqdiv∂_hudy=:ML_+M_L. (.)

On the other hand, similarly to (.) and (.), the two integralsML

 andML can be

estimated as follows:

ML_ μ∂_hN_uh+∂_hN_qh_∂_hu_√EHDL_{, (.)}

ML_= –

∂_hq∂_hDh_udy∇q ˜Au√EHDL_{, (.)}

where we have used (.)in (.). Consequently, putting the above two estimates into

(.), we obtain Lemma . for the casej= .

3.4 Stokes problem and stability condition

In this subsection, we use the regularity theory of the Stokes problem to derive more esti-mates of (η,u). To this end, we rewrite (.)and (.)as the following Stokes problem:

⎧ ⎪ ⎨ ⎪ ⎩

–w+∇q

=λM¯ · ∇(M¯ · ∇η) –λmη+ ρω(ue–ue) –ρut+μNuh+Nqh,

divw=μDh

u+λmdivη,

(.)

coupled with boundary condition

ω|∂= , (.)

Now, applying∂_hkto (.) and (.), we get

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

–∂_hkw+∇∂_hkq

=∂_hk(λM¯ · ∇(M¯ · ∇η) –λmη+ ρω(ue–ue)) –ρ∂_hkut+μ∂_hkNuh+Nqh,

div∂_hkw=μ∂_hkDh

u+λmdiv∂hkη, ∂_hkω|∂= .

Then we apply the classical regularity theory to the Stokes problem as in [], Proposition ., to deduce that

ω_k,i–k++∇q_k,i–k∇η_k+,i–k+(u,ut)_k,i–k+Sωk,i, (.)

where

Sω

k,i:=Nuh,Nqh 

k,i–k+D h u,divη

 k,i–k+.

In addition, applying∂_tkto (.)-(.), we see that ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

–μ∂k

tu+∇∂tkq

=λM¯ ·∂tk∇(M¯ · ∇η) + ρω∂tk(ue–ue) –ρ∂tk+u+μ∂tkNuh+∂tkNqh,

div∂_tku=∂_tkDh u, ∂_tku|∂= .

Hence, we apply again the classical regularity theory to the Stokes problem to get

∂_tku_i–k++∇∂_tkq_i–k∂_tk∇η,u,ut  i–k+S

u

k,i, (.)

whereSu

k,i:=∂tk(Nuh,Nqh)i–k+∂tkDhui–k+. As a result of (.) and (.), one has the

following estimates.

Lemma . We have

d dtη



,∗+c(η,u)+∇q  

(u,ut)  +η



,+EHDL, (.)

u_+∇q_η_+(u,ut)  +E

H_EL_{, (.)}

ut_+∇qt_u_+(ut,utt)  +E

H_DL_{, (.)}

where EL_:=EL_–∇η

,andη,∗is equivalent toη.

Proof Noting that, by virtue of (.),

ω_k,i–k+=λmη,μu  k,i–k++

λmμ

 d dtη

 k,i–k+,

we deduce from (.) that

d dtη



k,i–k++c(η,u) 

k,i–k++∇q  k,i–k

u_k,i–k+η_{k+,i–k+}+uti +S

ω

In particular, we take (i,k) = (, ) and (i,k) = (, ) to get d

dtη



+c(η,u)  +∇q

 

(u,ut)  +η

 ,+S

ω

, (.)

and

d dtη



,+c(η,u) 

,+∇q  ,

(u,ut)  +η



,+Sω,. (.)

On the other hand, it is easy to show that

S_,ω +S_,ω ≤N_uh,N_qh_+Dh_u,divη  ˜A

 

u_+∇q_+η_

EH_EL_EH_DL_{. (.)}

Thus we immediately obtain (.) from (.)-(.).

Now we turn to the derivation of (.). In view of (.) with (i,k) = (, ), we have

u_+∇q_η_+(u,ut)  +S

u ,.

On the other hand, we can use (.) to infer that

Su_,=N_uh,N_qh_+Dh_u_EHEL.

Hence, (.) follows from the above two estimates.

Finally, to show (.), we take (i,k) = (, ) in (.) to deduce that

ut_+∇qt_∇u,ut,utt  +S

u

,u+(ut,utt)  +S

u ,.

Keeping in mind that

Su_,∂t

N_uh,N_qh_+∂tDhu  

˜At 

u_+∇q_+ ˜A_ut_+∇qt_EHDL,

we get (.) from the above two estimates.

3.5 Lower-order energy inequality

Now, we are able to build the lower-order energy inequality. In what follows, the letterscL i

andi= , . . . ,  will denote generic positive constants which may depend on the domain

and some physical parameters in the transformed MHD equations (.).

Proposition . Under the assumption(.),ifδis sufficiently small,then there is an

en-ergy functionalE˜Lwhich is equivalent toEL,such that

d dtE˜

Proof We chooseδso small that

_∇∂_tju_∇_A∂_tju_, ≤j≤. (.)

Then, thanks to (.), (.) and (.), we deduce from (.) and Lemmas .-. that there are constantscL,cLandσ≥, such that

d dtE˜

L

 +D˜L≤cL( +σ) √

EHDL _{for anyσ≥σ}

, (.)

whereσdepends on the domain and the known physical parameters, and

˜

EL  :=

√

ρut_+λ ¯M· ∇u_+σ 

k=

√ρu_k,+λ ¯M· ∇η_k,

+



k= α+α=k

ρ∂α  ∂

α  η·∂

α  ∂

α  udy+

μ

∇η

 k,

,

˜

DL  :=cL

ut_+



k=

(η,M¯ · ∇η)_k,+u_k,

.

Utilizing (.), (.), the interpolation inequality, and the estimate

η_k,η_k+,+ ¯M· ∇η_k, for any ≤k≤, (.)

we find that

d dt

η_,∗+cL_∇_Aut_+cL_(η,u)_+∇q_+utt_

≤cL_(u,ut)  +η



,+ ¯M· ∇η,+ √

EHDL_{. (.)}

Now, multiplying (.) bycL/(cL) and adding the resulting inequality to (.), we obtain

d dtE˜

L

 +D˜L≤cL( +σ) √

EHDL_{, (.)}

whereE˜_LandD˜Lare defined by

˜

EL

 :=E˜L+cL

η_,∗+cL_∇_Aut

/cL_,

˜

DL_:=D˜L

/ +cLcL(η,u)  +∇q



+utt

/cL_.

On the other hand, from (.) we getDL_D˜L₊√_EHDL_{, which impliesD}L_D˜L_for

sufficiently smallδ. Therefore, (.) can be rewritten as follows:

d dtE˜

L

 +cLDL≤cL( +σ) √

EHDL_{. (.)}

that

∇η_,+η_+ut_E˜_L.

In view of (.), we further have

EL_E˜L  +

√

EHEL_,

which impliesEL_E˜L

 for sufficiently smallδ. In addition, obviouslyE˜LEL. Hence, the

energy functionalE˜_Lis equivalent toEL_{. Finally, lettingδ≤c}L

/cL( +σ) and notingE˜L=

E˜_L/cL_, we see that (.) immediately implies (.). Obviously, the energy functionalE˜L

is still equivalent toEL. This completes the proof.

4 Higher-order and highest-order energy inequalities

In this section, we derive the estimates (.) and (.) for the transformed magnetic RT problem. First we shall establish the higher-order version of Lemmas .-. in the following:

Lemma . We have

d dt

_√

ρ∂_tju_+λM¯ · ∇∂tj–u  +

∇∂_tj–q·Dt,j_u dy

+μ∇_A∂_tju_

√ELDH_{, j= , .}

Proof We only prove the casej= , and the casej=  can be shown similarly. Multiplying (.)withj=  by∂tu, integrating (by parts) the resulting equation over, and using

(.)and (.), we obtain

 

d dt

√

ρ∂_tu_+λ ¯M· ∇utt_+μ∇A∂_tu_

=

∂_tqdivDt,_u dy+μ

N_ut,·∂_tudy+

N_qt,·∂_tudy:=



k=

I_kH. (.)

On the other hand, the integralsIH,IandIHcan be estimated as follows:

I_H= –

∇∂_tq·Dt,_u dy= –d dt

∇∂_tq·Dt,_u dy+

∇∂_tq·∂tDt,u dy

≤–d dt

∇∂_tq·Dt,_u dy

+c∇∂_tq_∂_tA_u+∂_tA_ut+Attutt+At∂tu

≤–d dt

∇∂_tq·Dt,_u dy+c√ELDH_,

I_H∂_tA_Au+AttAtu+Aut

+At

Attu+Atut+Autt

I_H∂_tA

∇q+Att∇qt+At∇qtt∂  tu

√

ELDH_.

Inserting the above three inequalities into (.), we get the desired conclusion

immedi-ately.

Lemma . We have

d dt

ρ∂_hkη·∂_hkudy+μ ∇∂

k hη

 

+λM¯ · ∇∂hkη  

√EL_(η,u) +D

H_+∂k hu



 for any k= and.

Proof We only show the casek= , since the derivation of the casek=  is similar. Since the derivation involves the norm∇q, we first estimate∇q. It follows from (.) that

u+∇qη+(u,ut)_+Nuh,Nqh+D h u.

The last two terms on the right-hand side of the above inequality can be bounded as fol-lows:

N_uh,N_qh_+Dh_u ˜Au+ ˜A∇q.

Hence, ifδis sufficiently small, then one gets from the above two estimates that

u+∇qη+√DH_{. (.)}

Now, applying∂

h to (.), multiplying the resulting equality by∂hη, we utilize (.)

to have

ρ∂t

∂_hη·∂_hu–μ∂_hηt+λM¯ · ∇ _¯

M· ∇∂_hη·∂_hη

=∂_hρω(ue–ue) +μNuh+Nqh–∇q

·∂_hη+ρ∂_hu. Integrating (by parts) the above identity over, we obtain

d dt

ρ∂_hη·∂_hudy+μ ∇∂

 hη

 

+λM¯ · ∇∂hη  

= ρω

∂_h(ue–ue)·∂hηdy+μ

∂_hN_uh·∂_hηdy–

∂_hN_qh·∂_hηdy

–

∂_hqdiv∂_hηdy+√ρ∂_hu_



k=

J_kH+∂_hu_, (.)

where the first four integrals on the right-hand side are denoted byJH

 , . . . ,JH, respectively.

On the other hand, the four integralsJH

 , . . . ,JHcan be bounded as follows:

J_H= –μ

∂_h(A˜jlA˜jk+A˜lk+A˜kl)∂kui

∂l∂hηidy ˜Au+ ˜Au+ ˜Auη, ˜Au+ ˜A

   ˜A

  u

  u

 

 + ˜Au

η,

√EL_(η,u)

, (.)

J_H ˜A∇q+ ˜A∇q+ ˜A∇qη ˜A∇q+ ˜A

   ˜A

  ∇q

  ∇q

 

 + ˜A∇q

η

√EL_η +DH

, (.)

J_H=

∂_hq·∂_hdivηdy∇qηη+η_ η∇qη+η√EL_η

+DH

, (.)

where the interpolation inequality (.) has been employed in (.) and (.).

Conse-quently, putting the above four estimates into (.), we obtain Lemma ..

Lemma . We have

d dt

√

ρ∂_hku_+λM¯ · ∇∂hkη  

+μ∇∂_hku_

√EL_(η,u) +D

H _{for k= and.}

Proof We only show the casek= , since the derivation of the casek=  is similar. Ap-plying∂_hto (.), multiplying the resulting equality by∂hu, we make use of (.)to

have

ρ∂_hut·∂hu–μ∂hu·∂hu–λm∂∂hη·∂hηt

=ρω∂_h(ue–ue) +μ∂hNuh+∂hNqh–∇∂hq

·∂_hu. Integrating (by parts) the above identity over, we get

 

d dt

ρ∂_hudy+λM¯ · ∇∂hη   +μ ∇∂  hu   =μ

∂_hN_uh·∂_hudy–

∂_hN_qh·∂_hudy+

∂_hqdiv∂_hudy=:



k=

MH_k. (.)

Analogously to (.)-(.), the three integralsMH

 ,MHandMHcan be bounded as

fol-lows:

MH_ = –

∂_h(A˜jlA˜jk+A˜lk+A˜kl)∂kui

·∂l∂hudy √

EL_(η,u) ,

MH_ √EL_(η,u) +D

H_,

MH_ =

∂_hq·∂_hDh_udy∇q ˜Au+ ˜Au+ ˜Au

∇qηu+ ˜A

   ˜A

  u

  u

 

 + ˜Au

√EL_η +DH

Substituting the above three inequalities into (.), we obtain Lemma ..

Lemma . The following estimates hold:



k=

∂_tku_–k+∇∂_tkq_–kη_+u,ut,utt,∂tu  +E

L_EH_{, (.)}

d dtη



,∗+η+ 

k=

∂_tku_–k+∇∂_tkq_–k

η_,+η,u,ut,utt,∂tu  +E

L_DH_{, (.)}



k=

∂_tku_–k+∇∂_tkq_–ku_+ut,utt,∂tu  +E

H_DH_{, (.)}

where the normη_,∗is equivalent toη_and EH_:=EH_–∇η ,.

Proof () We begin with the derivation of (.). Taking (i,k) = (, ) in the Stokes estimate (.), we have

u_+∇q_η_+(u,ut)_+Su,.

Noting that

Su_,=N_uh,N_qh_+Dh_u_

˜A

u+ ˜Au+ ˜A∇q+ ˜A∇qELEH,

we get

u_+∇q_η_+(u,ut)_+ELEH. (.)

Using the recursion formula (.) fori=  fromk=  to , we obtain



k=

∂_tku_–k+∇∂_tkq_–k ∂_t(u,ut)_+ 

k=

∂_tk–u_–k+S_k,u .

On the other hand,Su

k,can be bounded from above by



k=

S_k,u =



k=

∂_tkN_uh_–k+∂_tkDh_u_–k+∂_tkN_qh_–k

˜A 

ut_+∇qt

+ ˜A_utt_+∇qtt_

+ ˜A

ut+ ˜At

u_+∇q_+ ˜At 

ut_+∇qt_

+ ˜At_u_+ ˜Att_u_+∇q_

Therefore,



k=

∂_tku_–k+∇∂_tkq_–k∂_t(u,ut)  +



k=

∂_tk–u_–k+EL_EH_{, (.)}

which, together with (.), yields

 k= ∂k tu 

–k+∇∂ k tq

 –k

η_+∂_t(u,ut)_+  k= ∂k– t u  –k+E

L_EH_{. (.)}

In view of the interpolation inequality (.), we have

u≤Cu+u and utCut+ut. (.)

Thus, inserting (.) into (.), one gets (.).

() We proceed to prove the estimate (.). Exploiting the recursion formula (.) for

i=  fromk=  to , we see that there are positive constantsck,k= , . . . , , such that

d dt



k=

ckη_k,–k+(η,u)_+∇q_ut+η,+ 

k=

Sω_k,,

which combined with (.) gives

d dt



k=

ckη_k,–k+c

η_+



k=

∂_tku_–k+∇∂_tkq_–k

η_,+∂_t(u,ut)  +  k= ∂k– t u  –k+E

L_EH_+Sω

,,

where we have used the fact thatSω

k,≤S

ω

,for ≤k≤. Since

Sω_,=N_uh,N_qh_+Dh_u,divη



˜A_u

+ ˜Au+ηη+ ˜A∇q+ ˜A∇q

EL_EH

andEH_≤DH_{, we further infer that}

d dt



k=

ckη_k,–k+c

η_+



k=

∂_tku_–k+∇∂_tkq_–k

η_,+∂_t(u,ut)  +



k=

∂_tk–u_–k+ELDH. (.)

Using the interpolation inequality (.), we get (.) from (.), whereη_,∗equals to



() Finally, we derive the estimate (.) for higher-order dissipation estimates. We use the recursion formula (.) fori=  fromk=  to  to deduce that



k=

∂_tku_–k+∇∂_tkq_–k ∂_t(u,ut)_+ 

k=

∂_tk–u_–k+Su_k,. (.)

Noting that



k=

S_k,u =



k=

∂_tkN_uh,N_qh_–k+∂_tkDh_u_–k

˜At_u

+ ˜Aut+ ˜At∇q+ ˜A∇qt

+ ˜Attu_+ ˜At ut_+ ˜Autt_

+ ˜Att_∇q_+ ˜At∇qt+ ˜A∇qtt EH_DH_,

we obtain (.) from (.).

Now we are in a position to build the higher-order and highest-order energy inequal-ities. In what follows, the letterscH_i andi= , . . . ,  will denote generic constants which

may depend on the domainand some physical parameters in the transformed MHD

equations (.).

Proposition . Under the assumption(.),ifδis sufficiently small,then there are two

normsE˜H_andη

,∗which are equivalent toEHandηrespectively,such that

d dtE˜

H_+DH√_EL_(η,u)

, (.)

d dtη



,∗+(η,u)E

H_+DH _{on(,T]. (.)}

Proof () We first prove (.). Similarly to (.), we make use of (.), (.), (.) and (.) to deduce from Lemmas .-. and (.) that there are constantscH

,cH,cH and α≥, such that

d dtE˜

H

 +D˜H ≤cH

( +α)√EL_(η,u) +D

H_+(η,u,u t)

 

for anyα≥α, (.)

whereαdepends on the domain and the known physical parameters, and

˜

EH  :=



k= _√

ρ∂_tku_+λM¯ · ∇∂tk–u  +

∇∂_tk–q·Dt,k_u dy

+

≤α+α≤

ρ∂α  ∂

α  η·∂

α  ∂

α  udy+



k= μ

∇η

 k,