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

Time-decay solutions of the initial-boundary value problem of rotating magnetohydrodynamic fluids College of Mathematics and We have investigated an initial-boundary problem for the perturbation equations of Computer Science, Fuzhou University, Fuzhou, 350108, China 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: ρv + ρv ·∇v + ∇(p + λ |M| /) + ρ(ω  × v)= μv + λ M ·∇M, t   M = M ·∇v – v ·∇M, (.) div v = div M =. Here the unknowns v = v(x, t), M := M(x, t)and p = p(x, t) denote the velocity, the magnetic field, and the pressure of the incompressible MHD fluid respectively; μ >, ρ and λ stand for the coefficients of the shear viscosity, the density constant, and the permeability of vacuum, respectively. ρ(ω  × v) represents the Coriolis force, and ω  = (, , 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 constraint div M = , it can be seen just as a restriction on the initial value of M since (div M) = due to (.) . t  © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 2 of 31 ¯ ¯ Let M := (m , m , m )be a constant vector with m =, 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 ρv + ρv ·∇v + ∇(q ˜ + λ |N + M| /) ⎪ t  ¯ ¯ = μv + λ (N + M) ·∇(N + M)+ρω(v e – v e ),      (.) ¯ ¯ N =(N + M) ·∇v – v ·∇(N + M), ⎪ t div v = div N =, wherewehaveusedthe relation ω  × v = ω(v e – v e ). For system (.), we impose the     initial and the boundary conditions: (v, N)| =(v , N )in ,(.) t=   v(·, t)| = for any t >, (.) where v and N should satisfy the compatibility conditions div v = div N =. 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 with h >,   where T := (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 consider ω =, 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 (.)-(.)withproperinitial data in H (). Our result also holds for the case ω = , thus improves Tan and Wang’s result in [], in which the suffi-  ciently small initial data at least belong to H (). 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 mapping ζ := ζ (y):  → ,suchthat   ∂ = ζ (∂)and det ∇ζ ≡ , (.)   Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 3 of 31 where ζ denotes the third component of ζ .Wedefine theflow map ζ as the solution to ζ (y, t)= v(ζ (y, t), t), (.) ζ (y,) = ζ . We denote the Eulerian coordinates by (x, t)with x = 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 that ζ (·, t) is invertible and  = ζ (, t). In other words, the Eulerian domain of the fluid is the image of  under mapping ζ . In view of the non- slip boundary condition v| =, we have ∂ = ζ (∂, t). In addition, since det ∇ζ =,we ∂  have det(∇ζ)= (.) due to div v =; see [], Proposition .. Now, we further define the Lagrangian unknowns by  + (u, p ˜, B)(y, t)= v, p + λ |M| /, M ζ (y, t), t for (y, t) ∈  × R.(.) Thus in Lagrangian coordinates the evolution equations for u, p ˜ and B read as ζ = u, ρu – μ u + ∇ p ˜ = λ B ·∇ B +ρω(u e – u e ), t A A  A     (.) B – B ·∇ u =, ⎪ t A div u =, with initial and boundary conditions (u, ζ – y)| = and (ζ , u, B)| =(ζ , u , B ). ∂ t=    Moreover, div B =  if the initial data ζ and B satisfy A   div B =. (.) A  T – Here A denotes the initial value of A,the matrix A := (A ) via A =(∇ζ ) :=  ij × – (∂ ζ ) , and the differential operators ∇ , div A and  are defined by ∇ f := j i A A A × T T (A ∂ f , A ∂ f , A ∂ f ) , div (X , X , X ) := A ∂ X ,and  f := div ∇ f for appro- k k k k k k A    lk k l A A A priate f and X. It should be noted that we have used the Einstein convention of summation over repeated indices, and ∂ = ∂ . In addition, in view of the definition of A and (.), we k y ∗ ∗ can see that A =(A ) ,where A is the algebraic complement minor of the (i, j)th entry × ij ij ∂ ζ .Since ∂ A = , we can get an important relation j i k ik div u = ∂ (A u )=, (.) A l kl k which will be used in the derivation of temporal derivative estimates. Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 4 of 31 Our next goal is to eliminate B by expressing it in terms of ζ . 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 of A,one has ∂ ζ A = A ∂ ζ = δ , i k kj ik k j ij where δ = for i = j,and δ = for i = j. Thus, applying A to (.) ,weobtain ij ij jl  A ∂ B = B A ∂ u A = B A ∂ (∂ ζ )A =–B A ∂ ζ ∂ A =–B ∂ A , jl t j i ik k j jl i ik t k j jl i ik k j t jl j t jl which implies that ∂ (A B )= (i.e.,(A B) =). Hence, t jl j t   A B = A B,(.) jl j jl j   which yields B = ∂ ζ A B , i.e., i l i jl j B = ∇ζ A B.(.) Here and in what follows, the notation f also denotes the initial data of the function f .To obtain the asymptotic stability in time, we naturally expect (ζ , B)converges to (y, M)as t →∞.(.) Thus (.) formally implies ¯ ¯ A B = M, i.e., B = M ·∇ζ.(.)    Putting the above expression of B into (.), we get B = M ·∇ζ.(.) Moreover, in view of (.), (.)and (.), the Lorentz force term can be represented by   ¯ ¯ ¯ (.) B ·∇ B = B A ∂ B = A B ∂ (M ·∇ζ)= M ·∇(M ·∇ζ ). A l lk k k lk l Summing up the above analyses, we can see that, under the initial conditions (.)and (.), one can use the relation (.)tochange(.) into the following Navier-Stokes sys- tem: ζ = u, ¯ ¯ ρu – μ u + ∇ p ˜ = λ M ·∇(M ·∇ζ)+ρω(u e – u e ), t A A      div u =, and B is defined by (.). Now, we introduce the shift functions η = ζ – y and q = p ˜ – p ¯.(.) Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 5 of 31 Then the evolution equations for the shift functions (η, q)and u read as η = u, ¯ ¯ ρu – μ u + ∇ q – λ M ·∇(M ·∇ζ)=ρω(u e – u e ), (.) t A A      div u =, – where A =(I + ∇η) , I =(δ ) . The associated initial and boundary conditions read as ij × follows: (η, u)| =(η , u ), (η, u)| =. (.) t=   ∂ It should be noted that the shift function q is the sum of the perturbed fluid and the mag- netic pressures in Lagrangian coordinates. Hence we still call q the 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: + p p ,p R := [, ∞), := , L := L ():= W ()for< p ≤∞,  , k k, H := W (), H := W (), · := · for k ≥ ,   H () α α α α k      ∂ denotes ∂ ∂ for any α + α = k, · := ∂ ∂ · ,   m,k h     k α +α =m   a  b means that a ≤ cb, where, and in what follows, the letter c denotes a generic constant which may depend on the domain  and some physical parameters, such as λ , M, g, μ and ρ in the MHD equations (.). It should be noted that a product space (X) of vector functions is still    denoted by X, for example, a vector function u ∈ (H ) is denoted by u ∈ H with norm  / u := ( u ) . Finally, we define some functionals: k  H k=   L  E := ∇η + (η, u) + (u , ∇q) , ,      L k   D := (η, M ·∇η, ∇u) + (η, u) + ∂ u + ∇q + ∇q , t   ,  –k k=     H   k k E := ∇η + η + ∂ u + ∇∂ q , ,  t t –k –k k= k=       H k k  D := (η, M ·∇η, ∇u) + (η, u) + ∂ u + ∇∂ q + ∇q , t t  ,  –k –k k= k= (η, u)(τ) G (t)= sup η(τ) , G (t)= dτ,    / ( + τ) ≤τ<t H H  L G (t)= sup E (τ)+ D (τ)dτ, G (t)= sup ( + τ) E (τ).   ≤τ<t ≤τ<t  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 6 of 31 Next, we introduce our main result. Theorem . Let  be a horizontally periodic domain with finite height, ω be an arbitrary real number, and m =. Then there is a sufficiently small constant δ >, 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., ∂ u(x,)| = for j =   ∂ 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(∞):= G (∞) ≤ c η + u .(.) k     k= Here the positive constants δ and c depend on the domain  and other known physical parameters λ , M, μ and ρ. Remark . Exploiting the inverse transformation of ζ , we can easily deduce from Theo- rem . the global well-posedness of the original MHD problem (.)-(.). More precisely, there is a sufficiently small constant δ >, such that, for any (v , N ) ∈ H satisfying the    following conditions: () there exists an invertible mapping ζ := ζ (x):  → ,suchthat(.)holds,where   T – A =(∇ζ ) ; ¯ ¯ () (M + N )(ζ )= M ·∇ζ ;      () ζ – x + v ≤ δ ;      () the initial data  , v , N satisfy necessary compatibility conditions (i.e.,    ∂ v(x,)| = for j = and ),  +  there exists a unique global solution (v, N) ∈ C (R , H ) to the original MHD problem (.)-(.) with an associated perturbation pressure q ˜.Moreover, (v, N, q ˜)enjoys the fol- lowing stability estimate:      k k sup N + ∂ v(t) + ∇∂ q ˜(t)  t t –k –k ≤t<∞ k= k=      + sup ( + t) (v, N) + (v , ∇q ˜) ≤ c ζ – x + v .(.) t       ≤t<∞ Now we briefly describe the basic idea in the proof of Theorem ..Bythe standard energy method, there are two functionals E and Q of (η, u) satisfying the lower-order energy inequality (see Proposition .) L L L E + D ≤ QD,(.) dt L L where the functional E is equivalent to E . Unfortunately, we can not close the energy estimates only based on (.), since Q can not be controlled by E .However,weobserve Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 7 of 31 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 (.). Since E contains η , we find that the higher-order energy at least includes η .Thus, similarto(.),   we establish the higher-order energy inequality (see Proposition .) H H ˜ L E + D ≤ E (η, u),(.) dt H H  where the functional E is equivalent to E .Moreover, thehighest-order norm (η, u) enjoys the highest-order energy inequality  H H η + (η, u)  E + D , (.) ,∗ dt where the norm η is equivalent to η . In the derivation of the apriori estimates, we ,∗  have Q  E ,and thus (.) implies (see Proposition .) L L E + D ≤ . (.) dt 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 ..InSection ,we first derive the lower-order energy inequality (.)for thetransformed MHDproblem. Then in Section  we derive the higher-order energy inequality (.)and thehighest- 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 theapriori estimates for the transformed MHD problem. To this end, let (η, u) be a solution of the transformed MHD problem with perturbed pressure q,suchthat G (T)+ sup E (τ) ≤ δ ∈ (, ) for some T >, (.) ≤τ ≤T where δ is 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 calculations, we shall repeatedly use Cauchy-Schwarz’s inequality, Hölder’s inequality, the embedding inequalities (see [], . Theorem) f p  f for  ≤ p ≤and f ∞  f ,(.) L  L  and the interpolation inequalities (see [], . Theorem) j j – i i f  f f ≤ C f +  f i (.) j   H  i Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 8 of 31 for any  ≤ j < i and any constant  > , where the constant C depends on the domain and . In addition, we shall also frequently use the following two estimates: fg  f g for j ≥(.) j j κ(j) and f  ∂ f for f ∈ H,(.)    where κ(j)= j for j ≥ and κ(j)= for j ≤ . 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 for A. To begin with, we give an expression of A.Using (.) ,wehave ∗ ∗ ∂ det(I + ∇η)= ∂ ∂ η A = A ∂ u,(.) t t j i j i ij ij ≤i,j≤ ≤i,j≤ where A is the algebraic complement minor of the (i, j)th entry in the matrix I + ∇η. ij Recalling the definition of A,wesee that A = A / det(I + ∇η). ij × Inserting this relation into (.), we get ∂ det(I + ∇η)= det(I + ∇η) A ∂ u =, t ij j i ≤i,j,≤ which, together with initial condition det(I + ∇η ) = , implies det(I + ∇η)=. Thus we obtain A = A.(.) ij × Now, exploiting (.) ,(.), (.)and (.), we easily see that A  + η + η for ≤ j ≤ , (.) j j+ j+ ∇A ≤ η + η  η for  ≤ j ≤ . j j+ j+ j+ Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 9 of 31 Similarly, we further deduce that i– i k ∂ A  ∂ ∇u for any  ≤ i ≤ and  ≤ j ≤ –i. t t j j k= ˜ ˜ Letting A := A – I,wenextbound A. To this end, we assume that δ is so small that the following expansion holds: T  i  T A = I – ∇η +(∇η) (–∇η) = I – ∇η +(∇η) A , i= whence T  T A =(∇η) A – ∇η. Using (.), (.)and (.), we find that A  ∇η for  ≤ j ≤ . j j 3.2 Temporal derivative estimates In this subsection, we try to control temporal derivatives. For this purpose, we apply ∂ to (.)toget j+ j ∂ η = ∂ u, t t j+ j j ρ∂ u – μ ∂ u + ∇ ∂ q A A t t t (.) j j t,j t,j ⎪ ¯ ¯ = λ ∂ M ·∇(M ·∇ζ)+ρω∂ (u e – u e )+ μN + N , ⎪      u q t t j t,j div ∂ u = div D , A u where j–m–n t,j n m N := ∂ A ∂ ∂ A ∂ ∂ u , il l ik k u t t t ≤m<j,≤n≤j j–l t,j l N := – ∂ A ∂ ∂ q , ik k q t × ≤l<j j–l j–l t,j l D := – C ∂ A ∂ u,(.) ki k u j t t × ≤l<j j–l C denotes the number of (j – l)-combinations from a given set S of j elements, andwehaveusedrelation(.)in(.). Then from (.) we show the following estimates: Lemma . It holds that for j = and , d √ j  j  j  ρ∂ u + λ M ·∇∂ η + μ ∇ ∂ u  E D,(.)  A t t t    dt d  √     L ∇ u + ρu  u + u + E D . (.) A t tt t     dt μ Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 10 of 31 Proof () We only prove (.)for j = , since the derivation of the case j = is similar. Multiplying (.) with j = by u , integrating (by parts) the resulting equality over ,  t and using (.) ,weget  d √    ρu + λ M · η + μ ∇ u t  t A t     dt =ρω ∂ (u e – u e ) · u dy + q div u dy t     t t A t t, t, + μ N · u dy + N · u dy t t u q := I.(.) k= L L The last three integrals I ,..., I can be estimated as follows:   L t, I := – ∇q · D dy  ∇q A  u   ∇q A u t t  t t  t    u L L E D , (.) L L I  A A u u  E D,(.)  t   t  L L I  A ∇q u  A ∇q u  E D,(.) t  L t  t   t  wherewehaveused(.) in (.). Consequently, the desired estimate (.) follows from (.)-(.). () Now we turn to the proof of (.). Multiplying (.) with j = by u , integrating  tt (by parts) the resulting equality over ,and using(.) ,weconclude μ d √   ∇ u + ρu A t tt    dt ¯ ¯ = ρω∂ (u e – u e ) · u + λ M ·∇(M ·∇u) · u dy t     tt  tt t, t, + q div u dy + μ N · u dy + N · u dy t A tt tt tt u q + μ ∇ u : ∇ u dy A t A t := J.(.) k= H H On the other hand, the five integrals J ,..., J can be bounded as follows:   J  u + u u ,(.) t   tt  H t, J =– ∇q · D dx  ∇q A u + A u t t  tt   t  t   u E D,(.) H L J  A A u u  E D,(.)  t   tt   Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 11 of 31 H L J  A ∇q u  E D,(.) t   tt  H  L J  A A u  E D.(.)  t  t   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: η = u, h h ¯ ¯ ρu – μu + ∇q – λ M ·∇(M ·∇η)=ρω(u e – u e )+ μN + N , (.) t      u q div u = D , where ˜ ˜ ˜ ˜ N := ∂ (A A + A + A )∂ u , l jl jk lk kl k i × h h ˜ ˜ N := –(A ∂ q) and D := A ∂ u . ik k × lk k l q u Then we have the following estimate on horizontal spatial derivatives of η. Lemma . It holds that d μ j j j  j  ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt  L  E D + u , ≤ j ≤ . j, Proof We only show the case j = ; the remaining three cases can be verified similarly.   Applying ∂ to (.) , multiplying the resulting equation by ∂ η,and then using(.) ,   h h we get      ¯ ¯ ρ∂ ∂ η · ∂ u – μ∂ η + λ M ·∇ M ·∇∂ η · ∂ η t t  h h h h h   h  h    = ρω∂ (u e – u e )+ μ∂ N + ∂ N – ∇∂ q · ∂ η + ρ ∂ u .     h h u h q h h h If we integrate (by parts) the above identity over ,weobtain d μ       ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt     h  h  =ρω ∂ (u e – u e ) · ∂ η dy + μ∂ N + ∂ N · ∂ η dy     h h h u h q h    + ∂ q div ∂ η dy + ρ∂ u h h h L  K + u ,(.) k , k= where the first three integrals on the right-hand side of the first equality in (.)are de- L L L noted by K , K and K ,respectively.    Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 12 of 31 We have the boundedness L   K ≤ ∂ u ∂ η.(.)  h h   Noting that  h  h ˜ ˜ ˜ ˜ μ∂ N + ∂ N  A u + A u + A ∇q + A ∇q         h u h q H L E D , we find that L  h  h  L K  μ∂ N + ∂ N ∂ η  E D.(.)  h u h q h   Next we estimate the third integral K .Tostart with,weanalyze thepropertyof div η. Since det(I + ∇η)=, we have by Sarrus’ rule div 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 ∂ η ).           Multiplying the above identity by a smooth test function φ, and then integrating (by parts) the resulting identity over ,wederivethat φ div η dy =– ∇φ · ψ dy, where ⎛ ⎞ η (∂ η + ∂ η )+ η (∂ η ∂ η – ∂ η ∂ η )               ⎜ ⎟ ψ := η ∂ η – η ∂ η + η (∂ η ∂ η – ∂ η ∂ η ) . ⎝ ⎠                –η ∂ η – η ∂ η + η (∂ η ∂ η – ∂ η ∂ η )                This means that div η = div ψ. Thus, it follows immediately that L    K =– ∂ ∇q · ∂ ψ dy =– ∂ ∇q · ∂ ψ dy  h h h  L ∇q ∂ ψ  η ∇q η  E D.(.)      Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 13 of 31 Now, substituting (.), (.)and (.)into(.), we immediately obtain the desired estimate for the case j =. Similarly, we also establish the following estimates of horizontal spatial derivatives of u: Lemma . We have d √    j j j ρ∂ u + λ M ·∇∂ η + μ ∇∂ u  E D , j = ,,. h h h    dt Proof We only prove the case j = ; the remaining two cases can be shown similarly. Ap-   plying ∂ to (.) , and multiplying the resulting equality by ∂ u, we make use of (.)   h h to get       ¯ ¯ ρ∂ u · ∂ u – μ∂ u · ∂ u – λ M ·∇ M ·∇∂ η · ∂ η t  t h h h h h h   h  h   = ρω∂ (u e – u e )+ μ∂ N + ∂ N – ∇∂ q · ∂ u.     h h u h q h h Integrating (by parts) the above identity over ,wehave  d μ       ρ ∂ u dy + λ M ·∇∂ η + ∇∂ u h h h    dt   h  h    L L = μ∂ N + ∂ N · ∂ u dy + ∂ q div ∂ u dy =: M + M.(.) h u h q h h h   L L On the other hand, similarly to (.)and (.), the two integrals M and M can be   estimated as follows: L  h  h  L M  μ∂ N + ∂ N ∂ u  E D,(.)  h u h q h   L   h L ˜ H M =– ∂ q∂ D dy  ∇q A u  E D,(.)     h h u wherewehaveused(.) in (.). Consequently, putting the above two estimates into (.), we obtain Lemma . for the case j =. 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  h h ¯ ¯ = λ M ·∇(M ·∇η)– λ m η +ρω(u e – u e )– ρu + μN + N , (.)       t u q div w = μD + λ m div η, u  coupled with boundary condition ω| =, (.) where ω = λ m η + μu.  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 14 of 31 Now, applying ∂ to (.)and (.), we get k k –∂ w + ∇∂ q h h k  k k h h ¯ ¯ = ∂ (λ M ·∇(M ·∇η)– λ m η +ρω(u e – u e )) – ρ∂ u + μ∂ N + N ,       t h  h h u q k k h  k div ∂ w = μ∂ D + λ m div ∂ η, ⎪  u  h h h ∂ ω| =. Then we apply the classical regularity theory to the Stokes problem as in [], Proposition ., to deduce that    ω ω + ∇q  ∇η + (u, u ) + S,(.) k,i–k+ k,i–k k+,i–k k,i k,i–k where   ω h h h S := N , N + D , div η . k,i u q u k,i–k k,i–k+ In addition, applying ∂ to (.) -(.) ,wesee that   k k –μ∂ u + ∇∂ q t t k k k+ k h k h ¯ ¯ = λ M · ∂ ∇(M ·∇η)+ρω∂ (u e – u e )– ρ∂ u + μ∂ N + ∂ N ,      t t t t u t q k k h div ∂ u = ∂ D , t t u ∂ u| =. Hence, we apply again the classical regularity theory to the Stokes problem to get    k k k  u ∂ u + ∇∂ q  ∂ ∇ η, u, u + S,(.) t t t k,i i–k+ i–k i–k u k h h  k h  where S := ∂ (N , N ) + ∂ D .Asaresult of (.)and (.), one has the k,i t u q i–k t u i–k+ following estimates. Lemma . We have      H L η + c (η, u) + ∇q  (u, u ) + η + E D,(.) ,∗  ,   dt    H L u + ∇q  η + (u, u ) + E E , (.)       H L u + ∇q  u + (u , u ) + E D,(.) t t t tt    L L  where E := E – ∇η and η is equivalent to η . ,∗  , Proof Noting that, by virtue of (.) , λ m μ d     ω = λ m η, μu + η , k,i–k+  k,i–k+ k,i–k+  dt we deduce from (.)that   η + c (η, u) + ∇q k,i–k+ k,i–k k,i–k+ dt    ω u + η + u + S.(.) k,i–k k+,i–k+ i k,i Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 15 of 31 In particular, we take (i, k)=(,) and (i, k)=(,) to get      ω η + c (η, u) + ∇q  (u, u ) + η + S (.)   , ,   dt and      ω η + c (η, u) + ∇q  (u, u ) + η + S.(.) , , , , ,  dt On the other hand, it is easy to show that   ω ω h h h     S + S ≤ N , N + D , div η  A u + ∇q + η , , u q u       H L H L E E  E D.(.) Thus we immediately obtain (.)from(.)-(.). Now we turn to the derivation of (.). In view of (.)with (i, k)=(,), we have    u u + ∇q  η + (u, u ) + S .    , On theother hand,wecan use(.)toinfer that   u h h h H L S = N , N + D  E E . , u q u   Hence, (.) follows from the above two estimates. Finally, to show (.), we take (i, k)=(,) in (.)todeducethat      u  u u + ∇q  ∇ u, u , u + S  u + (u , u ) + S . t t t tt t tt   ,  ,   Keeping in mind that   u h h h S  ∂ N , N + ∂ D t t , u q u         H L ˜ ˜ A u + ∇q + A u + ∇q  E D , t t t       we get (.)fromthe abovetwo estimates. 3.5 Lower-order energy inequality Now, we are able to build the lower-order energy inequality. In what follows, the letters c and i = ,..., will denote generic positive constants which may depend on the domain and some physical parameters in the transformed MHD equations (.). Proposition . Under the assumption (.), if δ is sufficiently small, then there is an en- L L ergy functional E which is equivalent to E , such that L L E + D ≤  on (, T]. (.) dt Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 16 of 31 Proof We choose δ so small that   j j ∇∂ u  ∇ ∂ u , ≤ j ≤ . (.) t A t   Then, thanks to (.), (.)and (.), we deduce from (.) and Lemmas .-. that L L there are constants c , c and σ ≥ , such that   L L L L ˜ ˜ H E + D ≤ c ( + σ ) E D for any σ ≥ σ,(.)    dt where σ depends on the domain and the known physical parameters, and √ √ L     ¯ ¯ E := ρu + λ M ·∇u + σ ρu + λ M ·∇η t      k, k, k= α α α α      + ρ∂ ∂ η · ∂ ∂ u dy + ∇η ,     k, k= α +α =k   L L   ˜  ¯ D := c u + (η, M ·∇η) + u .    k, k, k= Utilizing (.), (.), the interpolation inequality, and the estimate    η  η + M ·∇η for any  ≤ k ≤ , (.) k, k+, k, we find that  L  L   η + c ∇ u + c (η, u) + ∇q + u A t tt ,∗      dt L   L ≤ c (u, u ) + η + M ·∇η + E D.(.)  , , L L Now, multiplying (.)by c /(c ) and adding the resulting inequality to (.), we obtain   L L L L ˜ ˜ (.) E + D ≤ c ( + σ ) E D ,   dt ˜ ˜ where E and D are defined by L L L  L  L ˜ ˜ E := E + c η + c ∇ u / c , A t    ,∗    L L L L   L ˜ ˜ D := D / + c c (η, u) + ∇q + u / c . tt       L L L L L ˜ H ˜ On the other hand, from (.)weget D  D + E D , which implies D  D for sufficiently small δ. Therefore, (.) can be rewritten as follows: L L L L L (.) E + c D ≤ c ( + σ ) E D .    dt L L Next we show that E can be controlled by E . By virtue of Cauchy-Schwarz’s inequality, (.)and (.), there is an appropriately large constant σ , depending on ρ, μ and ,such Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 17 of 31 that    L ∇η + η + u  E . ,    In view of (.), we further have L L L E  E + E E , L L L L ˜ ˜ which implies E  E for sufficiently small δ. In addition, obviously E  E .Hence,the   L L L L L ˜ ˜ energy functional E is equivalent to E .Finally,letting δ ≤ c /c ( + σ ) and noting E =    L L L ˜ ˜ E /c ,wesee that (.) immediately implies (.). Obviously, the energy functional E   is still equivalent to E . This completes the proof. 4 Higher-order and highest-order energy inequalities In this section, we derive the estimates (.)and (.)for thetransformed magnetic RT problem. First we shall establish the higher-order version of Lemmas .-. in the following: Lemma . We have d √    j j– j– j t,j ρ∂ u + λ M ·∇∂ u + ∇∂ q · D dy + μ ∇ ∂ u  A t t t t    dt E D , j =,. Proof We only prove the case j =, and the case j =  can be shown similarly. Multiplying (.) with j = by ∂ u, integrating (by parts) the resulting equation over ,and using (.) and (.) ,weobtain    d √      ρ∂ u + λ M ·∇u + μ ∇ ∂ u  tt A t  t    dt  t, t,  t,  H = ∂ q div D dy + μ N · ∂ u dy + N · ∂ u dy := I.(.) t u u t q t k k= H H On the other hand, the integrals I , I and I can be estimated as follows:   H  t,  t,  t, I =– ∇∂ q · D dy =– ∇∂ q · D dy + ∇∂ q · ∂ D dy  t u t u t u dt  t, ≤ – ∇∂ q · D dy t u dt     + c ∇∂ q ∂ A u + ∂ A u + A u + A ∂ u  t  tt  tt  t  t t t t      t, H ≤ – ∇∂ q · D dy + c E D , t u dt H  I  ∂ A A u + A A u + A u   tt  t    t   t + A A u + A u + A u t  tt   t  t   tt    + A ∂ A u + A u + A u + A u ∂ u   tt  t  tt  t  t  tt  t t   E D , Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 18 of 31 H   H I  ∂ A ∇q + A ∇q + A ∇q ∂ u  E D .  tt  t  t  tt   t t   Inserting the above three inequalities into (.), we get the desired conclusion immedi- ately. Lemma . We have d μ   k k k k ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt    H k E (η, u) + D + ∂ u for any k = and .   Proof We only show the case k = , since the derivation of the case k = is similar. Since the derivation involves the norm ∇q ,wefirstestimate ∇q . It follows from (.)   that h h h u + ∇q  η + (u, u ) + N , N + D .    t u q u    The last two terms on the right-hand side of the above inequality can be bounded as fol- lows: h h h ˜ ˜ N , N + D  A u + A ∇q .     u q u   Hence, if δ is sufficiently small, then one gets from the above two estimates that u + ∇q  η + D.(.)      Now, applying ∂ to (.) , multiplying the resulting equality by ∂ η,weutilize (.)   h h to have      ¯ ¯ ρ∂ ∂ η · ∂ u – μ∂ η + λ M ·∇ M ·∇∂ η · ∂ η t t  h h h h h  h h   = ∂ ρω(u e – u e )+ μN + N – ∇q · ∂ η + ρ ∂ u .     h u q h h Integrating (by parts) the above identity over ,weobtain d μ       ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt     h   h  =ρω ∂ (u e – u e ) · ∂ η dy + μ ∂ N · ∂ η dy – ∂ N · ∂ η dy     h h h u h h q h    – ∂ q div ∂ η dy + ρ∂ u h h h H  J + ∂ u,(.) k h k= H H where the first four integrals on the right-hand side are denoted by J ,..., J ,respectively.   H H On the other hand, the four integrals J ,..., J can be bounded as follows:   H   J  ∂ η ∂ u , (.)  h h   Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 19 of 31 H   ˜ ˜ ˜ ˜ J =–μ ∂ (A A + A + A )∂ u ∂ ∂ η dy jl jk lk kl k i l i  h h ˜ ˜ ˜ A u + A u + A u η       ,         ˜ ˜ ˜ ˜ A u + A A u u + A u η     ,     E (η, u),(.) ˜ ˜ ˜ J  A ∇q + A ∇q + A ∇q η                ˜ ˜ ˜ ˜ A ∇q + A A ∇q ∇q + A ∇q η           H E η + D , (.) H    J = ∂ q · ∂ div η dy  ∇q η η + η     h h   H η ∇q η + η  E η + D,(.)     where the interpolation inequality (.) has been employed in (.)and (.). Conse- quently, putting the above four estimates into (.), we obtain Lemma .. Lemma . We have d √    k k k ρ∂ u + λ M ·∇∂ η + μ ∇∂ u h h h    dt E (η, u) + D for k = and . Proof We only show the case k = , since the derivation of the case k = is similar. Ap-   plying ∂ to (.) , multiplying the resulting equality by ∂ u, we make use of (.) to   h h have         ρ∂ u · ∂ u – μ∂ u · ∂ u – λ m ∂ ∂ η · ∂ η t  t h h h h  h h   h  h   = ρω∂ (u e – u e )+ μ∂ N + ∂ N – ∇∂ q · ∂ u.     h h u h q h h Integrating (by parts) the above identity over ,weget  d μ       ρ ∂ u dy + λ M ·∇∂ η + ∇∂ u h h h    dt   h   h    H = μ ∂ N · ∂ u dy – ∂ N · ∂ u dy + ∂ q div ∂ u dy =: M.(.) h u h h q h h h k k= H H H Analogously to (.)-(.), the three integrals M , M and M can be bounded as fol-    lows: H   ˜ ˜ ˜ ˜ M =– ∂ (A A + A + A )∂ u · ∂ ∂ u dy  E (η, u) , jl jk lk kl k i l  h h H H M  E (η, u) + D , H   h ˜ ˜ ˜ M = ∂ q · ∂ D dy  ∇q A u + A u + A u         h h u     √      H ˜ ˜ ˜ ∇q η u + A A u u + A u  E η + D .           Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 20 of 31 Substituting the above three inequalities into (.), we obtain Lemma .. Lemma . The following estimates hold:    k k   L H ∂ u + ∇∂ q  η + u, u , u , ∂ u + E E,(.) t tt t t  t –k –k  k=     k k η + η + ∂ u + ∇∂ q ,∗  t t –k –k dt k=   L H η + η, u, u , u , ∂ u + E D,(.) t tt , t    k k   H H ∂ u + ∇∂ q  u + u , u , ∂ u + E D,(.) t tt t t  t –k –k  k=   H H  where the norm η is equivalent to η and E := E – ∇η . ,∗  , Proof () We begin with the derivation of (.). Taking (i, k) = (, ) in the Stokes estimate (.), we have    u u + ∇q  η + (u, u ) + S .    , Noting that   u h h h S = N , N + D , u q u           L H ˜ ˜ ˜ ˜ A u + A u + A ∇q + A ∇q  E E ,         we get    L H u + ∇q  η + (u, u ) + E E.(.)    Using the recursion formula (.)for i = from k = to , we obtain       k k  k– u ∂ u + ∇∂ q  ∂ (u, u ) + ∂ u + S . t t t t k, –k –k  –k k= k= On the other hand, S can be bounded from above by k,      u k h k h k h S = ∂ N + ∂ D + ∂ N k, t u t u t q –k –k –k k= k=       ˜ ˜ A u + ∇q + A u + ∇q t t tt tt               ˜ ˜ ˜ + A u + A u + ∇q + A u + ∇q t t t t t              ˜ ˜ + A u + A u + ∇q t tt      L H E E . Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 21 of 31 Therefore,       k k  k– L H ∂ u + ∇∂ q  ∂ (u, u ) + ∂ u + E E , (.) t t t t –k –k  –k k= k= which, together with (.), yields       k k   k– L H ∂ u + ∇∂ q  η + ∂ (u, u ) + ∂ u + E E .(.) t t  t t –k –k  –k k= k= In view of the interpolation inequality (.), we have u ≤ C u +  u and u  C u +  u . (.)     t   t  t  Thus, inserting (.)into(.), one gets (.). () We proceed to prove the estimate (.). Exploiting the recursion formula (.)for i = from k =  to , we see that there are positive constants c , k = ,...,, such that       ω c η + (η, u) + ∇q  u + η + S , k t k,–k   , k, dt k= k= which combined with (.)gives       k k c η + c η + ∂ u + ∇∂ q k,–k  t t –k –k dt k= k=     k– L H ω η + ∂ (u, u ) + ∂ u + E E + S , , t t ,  –k k= ω ω wherewehaveusedthe fact that S ≤ S for  ≤ k ≤ . Since k, ,   ω h h h S = N , N + D , div η , u q u             ˜ ˜ ˜ ˜ A u + A u + η η + A ∇q + A ∇q           L H E E H H and E ≤ D ,wefurther inferthat       k k c η + c η + ∂ u + ∇∂ q k,–k  t t –k –k dt k= k=     k– L H η + ∂ (u, u ) + ∂ u + E D.(.) , t t  –k k= Using the interpolation inequality (.), we get (.)from(.), where η equals to ,∗ c η multiplied by some positive constant. k= k,–k Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 22 of 31 () Finally, we derive the estimate (.) for higher-order dissipation estimates. We use the recursion formula (.)for i = from k = to  to deduce that       k k  k– u ∂ u + ∇∂ q  ∂ (u, u ) + ∂ u + S .(.) t t t t k, –k –k  –k k= k= Noting that     u k h h k h S = ∂ N , N + ∂ D k, t u q t u –k –k k= k=         ˜ ˜ ˜ ˜ A u + A u + A ∇q + A ∇q t t t t               ˜ ˜ ˜ + A u + A u + A u tt t t tt             ˜ ˜ ˜ + A ∇q + A ∇q + A ∇q tt t t tt       H H E D , we obtain (.)from (.). Now we are in a position to build the higher-order and highest-order energy inequal- ities. In what follows, the letters c and i = ,..., will denote generic constants which may depend on the domain  and some physical parameters in the transformed MHD equations (.). Proposition . Under the assumption (.), if δ is sufficiently small, then there are two H  H  norms E and η which are equivalent to E and η respectively, such that ,∗  H H E + D  E (η, u),(.) dt  H H η + (η, u)  E + D on (, T]. (.) ,∗ dt Proof () We first prove (.). Similarly to (.), we make use of (.), (.), (.)and H H H (.) to deduce from Lemmas .-. and (.) that there are constants c , c , c and    α ≥ , such that   H H H H ˜ ˜ E + D ≤ c ( + α) E (η, u) + D + (η, u, u ) for any α ≥ α ,(.) t       dt where α depends on the domain and the known physical parameters, and   H k k– k– t,k ˜    ¯ E := ρ∂ u + λ M ·∇∂ u + ∇∂ q · D dy  t t t u   k= α α α α      + ρ∂ ∂ η · ∂ ∂ u dy + ∇η     k, ≤α +α ≤ k=     H  + α ρu + λ M ·∇η + c η , , ,  ,∗ Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 23 of 31     H H k  D := c ∂ u + (η, M ·∇η, ∇u) + η   t   k, k= k=   k k + ∂ u + ∇∂ q . t t –k –k k= Moreover, by (.) and Proposition . we find that H H H H ˜ ˜ L E + D ≤ c ( + σ ) E (η, u) + D,(.)   dt H H H L H H H L H H H H ˜ ˜ ˜ ˜ ˜ ˜ where E := E + c E and D := D + c D .By (.), we have D  D + E D .So,      (.)can be rewrittenas H H H H ˜ L E + c D ≤ c ( + σ ) E (η, u) for sufficiently small δ.(.)    dt H H Next, we show that E can be controlled by E . Keeping in mind that   k– t,k k– t,k ∇∂ q · D dy  ∇∂ q D t u t u   k= k= E (u, u ) ∂ A + A u t t  tt    k= E,(.) we use Cauchy-Schwarz’s inequality, (.)and (.) to infer that there is an appropriately large constant α, depending on ρ, μ and ,suchthat   k H H ˜  ∇η + η + ∂ u  E + E . ,  t  k= H H Recalling (.)and thefactthat E ≤ E ,wehave H H L H H H H ˜ ˜ E  E + E E + E ≤ E + E ,   H H which implies E  E for sufficiently small δ. H H H ˜ ˜ On the other hand, E  E obviously. Hence, the energy functional E is equivalent to   H H H H H H ˜ ˜ E .Finally,letting δ ≤ c /c (+ σ ) and denoting E =E /c ,wesee that (.)implies     (.). () We proceed to derive the highest-order estimate (.). We start with employing the recursion formula (.)with i = from k =  to  to find that there are positive constants d (k = ,,), such that     ω d η + (η, u)  u + η + u + S . k t k,–k  ,  , dt k= Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 24 of 31 On the other hand, by virtue of (.),   ω h h h S = N , N + D , div η , u q u           H H  ˜ ˜ ˜ A u + A u + A ∇q + η η  D + E η .          Thus, we conclude  H H H  d η + (η, u)  E + D + E η . k,–k  dt k=   Hence, (.) holds by defining η :=  d η , provided δ is sufficiently ,∗ k= k,–k small. The following lemma will be needed in the next section. Lemma . Under the assumption (.), if δ is sufficiently small, then H   E  E := η + u .(.)   Proof In view of (.), we have H   L H E  η + u, u , u , ∂ u + E E , t tt  t which results in H   E  η + u, u , u , ∂ u (.) t tt  t   for sufficiently small δ. Below we show that the L -norm of u and ∂ u can be controlled by E. () First, to bound u ,wemultiply (.) with j = by u and integrate the resulting t  t equation over  to obtain ¯ ¯ ρu = μ u + λ M ·∇(M ·∇u)+ρω∂ (u e – u e ) · u dy t A  t     t t, – ∇q · D dy t, E + ∇q D , where the last term on the right-hand side can be bounded from above by E.Hence, u  E.(.) () Then we control the term u . Multiplying (.) with j =by u and integrating the tt  tt resulting equation over ,weinfer that  t, t, ¯ ¯ ρu = μ ∂ u + λ M ·∇(M ·∇u)+ρω(u e – u e )+ μN + N tt A t       u q · ∂ u dy t, – ∇q · D dy, u Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 25 of 31 which yields    t, t, t, u  (u, u ) + N , N + ∇q D . tt t t   u q u    Noting that t, t, N , N  E u q and t,  ∇q D  A u + A u  u + E, t  t  t  tt   t u  we see that   u  u + E.(.) tt t      () Finally, we estimate the term ∂ u.Ifwemultiply (.) with j = by ∂ u in L (), t t we obtain   ¯ ¯ ρ∂ u = μ ∂ u + λ M ·∇(M ·∇u ) A  t t t  t, t,  +ρω∂ (u e – u e )+ μN + N · ∂ u dy     t u q t t, – ∇q · D dy, tt whence      t, t, t, ∂ u  u + (u , u ) + N , N + ∇q D . t tt tt  t  u q u     On the other hand, it is easy to show that t, t,  N , N  u + E u q  and t,   ∇q D  u + u + E. tt  t tt u   Therefore,    ∂ u  u + u + E.(.) t tt t   Now, we control the term u in (.). By (.) with j =, we deduce that tt       t, t, u  u + ∇q + ∇ u, N , N , tt t t    u q where the last term on the right-hand side can be bounded by E.Consequently,    u  u + ∇q + E.(.) tt t t    Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 26 of 31 To estimate ∇q ,werewrite (.) as follows: t  q = f , t  with boundary condition ∇q · ν = f · ν on ∂, t  where h h ¯ ¯ f := div μ∂ N + ∂ N – ρu + μu + λ M ·∇(M ·∇u)+ρω∂ (u e – u e ) ,  t t tt t  t     u q h h ¯ ¯ f := μ∂ N + ∂ N + μu + λ M ·∇(M ·∇u)+ρω∂ (u e – u e ).  t t t  t     u q In view of the standard elliptic regularity estimates [], we find that      ∇q  f + f ≤ u + u + E. t   t tt      Putting the above inequality into (.), we obtain    u  u + u + E, tt t tt    which, together with the interpolation inequality (.), yields    u  u + u + E.(.) tt t tt    Consequently, from (.)and (.) it follows that    ∂ u  u + u + E.(.) tt t t   Next, we control u in (.). Using (.) ,wederivethat t      u   u, ∇ η, u, ∇ q  ∇q + E,(.) t A A   where, by virtue of (.) , ∇q satisfies q = f with boundary condition ∇q · ν = f · ν on ∂, and h h ¯ ¯ f := div μN + N – ρu + μu + λ M ·∇(M ·∇η)+ρω(u e – u e ) ,  t      u q h h ¯ ¯ f := μN + N + μu + λ M ·∇(M ·∇η) .   u q Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 27 of 31 Thus, we can use the elliptic regularity estimate to get     ∇q  f + f  u + E.   t     Hence, from (.)weget   u  u + E, t t   which, together with the interpolation inequality (.), implies that   u  u + E.(.) t t   Inserting (.)into(.), we conclude   ∂ u  (u , u ) + E.(.) t tt   () Now we are able to show (.). Summing up the estimates (.), (.), (.)and (.), we arrive at ∂ u  E, k= which, combined with (.), gives (.). 5 Proof of Theorem 2.1 This section is devoted to the proof of Theorem .. Roughly speaking, Theorem . is shown by combining the apriori stability estimate (.) and the local well-posedness of the transformed MHD problem. Before we derive the apriori stability estimate (.), we begin with estimating the terms G ,..., G .     Using (.), and recalling the equivalence of η and η ,wededucethat ,∗    –t –(t–τ) H H η  η e + e E (τ)+ D (τ) dτ   t t  –t H –(t–τ) H η e + sup E (τ) e dτ + D (τ)dτ ≤τ ≤t    –t η e + G (t),   which yields G (t)  η + G (t). (.)    –/ Multiplying (.)by ( + t) ,weget    H H η η d  (η, u) E D ,∗ ,∗  + +  + , / / / / / dt ( + t)  ( + t) ( + t) ( + t) ( + t) Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 28 of 31 which implies that G (t)  η + G (t). (.)    An integration of (.)withrespect to t gives G (t)  E () + E (τ) (η, u)(τ) dτ. Let G (t):= G (t)+ sup E (τ)+ G (t).    ≤τ ≤t From now on, we further assume G (T) ≤ δ, which is a stronger requirement than (.). Thus, we make use of (.)tofind that H –/ G (t)  E () + δ( + τ) (η, u)(τ) dτ H  E () + δ η + G (t) ,   which implies  H G (t)  η + E (). (.)   Finally, we show the time decay behavior of G (t), noting that E can be controlled by L L D ,exceptthe term ∇η in E .Todealwith ∇η ,weuse (.)toget , ,     ∇η  η η . , , , On the other hand, we combine (.)with(.)toget L  L   H E + η  E + η  η + E ().    Thus,           L L L L  L  H     E  E  D E + η  D η + E () .   Putting the above estimate into the lower-order energy inequality (.), we obtain d (E ) E +  , / dt which yields  H I η + E ()   L L  E  E L /   ((I /E ()) + t/) + t  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 29 of 31 H  with I := c(E () + η ) for some positive constant c. Therefore,    H G (t)  η + E (). (.)   Now we sum up the estimates (.)-(.)toconcludethat  H   G(t):= G (t)  η + E ()  η + u , k       k= where (.) has been also used. Consequently, we have proved the following apriori stability estimate. Proposition . Let (η, u) be a solution of the transformed MHD problem with an associ- ated perturbation pressure q. Then there is a sufficiently small δ , such that (η, u, q) enjoys the following apriori stability estimate:   G(T ) ≤ C η + u , (.)       provided that G (T ) ≤ δ for some T >. Here C ≥  denotes a constant depending on      the domain  and other physical parameters in the transformed MHD equations. In view of theapriori stability estimate in Proposition . and the following result of local existence of a small solution to the transformed MHD problem, we immediately obtain Theorem .. Proposition . There is a sufficiently small δ , such that for any given initial data   (η , u ) ∈ H × H satisfying     η + u ≤ δ      and the compatibility conditions (i.e., ∂ u(x,)| =, j =,), there exist a T := T (δ )> ∂    which depends on δ , the domain  and other known physical parameters, and a unique    classical solution (η, u) ∈ C ([, T ], H × H ) to the transformed MHD problem (.), i  –i (.) with an associated perturbation pressure q. Moreover, ∂ u ∈ C ([, T ], H ) for   H    ≤ i ≤ , q ∈ C ([, T ], H ), E ()  η + u , and         H H sup η(τ) + E (τ) + D (τ)+ u(τ) + q(τ) dτ < ∞,    ≤τ ≤T and G(t) is continuous on [, T ]. Proof Thetransformed MHDproblem is very similartothe surfacewaveproblem (.)in []. Moreover, the current problem is indeed simpler than the surface wave problem due to the non-slip boundary condition u| = . Using the standard method in [], one can easily establish Proposition ., hence we omit its proof here. In addition, the continuity,     such as (η, u, q) ∈ C ([, T], H ×H ×H ), G(t) and so on, can be verified by using the reg- ularity of (η, u, q), the transformed MHD equations and a standard regularized method.  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 30 of 31 6Conclusion We have proved the existence of a unique time-decay solution to the initial-boundary problem (.)-(.) of rotating MHD fluids in Lagrangian coordinates, which, together with the inverse transformation of coordinates, implies the existence of a time-decay so- lution to the original initial-boundary problem (.)-(.) with proper initial data in H (). Ourresultalsoholds forthe case ω = (i.e., the absence of rotation), thus it improves Tan and Wang’s result in [], in which the sufficiently small initial data at least belongs to  H (). Hence our result reveals that rotation does not affect the existence of solutions of rotating MHD fluids. We mention that the phenomenon of rotating MHD fluids widely ex- ists in nature, so our result has potential applications. In addition, based on Theorem ., we will further study the Rayleigh-Taylor problem of rotating MHD fluids in the future; please referto[–] for relevant results on the Rayleigh-Taylor problem. Funding The current work is partially supported by NSFC under Grant Nos. 11501116 and 11671086, and the Education Department of Fujian Province under Grant No. JA15063. Competing interests The authors declare that there is no conflict of interest regarding the publication of this manuscript. The authors declare that they have no competing interests. Authors’ contributions All authors have made the same contribution and finalized the current version of this manuscript. They both read and approved the final manuscript. 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Jiang, F, Jiang, S, Wang, Y: On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations. Commun. Partial Differ. Equ. 39(3), 399-438 (2014) 13. Wang, Y: Critical magnetic number in the MHD Rayleigh-Taylor instability. J. Math. Phys. 53, 073701 (2012) 14. Majda, AJ, Bertozzi, AL: Vorticity and Incompressible Flow. Cambridge university press, Cambridge (2002) 15. Guo, Y, Tice, I: Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6(6), 1429-1533 (2013) 16. Adams, RA, John, JFF: Sobolev Space. Academic Press, New York (2005) 17. Jiang, F, Jiang, S: Nonlinear stability and instablity in Rayleight-Taylor problem of stratisfied compressible MHD fluids (2017). arXiv:1702.07529 18. Temam, R: Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001) Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 31 of 31 19. 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Huang, G, Jiang, F, Wang, W: On the nonlinear Rayleigh-Taylor instability of nonhomogeneous incompressible viscoelastic fluids under L -norm. J. Math. Anal. Appl. 455, 873-904 (2017). doi:10.1016/j.jmaa.2017.06.022 26. Jiang, F, Jiang, S, Wu, G: On stabilizing effect of elasticity in the Rayleigh-Taylor problem of stratified viscoelastic fluids. J. Funct. Anal. 272, 3763-3824 (2017) 27. Jiang, F, Jiang, S: On linear instability and stability of the Rayleigh-Taylor problem in magnetohydrodynamics. J. Math. Fluid Mech. 17, 639-668 (2015) 28. Jiang, F: An improved result on Rayleigh-Taylor instability of nonhomogeneous incompressible viscous flows. Commun. Math. Sci. 14, 1269-1281 (2016) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Boundary Value Problems Springer Journals

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

Wang, Weiwei; Zhao, Youyi
Boundary Value Problems , Volume 2017 (1) – Aug 11, 2017
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Springer Journals
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Copyright © 2017 by The Author(s)
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general
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1687-2770
DOI
10.1186/s13661-017-0845-2
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Abstract

College of Mathematics and We have investigated an initial-boundary problem for the perturbation equations of Computer Science, Fuzhou University, Fuzhou, 350108, China 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: ρv + ρv ·∇v + ∇(p + λ |M| /) + ρ(ω  × v)= μv + λ M ·∇M, t   M = M ·∇v – v ·∇M, (.) div v = div M =. Here the unknowns v = v(x, t), M := M(x, t)and p = p(x, t) denote the velocity, the magnetic field, and the pressure of the incompressible MHD fluid respectively; μ >, ρ and λ stand for the coefficients of the shear viscosity, the density constant, and the permeability of vacuum, respectively. ρ(ω  × v) represents the Coriolis force, and ω  = (, , 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 constraint div M = , it can be seen just as a restriction on the initial value of M since (div M) = due to (.) . t  © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 2 of 31 ¯ ¯ Let M := (m , m , m )be a constant vector with m =, 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 ρv + ρv ·∇v + ∇(q ˜ + λ |N + M| /) ⎪ t  ¯ ¯ = μv + λ (N + M) ·∇(N + M)+ρω(v e – v e ),      (.) ¯ ¯ N =(N + M) ·∇v – v ·∇(N + M), ⎪ t div v = div N =, wherewehaveusedthe relation ω  × v = ω(v e – v e ). For system (.), we impose the     initial and the boundary conditions: (v, N)| =(v , N )in ,(.) t=   v(·, t)| = for any t >, (.) where v and N should satisfy the compatibility conditions div v = div N =. 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 with h >,   where T := (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 consider ω =, 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 (.)-(.)withproperinitial data in H (). Our result also holds for the case ω = , thus improves Tan and Wang’s result in [], in which the suffi-  ciently small initial data at least belong to H (). 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 mapping ζ := ζ (y):  → ,suchthat   ∂ = ζ (∂)and det ∇ζ ≡ , (.)   Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 3 of 31 where ζ denotes the third component of ζ .Wedefine theflow map ζ as the solution to ζ (y, t)= v(ζ (y, t), t), (.) ζ (y,) = ζ . We denote the Eulerian coordinates by (x, t)with x = 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 that ζ (·, t) is invertible and  = ζ (, t). In other words, the Eulerian domain of the fluid is the image of  under mapping ζ . In view of the non- slip boundary condition v| =, we have ∂ = ζ (∂, t). In addition, since det ∇ζ =,we ∂  have det(∇ζ)= (.) due to div v =; see [], Proposition .. Now, we further define the Lagrangian unknowns by  + (u, p ˜, B)(y, t)= v, p + λ |M| /, M ζ (y, t), t for (y, t) ∈  × R.(.) Thus in Lagrangian coordinates the evolution equations for u, p ˜ and B read as ζ = u, ρu – μ u + ∇ p ˜ = λ B ·∇ B +ρω(u e – u e ), t A A  A     (.) B – B ·∇ u =, ⎪ t A div u =, with initial and boundary conditions (u, ζ – y)| = and (ζ , u, B)| =(ζ , u , B ). ∂ t=    Moreover, div B =  if the initial data ζ and B satisfy A   div B =. (.) A  T – Here A denotes the initial value of A,the matrix A := (A ) via A =(∇ζ ) :=  ij × – (∂ ζ ) , and the differential operators ∇ , div A and  are defined by ∇ f := j i A A A × T T (A ∂ f , A ∂ f , A ∂ f ) , div (X , X , X ) := A ∂ X ,and  f := div ∇ f for appro- k k k k k k A    lk k l A A A priate f and X. It should be noted that we have used the Einstein convention of summation over repeated indices, and ∂ = ∂ . In addition, in view of the definition of A and (.), we k y ∗ ∗ can see that A =(A ) ,where A is the algebraic complement minor of the (i, j)th entry × ij ij ∂ ζ .Since ∂ A = , we can get an important relation j i k ik div u = ∂ (A u )=, (.) A l kl k which will be used in the derivation of temporal derivative estimates. Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 4 of 31 Our next goal is to eliminate B by expressing it in terms of ζ . 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 of A,one has ∂ ζ A = A ∂ ζ = δ , i k kj ik k j ij where δ = for i = j,and δ = for i = j. Thus, applying A to (.) ,weobtain ij ij jl  A ∂ B = B A ∂ u A = B A ∂ (∂ ζ )A =–B A ∂ ζ ∂ A =–B ∂ A , jl t j i ik k j jl i ik t k j jl i ik k j t jl j t jl which implies that ∂ (A B )= (i.e.,(A B) =). Hence, t jl j t   A B = A B,(.) jl j jl j   which yields B = ∂ ζ A B , i.e., i l i jl j B = ∇ζ A B.(.) Here and in what follows, the notation f also denotes the initial data of the function f .To obtain the asymptotic stability in time, we naturally expect (ζ , B)converges to (y, M)as t →∞.(.) Thus (.) formally implies ¯ ¯ A B = M, i.e., B = M ·∇ζ.(.)    Putting the above expression of B into (.), we get B = M ·∇ζ.(.) Moreover, in view of (.), (.)and (.), the Lorentz force term can be represented by   ¯ ¯ ¯ (.) B ·∇ B = B A ∂ B = A B ∂ (M ·∇ζ)= M ·∇(M ·∇ζ ). A l lk k k lk l Summing up the above analyses, we can see that, under the initial conditions (.)and (.), one can use the relation (.)tochange(.) into the following Navier-Stokes sys- tem: ζ = u, ¯ ¯ ρu – μ u + ∇ p ˜ = λ M ·∇(M ·∇ζ)+ρω(u e – u e ), t A A      div u =, and B is defined by (.). Now, we introduce the shift functions η = ζ – y and q = p ˜ – p ¯.(.) Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 5 of 31 Then the evolution equations for the shift functions (η, q)and u read as η = u, ¯ ¯ ρu – μ u + ∇ q – λ M ·∇(M ·∇ζ)=ρω(u e – u e ), (.) t A A      div u =, – where A =(I + ∇η) , I =(δ ) . The associated initial and boundary conditions read as ij × follows: (η, u)| =(η , u ), (η, u)| =. (.) t=   ∂ It should be noted that the shift function q is the sum of the perturbed fluid and the mag- netic pressures in Lagrangian coordinates. Hence we still call q the 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: + p p ,p R := [, ∞), := , L := L ():= W ()for< p ≤∞,  , k k, H := W (), H := W (), · := · for k ≥ ,   H () α α α α k      ∂ denotes ∂ ∂ for any α + α = k, · := ∂ ∂ · ,   m,k h     k α +α =m   a  b means that a ≤ cb, where, and in what follows, the letter c denotes a generic constant which may depend on the domain  and some physical parameters, such as λ , M, g, μ and ρ in the MHD equations (.). It should be noted that a product space (X) of vector functions is still    denoted by X, for example, a vector function u ∈ (H ) is denoted by u ∈ H with norm  / u := ( u ) . Finally, we define some functionals: k  H k=   L  E := ∇η + (η, u) + (u , ∇q) , ,      L k   D := (η, M ·∇η, ∇u) + (η, u) + ∂ u + ∇q + ∇q , t   ,  –k k=     H   k k E := ∇η + η + ∂ u + ∇∂ q , ,  t t –k –k k= k=       H k k  D := (η, M ·∇η, ∇u) + (η, u) + ∂ u + ∇∂ q + ∇q , t t  ,  –k –k k= k= (η, u)(τ) G (t)= sup η(τ) , G (t)= dτ,    / ( + τ) ≤τ<t H H  L G (t)= sup E (τ)+ D (τ)dτ, G (t)= sup ( + τ) E (τ).   ≤τ<t ≤τ<t  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 6 of 31 Next, we introduce our main result. Theorem . Let  be a horizontally periodic domain with finite height, ω be an arbitrary real number, and m =. Then there is a sufficiently small constant δ >, 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., ∂ u(x,)| = for j =   ∂ 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(∞):= G (∞) ≤ c η + u .(.) k     k= Here the positive constants δ and c depend on the domain  and other known physical parameters λ , M, μ and ρ. Remark . Exploiting the inverse transformation of ζ , we can easily deduce from Theo- rem . the global well-posedness of the original MHD problem (.)-(.). More precisely, there is a sufficiently small constant δ >, such that, for any (v , N ) ∈ H satisfying the    following conditions: () there exists an invertible mapping ζ := ζ (x):  → ,suchthat(.)holds,where   T – A =(∇ζ ) ; ¯ ¯ () (M + N )(ζ )= M ·∇ζ ;      () ζ – x + v ≤ δ ;      () the initial data  , v , N satisfy necessary compatibility conditions (i.e.,    ∂ v(x,)| = for j = and ),  +  there exists a unique global solution (v, N) ∈ C (R , H ) to the original MHD problem (.)-(.) with an associated perturbation pressure q ˜.Moreover, (v, N, q ˜)enjoys the fol- lowing stability estimate:      k k sup N + ∂ v(t) + ∇∂ q ˜(t)  t t –k –k ≤t<∞ k= k=      + sup ( + t) (v, N) + (v , ∇q ˜) ≤ c ζ – x + v .(.) t       ≤t<∞ Now we briefly describe the basic idea in the proof of Theorem ..Bythe standard energy method, there are two functionals E and Q of (η, u) satisfying the lower-order energy inequality (see Proposition .) L L L E + D ≤ QD,(.) dt L L where the functional E is equivalent to E . Unfortunately, we can not close the energy estimates only based on (.), since Q can not be controlled by E .However,weobserve Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 7 of 31 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 (.). Since E contains η , we find that the higher-order energy at least includes η .Thus, similarto(.),   we establish the higher-order energy inequality (see Proposition .) H H ˜ L E + D ≤ E (η, u),(.) dt H H  where the functional E is equivalent to E .Moreover, thehighest-order norm (η, u) enjoys the highest-order energy inequality  H H η + (η, u)  E + D , (.) ,∗ dt where the norm η is equivalent to η . In the derivation of the apriori estimates, we ,∗  have Q  E ,and thus (.) implies (see Proposition .) L L E + D ≤ . (.) dt 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 ..InSection ,we first derive the lower-order energy inequality (.)for thetransformed MHDproblem. Then in Section  we derive the higher-order energy inequality (.)and thehighest- 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 theapriori estimates for the transformed MHD problem. To this end, let (η, u) be a solution of the transformed MHD problem with perturbed pressure q,suchthat G (T)+ sup E (τ) ≤ δ ∈ (, ) for some T >, (.) ≤τ ≤T where δ is 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 calculations, we shall repeatedly use Cauchy-Schwarz’s inequality, Hölder’s inequality, the embedding inequalities (see [], . Theorem) f p  f for  ≤ p ≤and f ∞  f ,(.) L  L  and the interpolation inequalities (see [], . Theorem) j j – i i f  f f ≤ C f +  f i (.) j   H  i Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 8 of 31 for any  ≤ j < i and any constant  > , where the constant C depends on the domain and . In addition, we shall also frequently use the following two estimates: fg  f g for j ≥(.) j j κ(j) and f  ∂ f for f ∈ H,(.)    where κ(j)= j for j ≥ and κ(j)= for j ≤ . 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 for A. To begin with, we give an expression of A.Using (.) ,wehave ∗ ∗ ∂ det(I + ∇η)= ∂ ∂ η A = A ∂ u,(.) t t j i j i ij ij ≤i,j≤ ≤i,j≤ where A is the algebraic complement minor of the (i, j)th entry in the matrix I + ∇η. ij Recalling the definition of A,wesee that A = A / det(I + ∇η). ij × Inserting this relation into (.), we get ∂ det(I + ∇η)= det(I + ∇η) A ∂ u =, t ij j i ≤i,j,≤ which, together with initial condition det(I + ∇η ) = , implies det(I + ∇η)=. Thus we obtain A = A.(.) ij × Now, exploiting (.) ,(.), (.)and (.), we easily see that A  + η + η for ≤ j ≤ , (.) j j+ j+ ∇A ≤ η + η  η for  ≤ j ≤ . j j+ j+ j+ Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 9 of 31 Similarly, we further deduce that i– i k ∂ A  ∂ ∇u for any  ≤ i ≤ and  ≤ j ≤ –i. t t j j k= ˜ ˜ Letting A := A – I,wenextbound A. To this end, we assume that δ is so small that the following expansion holds: T  i  T A = I – ∇η +(∇η) (–∇η) = I – ∇η +(∇η) A , i= whence T  T A =(∇η) A – ∇η. Using (.), (.)and (.), we find that A  ∇η for  ≤ j ≤ . j j 3.2 Temporal derivative estimates In this subsection, we try to control temporal derivatives. For this purpose, we apply ∂ to (.)toget j+ j ∂ η = ∂ u, t t j+ j j ρ∂ u – μ ∂ u + ∇ ∂ q A A t t t (.) j j t,j t,j ⎪ ¯ ¯ = λ ∂ M ·∇(M ·∇ζ)+ρω∂ (u e – u e )+ μN + N , ⎪      u q t t j t,j div ∂ u = div D , A u where j–m–n t,j n m N := ∂ A ∂ ∂ A ∂ ∂ u , il l ik k u t t t ≤m<j,≤n≤j j–l t,j l N := – ∂ A ∂ ∂ q , ik k q t × ≤l<j j–l j–l t,j l D := – C ∂ A ∂ u,(.) ki k u j t t × ≤l<j j–l C denotes the number of (j – l)-combinations from a given set S of j elements, andwehaveusedrelation(.)in(.). Then from (.) we show the following estimates: Lemma . It holds that for j = and , d √ j  j  j  ρ∂ u + λ M ·∇∂ η + μ ∇ ∂ u  E D,(.)  A t t t    dt d  √     L ∇ u + ρu  u + u + E D . (.) A t tt t     dt μ Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 10 of 31 Proof () We only prove (.)for j = , since the derivation of the case j = is similar. Multiplying (.) with j = by u , integrating (by parts) the resulting equality over ,  t and using (.) ,weget  d √    ρu + λ M · η + μ ∇ u t  t A t     dt =ρω ∂ (u e – u e ) · u dy + q div u dy t     t t A t t, t, + μ N · u dy + N · u dy t t u q := I.(.) k= L L The last three integrals I ,..., I can be estimated as follows:   L t, I := – ∇q · D dy  ∇q A  u   ∇q A u t t  t t  t    u L L E D , (.) L L I  A A u u  E D,(.)  t   t  L L I  A ∇q u  A ∇q u  E D,(.) t  L t  t   t  wherewehaveused(.) in (.). Consequently, the desired estimate (.) follows from (.)-(.). () Now we turn to the proof of (.). Multiplying (.) with j = by u , integrating  tt (by parts) the resulting equality over ,and using(.) ,weconclude μ d √   ∇ u + ρu A t tt    dt ¯ ¯ = ρω∂ (u e – u e ) · u + λ M ·∇(M ·∇u) · u dy t     tt  tt t, t, + q div u dy + μ N · u dy + N · u dy t A tt tt tt u q + μ ∇ u : ∇ u dy A t A t := J.(.) k= H H On the other hand, the five integrals J ,..., J can be bounded as follows:   J  u + u u ,(.) t   tt  H t, J =– ∇q · D dx  ∇q A u + A u t t  tt   t  t   u E D,(.) H L J  A A u u  E D,(.)  t   tt   Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 11 of 31 H L J  A ∇q u  E D,(.) t   tt  H  L J  A A u  E D.(.)  t  t   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: η = u, h h ¯ ¯ ρu – μu + ∇q – λ M ·∇(M ·∇η)=ρω(u e – u e )+ μN + N , (.) t      u q div u = D , where ˜ ˜ ˜ ˜ N := ∂ (A A + A + A )∂ u , l jl jk lk kl k i × h h ˜ ˜ N := –(A ∂ q) and D := A ∂ u . ik k × lk k l q u Then we have the following estimate on horizontal spatial derivatives of η. Lemma . It holds that d μ j j j  j  ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt  L  E D + u , ≤ j ≤ . j, Proof We only show the case j = ; the remaining three cases can be verified similarly.   Applying ∂ to (.) , multiplying the resulting equation by ∂ η,and then using(.) ,   h h we get      ¯ ¯ ρ∂ ∂ η · ∂ u – μ∂ η + λ M ·∇ M ·∇∂ η · ∂ η t t  h h h h h   h  h    = ρω∂ (u e – u e )+ μ∂ N + ∂ N – ∇∂ q · ∂ η + ρ ∂ u .     h h u h q h h h If we integrate (by parts) the above identity over ,weobtain d μ       ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt     h  h  =ρω ∂ (u e – u e ) · ∂ η dy + μ∂ N + ∂ N · ∂ η dy     h h h u h q h    + ∂ q div ∂ η dy + ρ∂ u h h h L  K + u ,(.) k , k= where the first three integrals on the right-hand side of the first equality in (.)are de- L L L noted by K , K and K ,respectively.    Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 12 of 31 We have the boundedness L   K ≤ ∂ u ∂ η.(.)  h h   Noting that  h  h ˜ ˜ ˜ ˜ μ∂ N + ∂ N  A u + A u + A ∇q + A ∇q         h u h q H L E D , we find that L  h  h  L K  μ∂ N + ∂ N ∂ η  E D.(.)  h u h q h   Next we estimate the third integral K .Tostart with,weanalyze thepropertyof div η. Since det(I + ∇η)=, we have by Sarrus’ rule div 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 ∂ η ).           Multiplying the above identity by a smooth test function φ, and then integrating (by parts) the resulting identity over ,wederivethat φ div η dy =– ∇φ · ψ dy, where ⎛ ⎞ η (∂ η + ∂ η )+ η (∂ η ∂ η – ∂ η ∂ η )               ⎜ ⎟ ψ := η ∂ η – η ∂ η + η (∂ η ∂ η – ∂ η ∂ η ) . ⎝ ⎠                –η ∂ η – η ∂ η + η (∂ η ∂ η – ∂ η ∂ η )                This means that div η = div ψ. Thus, it follows immediately that L    K =– ∂ ∇q · ∂ ψ dy =– ∂ ∇q · ∂ ψ dy  h h h  L ∇q ∂ ψ  η ∇q η  E D.(.)      Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 13 of 31 Now, substituting (.), (.)and (.)into(.), we immediately obtain the desired estimate for the case j =. Similarly, we also establish the following estimates of horizontal spatial derivatives of u: Lemma . We have d √    j j j ρ∂ u + λ M ·∇∂ η + μ ∇∂ u  E D , j = ,,. h h h    dt Proof We only prove the case j = ; the remaining two cases can be shown similarly. Ap-   plying ∂ to (.) , and multiplying the resulting equality by ∂ u, we make use of (.)   h h to get       ¯ ¯ ρ∂ u · ∂ u – μ∂ u · ∂ u – λ M ·∇ M ·∇∂ η · ∂ η t  t h h h h h h   h  h   = ρω∂ (u e – u e )+ μ∂ N + ∂ N – ∇∂ q · ∂ u.     h h u h q h h Integrating (by parts) the above identity over ,wehave  d μ       ρ ∂ u dy + λ M ·∇∂ η + ∇∂ u h h h    dt   h  h    L L = μ∂ N + ∂ N · ∂ u dy + ∂ q div ∂ u dy =: M + M.(.) h u h q h h h   L L On the other hand, similarly to (.)and (.), the two integrals M and M can be   estimated as follows: L  h  h  L M  μ∂ N + ∂ N ∂ u  E D,(.)  h u h q h   L   h L ˜ H M =– ∂ q∂ D dy  ∇q A u  E D,(.)     h h u wherewehaveused(.) in (.). Consequently, putting the above two estimates into (.), we obtain Lemma . for the case j =. 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  h h ¯ ¯ = λ M ·∇(M ·∇η)– λ m η +ρω(u e – u e )– ρu + μN + N , (.)       t u q div w = μD + λ m div η, u  coupled with boundary condition ω| =, (.) where ω = λ m η + μu.  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 14 of 31 Now, applying ∂ to (.)and (.), we get k k –∂ w + ∇∂ q h h k  k k h h ¯ ¯ = ∂ (λ M ·∇(M ·∇η)– λ m η +ρω(u e – u e )) – ρ∂ u + μ∂ N + N ,       t h  h h u q k k h  k div ∂ w = μ∂ D + λ m div ∂ η, ⎪  u  h h h ∂ ω| =. Then we apply the classical regularity theory to the Stokes problem as in [], Proposition ., to deduce that    ω ω + ∇q  ∇η + (u, u ) + S,(.) k,i–k+ k,i–k k+,i–k k,i k,i–k where   ω h h h S := N , N + D , div η . k,i u q u k,i–k k,i–k+ In addition, applying ∂ to (.) -(.) ,wesee that   k k –μ∂ u + ∇∂ q t t k k k+ k h k h ¯ ¯ = λ M · ∂ ∇(M ·∇η)+ρω∂ (u e – u e )– ρ∂ u + μ∂ N + ∂ N ,      t t t t u t q k k h div ∂ u = ∂ D , t t u ∂ u| =. Hence, we apply again the classical regularity theory to the Stokes problem to get    k k k  u ∂ u + ∇∂ q  ∂ ∇ η, u, u + S,(.) t t t k,i i–k+ i–k i–k u k h h  k h  where S := ∂ (N , N ) + ∂ D .Asaresult of (.)and (.), one has the k,i t u q i–k t u i–k+ following estimates. Lemma . We have      H L η + c (η, u) + ∇q  (u, u ) + η + E D,(.) ,∗  ,   dt    H L u + ∇q  η + (u, u ) + E E , (.)       H L u + ∇q  u + (u , u ) + E D,(.) t t t tt    L L  where E := E – ∇η and η is equivalent to η . ,∗  , Proof Noting that, by virtue of (.) , λ m μ d     ω = λ m η, μu + η , k,i–k+  k,i–k+ k,i–k+  dt we deduce from (.)that   η + c (η, u) + ∇q k,i–k+ k,i–k k,i–k+ dt    ω u + η + u + S.(.) k,i–k k+,i–k+ i k,i Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 15 of 31 In particular, we take (i, k)=(,) and (i, k)=(,) to get      ω η + c (η, u) + ∇q  (u, u ) + η + S (.)   , ,   dt and      ω η + c (η, u) + ∇q  (u, u ) + η + S.(.) , , , , ,  dt On the other hand, it is easy to show that   ω ω h h h     S + S ≤ N , N + D , div η  A u + ∇q + η , , u q u       H L H L E E  E D.(.) Thus we immediately obtain (.)from(.)-(.). Now we turn to the derivation of (.). In view of (.)with (i, k)=(,), we have    u u + ∇q  η + (u, u ) + S .    , On theother hand,wecan use(.)toinfer that   u h h h H L S = N , N + D  E E . , u q u   Hence, (.) follows from the above two estimates. Finally, to show (.), we take (i, k)=(,) in (.)todeducethat      u  u u + ∇q  ∇ u, u , u + S  u + (u , u ) + S . t t t tt t tt   ,  ,   Keeping in mind that   u h h h S  ∂ N , N + ∂ D t t , u q u         H L ˜ ˜ A u + ∇q + A u + ∇q  E D , t t t       we get (.)fromthe abovetwo estimates. 3.5 Lower-order energy inequality Now, we are able to build the lower-order energy inequality. In what follows, the letters c and i = ,..., will denote generic positive constants which may depend on the domain and some physical parameters in the transformed MHD equations (.). Proposition . Under the assumption (.), if δ is sufficiently small, then there is an en- L L ergy functional E which is equivalent to E , such that L L E + D ≤  on (, T]. (.) dt Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 16 of 31 Proof We choose δ so small that   j j ∇∂ u  ∇ ∂ u , ≤ j ≤ . (.) t A t   Then, thanks to (.), (.)and (.), we deduce from (.) and Lemmas .-. that L L there are constants c , c and σ ≥ , such that   L L L L ˜ ˜ H E + D ≤ c ( + σ ) E D for any σ ≥ σ,(.)    dt where σ depends on the domain and the known physical parameters, and √ √ L     ¯ ¯ E := ρu + λ M ·∇u + σ ρu + λ M ·∇η t      k, k, k= α α α α      + ρ∂ ∂ η · ∂ ∂ u dy + ∇η ,     k, k= α +α =k   L L   ˜  ¯ D := c u + (η, M ·∇η) + u .    k, k, k= Utilizing (.), (.), the interpolation inequality, and the estimate    η  η + M ·∇η for any  ≤ k ≤ , (.) k, k+, k, we find that  L  L   η + c ∇ u + c (η, u) + ∇q + u A t tt ,∗      dt L   L ≤ c (u, u ) + η + M ·∇η + E D.(.)  , , L L Now, multiplying (.)by c /(c ) and adding the resulting inequality to (.), we obtain   L L L L ˜ ˜ (.) E + D ≤ c ( + σ ) E D ,   dt ˜ ˜ where E and D are defined by L L L  L  L ˜ ˜ E := E + c η + c ∇ u / c , A t    ,∗    L L L L   L ˜ ˜ D := D / + c c (η, u) + ∇q + u / c . tt       L L L L L ˜ H ˜ On the other hand, from (.)weget D  D + E D , which implies D  D for sufficiently small δ. Therefore, (.) can be rewritten as follows: L L L L L (.) E + c D ≤ c ( + σ ) E D .    dt L L Next we show that E can be controlled by E . By virtue of Cauchy-Schwarz’s inequality, (.)and (.), there is an appropriately large constant σ , depending on ρ, μ and ,such Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 17 of 31 that    L ∇η + η + u  E . ,    In view of (.), we further have L L L E  E + E E , L L L L ˜ ˜ which implies E  E for sufficiently small δ. In addition, obviously E  E .Hence,the   L L L L L ˜ ˜ energy functional E is equivalent to E .Finally,letting δ ≤ c /c ( + σ ) and noting E =    L L L ˜ ˜ E /c ,wesee that (.) immediately implies (.). Obviously, the energy functional E   is still equivalent to E . This completes the proof. 4 Higher-order and highest-order energy inequalities In this section, we derive the estimates (.)and (.)for thetransformed magnetic RT problem. First we shall establish the higher-order version of Lemmas .-. in the following: Lemma . We have d √    j j– j– j t,j ρ∂ u + λ M ·∇∂ u + ∇∂ q · D dy + μ ∇ ∂ u  A t t t t    dt E D , j =,. Proof We only prove the case j =, and the case j =  can be shown similarly. Multiplying (.) with j = by ∂ u, integrating (by parts) the resulting equation over ,and using (.) and (.) ,weobtain    d √      ρ∂ u + λ M ·∇u + μ ∇ ∂ u  tt A t  t    dt  t, t,  t,  H = ∂ q div D dy + μ N · ∂ u dy + N · ∂ u dy := I.(.) t u u t q t k k= H H On the other hand, the integrals I , I and I can be estimated as follows:   H  t,  t,  t, I =– ∇∂ q · D dy =– ∇∂ q · D dy + ∇∂ q · ∂ D dy  t u t u t u dt  t, ≤ – ∇∂ q · D dy t u dt     + c ∇∂ q ∂ A u + ∂ A u + A u + A ∂ u  t  tt  tt  t  t t t t      t, H ≤ – ∇∂ q · D dy + c E D , t u dt H  I  ∂ A A u + A A u + A u   tt  t    t   t + A A u + A u + A u t  tt   t  t   tt    + A ∂ A u + A u + A u + A u ∂ u   tt  t  tt  t  t  tt  t t   E D , Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 18 of 31 H   H I  ∂ A ∇q + A ∇q + A ∇q ∂ u  E D .  tt  t  t  tt   t t   Inserting the above three inequalities into (.), we get the desired conclusion immedi- ately. Lemma . We have d μ   k k k k ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt    H k E (η, u) + D + ∂ u for any k = and .   Proof We only show the case k = , since the derivation of the case k = is similar. Since the derivation involves the norm ∇q ,wefirstestimate ∇q . It follows from (.)   that h h h u + ∇q  η + (u, u ) + N , N + D .    t u q u    The last two terms on the right-hand side of the above inequality can be bounded as fol- lows: h h h ˜ ˜ N , N + D  A u + A ∇q .     u q u   Hence, if δ is sufficiently small, then one gets from the above two estimates that u + ∇q  η + D.(.)      Now, applying ∂ to (.) , multiplying the resulting equality by ∂ η,weutilize (.)   h h to have      ¯ ¯ ρ∂ ∂ η · ∂ u – μ∂ η + λ M ·∇ M ·∇∂ η · ∂ η t t  h h h h h  h h   = ∂ ρω(u e – u e )+ μN + N – ∇q · ∂ η + ρ ∂ u .     h u q h h Integrating (by parts) the above identity over ,weobtain d μ       ρ∂ η · ∂ u dy + ∇∂ η + λ M ·∇∂ η h h h h   dt     h   h  =ρω ∂ (u e – u e ) · ∂ η dy + μ ∂ N · ∂ η dy – ∂ N · ∂ η dy     h h h u h h q h    – ∂ q div ∂ η dy + ρ∂ u h h h H  J + ∂ u,(.) k h k= H H where the first four integrals on the right-hand side are denoted by J ,..., J ,respectively.   H H On the other hand, the four integrals J ,..., J can be bounded as follows:   H   J  ∂ η ∂ u , (.)  h h   Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 19 of 31 H   ˜ ˜ ˜ ˜ J =–μ ∂ (A A + A + A )∂ u ∂ ∂ η dy jl jk lk kl k i l i  h h ˜ ˜ ˜ A u + A u + A u η       ,         ˜ ˜ ˜ ˜ A u + A A u u + A u η     ,     E (η, u),(.) ˜ ˜ ˜ J  A ∇q + A ∇q + A ∇q η                ˜ ˜ ˜ ˜ A ∇q + A A ∇q ∇q + A ∇q η           H E η + D , (.) H    J = ∂ q · ∂ div η dy  ∇q η η + η     h h   H η ∇q η + η  E η + D,(.)     where the interpolation inequality (.) has been employed in (.)and (.). Conse- quently, putting the above four estimates into (.), we obtain Lemma .. Lemma . We have d √    k k k ρ∂ u + λ M ·∇∂ η + μ ∇∂ u h h h    dt E (η, u) + D for k = and . Proof We only show the case k = , since the derivation of the case k = is similar. Ap-   plying ∂ to (.) , multiplying the resulting equality by ∂ u, we make use of (.) to   h h have         ρ∂ u · ∂ u – μ∂ u · ∂ u – λ m ∂ ∂ η · ∂ η t  t h h h h  h h   h  h   = ρω∂ (u e – u e )+ μ∂ N + ∂ N – ∇∂ q · ∂ u.     h h u h q h h Integrating (by parts) the above identity over ,weget  d μ       ρ ∂ u dy + λ M ·∇∂ η + ∇∂ u h h h    dt   h   h    H = μ ∂ N · ∂ u dy – ∂ N · ∂ u dy + ∂ q div ∂ u dy =: M.(.) h u h h q h h h k k= H H H Analogously to (.)-(.), the three integrals M , M and M can be bounded as fol-    lows: H   ˜ ˜ ˜ ˜ M =– ∂ (A A + A + A )∂ u · ∂ ∂ u dy  E (η, u) , jl jk lk kl k i l  h h H H M  E (η, u) + D , H   h ˜ ˜ ˜ M = ∂ q · ∂ D dy  ∇q A u + A u + A u         h h u     √      H ˜ ˜ ˜ ∇q η u + A A u u + A u  E η + D .           Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 20 of 31 Substituting the above three inequalities into (.), we obtain Lemma .. Lemma . The following estimates hold:    k k   L H ∂ u + ∇∂ q  η + u, u , u , ∂ u + E E,(.) t tt t t  t –k –k  k=     k k η + η + ∂ u + ∇∂ q ,∗  t t –k –k dt k=   L H η + η, u, u , u , ∂ u + E D,(.) t tt , t    k k   H H ∂ u + ∇∂ q  u + u , u , ∂ u + E D,(.) t tt t t  t –k –k  k=   H H  where the norm η is equivalent to η and E := E – ∇η . ,∗  , Proof () We begin with the derivation of (.). Taking (i, k) = (, ) in the Stokes estimate (.), we have    u u + ∇q  η + (u, u ) + S .    , Noting that   u h h h S = N , N + D , u q u           L H ˜ ˜ ˜ ˜ A u + A u + A ∇q + A ∇q  E E ,         we get    L H u + ∇q  η + (u, u ) + E E.(.)    Using the recursion formula (.)for i = from k = to , we obtain       k k  k– u ∂ u + ∇∂ q  ∂ (u, u ) + ∂ u + S . t t t t k, –k –k  –k k= k= On the other hand, S can be bounded from above by k,      u k h k h k h S = ∂ N + ∂ D + ∂ N k, t u t u t q –k –k –k k= k=       ˜ ˜ A u + ∇q + A u + ∇q t t tt tt               ˜ ˜ ˜ + A u + A u + ∇q + A u + ∇q t t t t t              ˜ ˜ + A u + A u + ∇q t tt      L H E E . Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 21 of 31 Therefore,       k k  k– L H ∂ u + ∇∂ q  ∂ (u, u ) + ∂ u + E E , (.) t t t t –k –k  –k k= k= which, together with (.), yields       k k   k– L H ∂ u + ∇∂ q  η + ∂ (u, u ) + ∂ u + E E .(.) t t  t t –k –k  –k k= k= In view of the interpolation inequality (.), we have u ≤ C u +  u and u  C u +  u . (.)     t   t  t  Thus, inserting (.)into(.), one gets (.). () We proceed to prove the estimate (.). Exploiting the recursion formula (.)for i = from k =  to , we see that there are positive constants c , k = ,...,, such that       ω c η + (η, u) + ∇q  u + η + S , k t k,–k   , k, dt k= k= which combined with (.)gives       k k c η + c η + ∂ u + ∇∂ q k,–k  t t –k –k dt k= k=     k– L H ω η + ∂ (u, u ) + ∂ u + E E + S , , t t ,  –k k= ω ω wherewehaveusedthe fact that S ≤ S for  ≤ k ≤ . Since k, ,   ω h h h S = N , N + D , div η , u q u             ˜ ˜ ˜ ˜ A u + A u + η η + A ∇q + A ∇q           L H E E H H and E ≤ D ,wefurther inferthat       k k c η + c η + ∂ u + ∇∂ q k,–k  t t –k –k dt k= k=     k– L H η + ∂ (u, u ) + ∂ u + E D.(.) , t t  –k k= Using the interpolation inequality (.), we get (.)from(.), where η equals to ,∗ c η multiplied by some positive constant. k= k,–k Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 22 of 31 () Finally, we derive the estimate (.) for higher-order dissipation estimates. We use the recursion formula (.)for i = from k = to  to deduce that       k k  k– u ∂ u + ∇∂ q  ∂ (u, u ) + ∂ u + S .(.) t t t t k, –k –k  –k k= k= Noting that     u k h h k h S = ∂ N , N + ∂ D k, t u q t u –k –k k= k=         ˜ ˜ ˜ ˜ A u + A u + A ∇q + A ∇q t t t t               ˜ ˜ ˜ + A u + A u + A u tt t t tt             ˜ ˜ ˜ + A ∇q + A ∇q + A ∇q tt t t tt       H H E D , we obtain (.)from (.). Now we are in a position to build the higher-order and highest-order energy inequal- ities. In what follows, the letters c and i = ,..., will denote generic constants which may depend on the domain  and some physical parameters in the transformed MHD equations (.). Proposition . Under the assumption (.), if δ is sufficiently small, then there are two H  H  norms E and η which are equivalent to E and η respectively, such that ,∗  H H E + D  E (η, u),(.) dt  H H η + (η, u)  E + D on (, T]. (.) ,∗ dt Proof () We first prove (.). Similarly to (.), we make use of (.), (.), (.)and H H H (.) to deduce from Lemmas .-. and (.) that there are constants c , c , c and    α ≥ , such that   H H H H ˜ ˜ E + D ≤ c ( + α) E (η, u) + D + (η, u, u ) for any α ≥ α ,(.) t       dt where α depends on the domain and the known physical parameters, and   H k k– k– t,k ˜    ¯ E := ρ∂ u + λ M ·∇∂ u + ∇∂ q · D dy  t t t u   k= α α α α      + ρ∂ ∂ η · ∂ ∂ u dy + ∇η     k, ≤α +α ≤ k=     H  + α ρu + λ M ·∇η + c η , , ,  ,∗ Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 23 of 31     H H k  D := c ∂ u + (η, M ·∇η, ∇u) + η   t   k, k= k=   k k + ∂ u + ∇∂ q . t t –k –k k= Moreover, by (.) and Proposition . we find that H H H H ˜ ˜ L E + D ≤ c ( + σ ) E (η, u) + D,(.)   dt H H H L H H H L H H H H ˜ ˜ ˜ ˜ ˜ ˜ where E := E + c E and D := D + c D .By (.), we have D  D + E D .So,      (.)can be rewrittenas H H H H ˜ L E + c D ≤ c ( + σ ) E (η, u) for sufficiently small δ.(.)    dt H H Next, we show that E can be controlled by E . Keeping in mind that   k– t,k k– t,k ∇∂ q · D dy  ∇∂ q D t u t u   k= k= E (u, u ) ∂ A + A u t t  tt    k= E,(.) we use Cauchy-Schwarz’s inequality, (.)and (.) to infer that there is an appropriately large constant α, depending on ρ, μ and ,suchthat   k H H ˜  ∇η + η + ∂ u  E + E . ,  t  k= H H Recalling (.)and thefactthat E ≤ E ,wehave H H L H H H H ˜ ˜ E  E + E E + E ≤ E + E ,   H H which implies E  E for sufficiently small δ. H H H ˜ ˜ On the other hand, E  E obviously. Hence, the energy functional E is equivalent to   H H H H H H ˜ ˜ E .Finally,letting δ ≤ c /c (+ σ ) and denoting E =E /c ,wesee that (.)implies     (.). () We proceed to derive the highest-order estimate (.). We start with employing the recursion formula (.)with i = from k =  to  to find that there are positive constants d (k = ,,), such that     ω d η + (η, u)  u + η + u + S . k t k,–k  ,  , dt k= Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 24 of 31 On the other hand, by virtue of (.),   ω h h h S = N , N + D , div η , u q u           H H  ˜ ˜ ˜ A u + A u + A ∇q + η η  D + E η .          Thus, we conclude  H H H  d η + (η, u)  E + D + E η . k,–k  dt k=   Hence, (.) holds by defining η :=  d η , provided δ is sufficiently ,∗ k= k,–k small. The following lemma will be needed in the next section. Lemma . Under the assumption (.), if δ is sufficiently small, then H   E  E := η + u .(.)   Proof In view of (.), we have H   L H E  η + u, u , u , ∂ u + E E , t tt  t which results in H   E  η + u, u , u , ∂ u (.) t tt  t   for sufficiently small δ. Below we show that the L -norm of u and ∂ u can be controlled by E. () First, to bound u ,wemultiply (.) with j = by u and integrate the resulting t  t equation over  to obtain ¯ ¯ ρu = μ u + λ M ·∇(M ·∇u)+ρω∂ (u e – u e ) · u dy t A  t     t t, – ∇q · D dy t, E + ∇q D , where the last term on the right-hand side can be bounded from above by E.Hence, u  E.(.) () Then we control the term u . Multiplying (.) with j =by u and integrating the tt  tt resulting equation over ,weinfer that  t, t, ¯ ¯ ρu = μ ∂ u + λ M ·∇(M ·∇u)+ρω(u e – u e )+ μN + N tt A t       u q · ∂ u dy t, – ∇q · D dy, u Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 25 of 31 which yields    t, t, t, u  (u, u ) + N , N + ∇q D . tt t t   u q u    Noting that t, t, N , N  E u q and t,  ∇q D  A u + A u  u + E, t  t  t  tt   t u  we see that   u  u + E.(.) tt t      () Finally, we estimate the term ∂ u.Ifwemultiply (.) with j = by ∂ u in L (), t t we obtain   ¯ ¯ ρ∂ u = μ ∂ u + λ M ·∇(M ·∇u ) A  t t t  t, t,  +ρω∂ (u e – u e )+ μN + N · ∂ u dy     t u q t t, – ∇q · D dy, tt whence      t, t, t, ∂ u  u + (u , u ) + N , N + ∇q D . t tt tt  t  u q u     On the other hand, it is easy to show that t, t,  N , N  u + E u q  and t,   ∇q D  u + u + E. tt  t tt u   Therefore,    ∂ u  u + u + E.(.) t tt t   Now, we control the term u in (.). By (.) with j =, we deduce that tt       t, t, u  u + ∇q + ∇ u, N , N , tt t t    u q where the last term on the right-hand side can be bounded by E.Consequently,    u  u + ∇q + E.(.) tt t t    Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 26 of 31 To estimate ∇q ,werewrite (.) as follows: t  q = f , t  with boundary condition ∇q · ν = f · ν on ∂, t  where h h ¯ ¯ f := div μ∂ N + ∂ N – ρu + μu + λ M ·∇(M ·∇u)+ρω∂ (u e – u e ) ,  t t tt t  t     u q h h ¯ ¯ f := μ∂ N + ∂ N + μu + λ M ·∇(M ·∇u)+ρω∂ (u e – u e ).  t t t  t     u q In view of the standard elliptic regularity estimates [], we find that      ∇q  f + f ≤ u + u + E. t   t tt      Putting the above inequality into (.), we obtain    u  u + u + E, tt t tt    which, together with the interpolation inequality (.), yields    u  u + u + E.(.) tt t tt    Consequently, from (.)and (.) it follows that    ∂ u  u + u + E.(.) tt t t   Next, we control u in (.). Using (.) ,wederivethat t      u   u, ∇ η, u, ∇ q  ∇q + E,(.) t A A   where, by virtue of (.) , ∇q satisfies q = f with boundary condition ∇q · ν = f · ν on ∂, and h h ¯ ¯ f := div μN + N – ρu + μu + λ M ·∇(M ·∇η)+ρω(u e – u e ) ,  t      u q h h ¯ ¯ f := μN + N + μu + λ M ·∇(M ·∇η) .   u q Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 27 of 31 Thus, we can use the elliptic regularity estimate to get     ∇q  f + f  u + E.   t     Hence, from (.)weget   u  u + E, t t   which, together with the interpolation inequality (.), implies that   u  u + E.(.) t t   Inserting (.)into(.), we conclude   ∂ u  (u , u ) + E.(.) t tt   () Now we are able to show (.). Summing up the estimates (.), (.), (.)and (.), we arrive at ∂ u  E, k= which, combined with (.), gives (.). 5 Proof of Theorem 2.1 This section is devoted to the proof of Theorem .. Roughly speaking, Theorem . is shown by combining the apriori stability estimate (.) and the local well-posedness of the transformed MHD problem. Before we derive the apriori stability estimate (.), we begin with estimating the terms G ,..., G .     Using (.), and recalling the equivalence of η and η ,wededucethat ,∗    –t –(t–τ) H H η  η e + e E (τ)+ D (τ) dτ   t t  –t H –(t–τ) H η e + sup E (τ) e dτ + D (τ)dτ ≤τ ≤t    –t η e + G (t),   which yields G (t)  η + G (t). (.)    –/ Multiplying (.)by ( + t) ,weget    H H η η d  (η, u) E D ,∗ ,∗  + +  + , / / / / / dt ( + t)  ( + t) ( + t) ( + t) ( + t) Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 28 of 31 which implies that G (t)  η + G (t). (.)    An integration of (.)withrespect to t gives G (t)  E () + E (τ) (η, u)(τ) dτ. Let G (t):= G (t)+ sup E (τ)+ G (t).    ≤τ ≤t From now on, we further assume G (T) ≤ δ, which is a stronger requirement than (.). Thus, we make use of (.)tofind that H –/ G (t)  E () + δ( + τ) (η, u)(τ) dτ H  E () + δ η + G (t) ,   which implies  H G (t)  η + E (). (.)   Finally, we show the time decay behavior of G (t), noting that E can be controlled by L L D ,exceptthe term ∇η in E .Todealwith ∇η ,weuse (.)toget , ,     ∇η  η η . , , , On the other hand, we combine (.)with(.)toget L  L   H E + η  E + η  η + E ().    Thus,           L L L L  L  H     E  E  D E + η  D η + E () .   Putting the above estimate into the lower-order energy inequality (.), we obtain d (E ) E +  , / dt which yields  H I η + E ()   L L  E  E L /   ((I /E ()) + t/) + t  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 29 of 31 H  with I := c(E () + η ) for some positive constant c. Therefore,    H G (t)  η + E (). (.)   Now we sum up the estimates (.)-(.)toconcludethat  H   G(t):= G (t)  η + E ()  η + u , k       k= where (.) has been also used. Consequently, we have proved the following apriori stability estimate. Proposition . Let (η, u) be a solution of the transformed MHD problem with an associ- ated perturbation pressure q. Then there is a sufficiently small δ , such that (η, u, q) enjoys the following apriori stability estimate:   G(T ) ≤ C η + u , (.)       provided that G (T ) ≤ δ for some T >. Here C ≥  denotes a constant depending on      the domain  and other physical parameters in the transformed MHD equations. In view of theapriori stability estimate in Proposition . and the following result of local existence of a small solution to the transformed MHD problem, we immediately obtain Theorem .. Proposition . There is a sufficiently small δ , such that for any given initial data   (η , u ) ∈ H × H satisfying     η + u ≤ δ      and the compatibility conditions (i.e., ∂ u(x,)| =, j =,), there exist a T := T (δ )> ∂    which depends on δ , the domain  and other known physical parameters, and a unique    classical solution (η, u) ∈ C ([, T ], H × H ) to the transformed MHD problem (.), i  –i (.) with an associated perturbation pressure q. Moreover, ∂ u ∈ C ([, T ], H ) for   H    ≤ i ≤ , q ∈ C ([, T ], H ), E ()  η + u , and         H H sup η(τ) + E (τ) + D (τ)+ u(τ) + q(τ) dτ < ∞,    ≤τ ≤T and G(t) is continuous on [, T ]. Proof Thetransformed MHDproblem is very similartothe surfacewaveproblem (.)in []. Moreover, the current problem is indeed simpler than the surface wave problem due to the non-slip boundary condition u| = . Using the standard method in [], one can easily establish Proposition ., hence we omit its proof here. In addition, the continuity,     such as (η, u, q) ∈ C ([, T], H ×H ×H ), G(t) and so on, can be verified by using the reg- ularity of (η, u, q), the transformed MHD equations and a standard regularized method.  Wang and Zhao Boundary Value Problems (2017) 2017:114 Page 30 of 31 6Conclusion We have proved the existence of a unique time-decay solution to the initial-boundary problem (.)-(.) of rotating MHD fluids in Lagrangian coordinates, which, together with the inverse transformation of coordinates, implies the existence of a time-decay so- lution to the original initial-boundary problem (.)-(.) with proper initial data in H (). Ourresultalsoholds forthe case ω = (i.e., the absence of rotation), thus it improves Tan and Wang’s result in [], in which the sufficiently small initial data at least belongs to  H (). Hence our result reveals that rotation does not affect the existence of solutions of rotating MHD fluids. We mention that the phenomenon of rotating MHD fluids widely ex- ists in nature, so our result has potential applications. In addition, based on Theorem ., we will further study the Rayleigh-Taylor problem of rotating MHD fluids in the future; please referto[–] for relevant results on the Rayleigh-Taylor problem. Funding The current work is partially supported by NSFC under Grant Nos. 11501116 and 11671086, and the Education Department of Fujian Province under Grant No. JA15063. Competing interests The authors declare that there is no conflict of interest regarding the publication of this manuscript. The authors declare that they have no competing interests. Authors’ contributions All authors have made the same contribution and finalized the current version of this manuscript. They both read and approved the final manuscript. 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