JUSTIFICATION OF PRANDTL ANSATZ FOR MHD BOUNDARY LAYER CHENG-JIE LIU, FENG XIE, AND TONG YANG Abstract.

arXiv:1704.00523v4 [math.AP] 7 Jul 2018

CHENG-JIE LIU, FENG XIE, AND TONG YANG

Abstract. As a continuation of [43], the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on velocity field and perfect conducting boundary condition on magnetic field. Under the assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give aL⁸ estimate on the error by multi-scale analysis.

1. Introduction and Main Results

For electrically conducting fluid such as plasmas and liquid metals, the system of magnetohydrodynam- ics(denoted by MHD) is a fundamental system to describe the motion of fluid under the influence of electro- magnetic field. The study on the MHD was initiated by Hannes Alfv´en who showed that magnetic field can induce currents in a moving conductive fluid with a new propagation mechanism along the magnetic field, called Alfv´en waves (see [3]). One important problem about MHD is to understand the inviscid and vanishing resistivity limit in a domain with boundary. The purpose of this paper is to justify this high Reynold numbers limit when the tangential magnetic field is not degenerate on the boundary.

To this end, consider the following two-dimensional (2D) incompressible viscous MHD equations in the domaintpt, x, yq|tą0, xPT, yPR_`u,

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Btu^ǫ` pu<sup>ǫ</sup>¨∇qu<sup>ǫ</sup>`∇p^ǫ´ pH^ǫ¨∇qH^ǫ“µǫ△u^ǫ, BtH^ǫ` pu^ǫ¨∇qH^ǫ´ pH^ǫ¨∇qu^ǫ“κǫ△H^ǫ,

∇¨u^ǫ“0, ∇¨H^ǫ“0.

(1. 1) Here u^ǫ“ pu^ǫ, v^ǫq andH^ǫ“ ph^ǫ, g^ǫqstand for the velocity field and magnetic field respectively, p^ǫ denotes the total pressure, the tangential variable is periodic: x P T, and the normal variable y P R_`. Also we assume µ, κ are positive constants, and the viscosity and resistivity coefficients are of the same order in a small parameterǫ. The initial data of (1. 1) is given by

pu^ǫ,H^ǫq|t“0“ pu0,H0qpx, yq “ pu0, v0, h0, g0qpx, yq (1. 2) independent ofǫ. The no-slip boundary condition is imposed on velocity field, and the perfectly conducting boundary condition on magnetic field:

u^ǫ|y“0“0, pByh^ǫ, g^ǫq|y“0“0. (1. 3) The initial-boundary value problem (1. 1)-(1. 3) with fixed ǫ ą 0 has been investigated and its global well-posedness is well known, see [11, 56] for instance. Let us also mention that there are vast literatures on the MHD system, in particular in the case without boundaries, cf. [1, 5–8, 27, 38, 39, 62] and the references therein.

In this paper, we are concerned with the asymptotic behavior of solutionspu^ǫ,H^ǫqto problem (1. 1)-(1. 3) as ǫÑ0. Formally, whenǫ“0, (1. 1) becomes the following incompressible ideal MHD system:

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Btu⁰` pu<sup>0</sup>¨∇qu<sup>0</sup>`∇p⁰´ pH⁰¨∇qH⁰“0, BtH⁰` pu⁰¨∇qH⁰´ pH⁰¨∇qu⁰“0,

∇¨u⁰“0, ∇¨H⁰“0,

(1. 4)

2000Mathematics Subject Classification. 76N20, 35A07, 35G31, 35M33.

Key words and phrases. MHD boundary layer, high Reynolds numbers limit, Prandtl boundary layer expansion,L⁸estimate.

1

where the velocity field u⁰ “ pu⁰, v⁰q and magnetic field H⁰ “ ph⁰, g⁰q. Naturally, we endow (1. 4) with homogeneous Dirichlet boundary condition on normal components of velocity and magnetic field:

pv⁰, g⁰q|y“0“0. (1. 5)

Note that such boundary condition (1. 5) is sufficient to solve (1. 4), since the boundary ty “ 0u is the streamline for (1. 4) under (1. 5).

Comparing the boundary conditions (1. 3) with (1. 5), there is a mismatch between the tangential com- ponent pu^ǫ, h^ǫqpt, x, yq and pu⁰, h⁰qpt, x, yq on the boundary ty “ 0u. According to the classical Prandtl boundary layer theory [51], there is a thin layer of width of order?

ǫ near the boundary, in which pu^ǫ, h^ǫq changes dramatically from its boundary data to the outer flow pu⁰_e, h⁰_eqpt, x, yq. In other words, there exist boundary layer profiles u⁰_bpt, x,?^y

ǫqand h^b₀pt, x,?^y

ǫq, such that the solution to the problem (1. 1)-(1. 3) is expected to have the form:

#pu^ǫ,H^ǫqpt, x, yq “ pu⁰,H⁰qpt, x, yq ` pu<sup>0</sup><sub>b</sub>,0, h<sup>0</sup><sub>b</sub>,0qpt, x,<sup>?y</sup><sub>ǫ</sub>q `op1q,

p^ǫpt, x, yq “ p⁰pt, x, yq `op1q, (1. 6)

where the error terms op1qtends to zero inL⁸-norm asǫtends to zero.

We are devoted to verify the Prandtl boundary layer expansion in (1. 6) for the MHD system (1. 1)-(1. 3).

Let us first review some mathematical results on the classical boundary layer theory. It is well known that in both physics and mathematics, the study on fluid around a rigid body with high Reynolds number is important and challenging, and the fluid motion exhibit rather complicated behaviors, especially near the surface of body. This is partially due to the appearance of boundary layers, whose formation is formally explained by Prandtl [51] in 1904. Prandtl also derived the simplified equations, the well-known Prandtl equations, from the incompressible Navier-Stokes equations with no-slip boundary condition on velocity, to describe the fluid motion in the boundary layer. Under the monotone assumption on the tangential velocity in the normal direction, Oleinik firstly in 1960s’ obtained the local existence of classical solutions of 2D Prandtl equations by using the Crocco transformation, cf. [49]. The result together with some other related works are well written in the classical book [50]. Recently, this well-posedness result was established in the Sobolev spaces by using energy methods in [2] and [47] independently. Moreover, by imposing an additional favorable condition on the pressure, a global in time weak solution was obtained in [61]. Some of these results were generalized to 3D case with special structure in [41] and [42]. In the absence of monotonicity condition, boundary separation can be observed, and so far we only gain the local-in-time solvability of the Prandtl equations in the analytic framework [28, 32, 33, 45, 53, 63] or Gevrey framework [17, 35–37]. To our knowledge, the solvability of Prandtl equations for general initial data in a Sobolev class is still open, although some interesting ill-posedness (or instability) results for Prandtl equations have been established, cf. [13, 14, 18, 20–22, 25, 34, 40, 44] and the references therein. On the other hand, the rigorous verification of the Prandtl boundary layer theory, i.e., the solution to the Navier-Stokes equations as a superposition of solutions to the Euler and Prandtl systems in vanishing viscosity limit, was achieved only for some specific cases, e.g., in the analytic framework in [53,54,57]. In 2014, the convergence problem in 2D case was studied by Maekawa [46] that requires that the vorticity of flow vanishes in a neighborhood of boundary initially, and such condition implies the analyticity of the initial data with respect to the tangential variable in the same region. In 2016, the authors in [16] improved the results of Sammartino & Caflisch [53, 54] in Gevery class. For the steady case, Guo & Nguyen in [26] justified the Prandtl boundary layer expansions for the steady Navier-Stokes flows over a moving plate, and similar results for steady flows were obtained in [29–31].

Very recently, G´erard-Varet & Maekawa in [15] shows the H¹ stability of shear flows of Prandtl type and verifies the Prandtl expansions for the steady Navier-Stokes equations with no-slip boundary condition. It is remarkable that the solvability of Prandtl equations does not necessarily imply the validity of corresponding Prandtl boundary layer expansions, see the counterexamples in [20, 23, 24].

For plasma, the boundary layer equations can be derived from the fundamental MHD system and they are more complicated than the classical Prandtl system because of the coupling of magnetic field with velocity field through the Maxwell equations. It should be emphasized that the MHD boundary layer is an important problem in study of plasma with fruitful results, cf. [4,11,12,19,52,58,60]. On one hand, if the magnetic field is transversal to the boundary, there are extensive discussions on the so-called Hartmann boundary layer, cf. [9, 10]. On the other hand, if the magnetic field is tangent to the boundary, it is just the case we are concerned with in this paper. Note that in physics, it is believed that the magnetic field has a stabilizing

effect on the boundary layer that could provide a mechanism for containment of some kind of instability and singularity. Recently, the same authors of this paper established the well-posedness of MHD boundary layer equations in weighted Sobolev spaces without monotonicity condition on the velocity in [43]. The key assumption is that tangential magnetic field is not degenerate on the boundary. As the continuation of [43], we study the high Reynolds numbers limit problem for (1. 1)-(1. 3). Precisely, by applying the multi-scale expansion ofpu^ǫ,H^ǫqin Sections 2 and 3, we will justify the validity of the Prandtl boundary layer theory in (1. 6) under the non-degeneracy condition on the initial tangential magnetic field on the boundary. As it is well known that the Alfv´en wave propagates along the magnetic field, this result in some sense justifies the physical phenomena that the Alfv´en wave along the boundary carries away the energy so that it stabilizes the boundary layer rigorously in mathematics.

We are now ready to state main result in this paper as follows.

Theorem 1.1. Let mě36 be an integer. Let the initial datapu0,H0qpx, yq PH^mpTˆR^`qsatisfy:

i) u0“ pu0, v0qis a divergence free vector field vanishing on the boundary, andH0“ ph0, g0qis a divergence free vector field tangent to the boundary;

ii) there exists a smallǫ0ą0such that the following ‘strong’ compatibility conditions hold for anyǫP r0, ǫ⁰s, Btⁱpu^ǫ, v^ǫ, g^ǫqp0q|y“0“0, Bt^i´1Byh^ǫp0q|y“0“0, 1ďiď rm

2s ´3,

where rks, k P R stands for the largest integer less than or equal to k, and Btⁱpu^ǫ, v^ǫ, h^ǫ, g^ǫqp0q is the i´th time derivative at tt “ 0u of any solution of (1. 1)-(1. 3), as calculated from (1. 1) to yield an expression in terms of derivatives ofpu0,H0q;

iii) the initial tangential magnetic filed is non-degenerate:

h0px,0q ěδ0ą0 for some constant δ0. (1. 7) Then there exists T_˚ ą0 independent of ǫsuch that, the problem (1. 1)-(1. 3)admits a solution pu^ǫ,H^ǫq in the time intervalr0, T_˚s, and there exists a smooth solutionpu⁰,H⁰qpt, x, yq PC`

r0, T_˚s, H^m˘

to the problem (1. 4)-(1. 5) with the initial datapu0,H0q, and a boundary layer profile

pu⁰_b, v_b⁰, h⁰_b, g⁰_bq PC`

r0, T_˚s ˆTˆR^`˘ , such that for any arbitrarily small σą0,

sup

0ďtďT˚

››pu^ǫ,H^ǫqpt, x, yq ´ pu⁰,H⁰qpt, x, yq ´` u⁰_b,?

ǫv⁰_b, h⁰_b,? ǫg_b⁰˘`

t, x, y

?ǫ˘››

L⁸_xy ď Cǫ^3{8´σ, (1. 8) where the constant Cą0is independent of ǫ.

Remark 1.1. The assumptions on the regularity and compatibility conditions of the initial data pu0,H0q are not optimal. Here, we mainly utilize such assumptions to verify Proposition 2.2 in Subsection 2.2. One may relax the requirement on the regularity and compatibility conditions.

Finally, we would like to comment why the justification of the high Reynolds numbers limit can be achieved for MHD in the framework of Sobolev space, but the corresponding problem for incompressible Navier-Stokes equations remains open. Note that after receiving the solvability of boundary layer equations, the key issue to verify the Pandtl ansatz is to control the interaction, mainly in the boundary layer, between the vorticity produced by the boundary layer and the outer vorticity generated by the initial one. For the Navier-Stokes equations, some essential cancellations are observed in [2] and [47] for recovering the loss of derivatives in the well-posedness theory of classical Prandtl equations. If we use these cancellations to govern the above interaction, it will destroy the divergence free structure of newly introduced unknown functions for velocity field, and then the uniform estimation on the pressure function becomes a challenging and unsolved problem.

However, for MHD system, the newly observed cancellation mechanism for MHD boundary layer equations in [43] not only can be utilized to control the desired interaction, but also preserves the divergence free condition of the newly defined unknown function for velocity field. It needs to be emphasized that in this analysis, the non-degeneracy condition on the tangential magnetic field plays an essential role.

This paper is organized as follows: In Section 2, we will construct a suitable approximation solution and derive some necessary estimates. In Section 3, the error of the approximation is estimated in L⁸-norm for the proof of Theorem 1.1. Finally, to make the paper self-contained, we will provide several proofs and computations in the appendix.

2. Construction of approximate solution

To prove Theorem 1.1, we need to construct high order approximate solutions to the problem (1. 1)-(1. 3).

Precisely, we take the forms of approximate solutions as follows:

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pu^a,H^aqpt, x, yq “ pu⁰,H⁰qpt, x, yq `` u⁰_b,?

ǫv_b⁰, h⁰_b,? ǫg⁰_b˘`

t, x,?^y ǫ

˘

`? ǫ”

pu¹,H¹qpt, x, yq `` u¹_b,?

ǫv¹_b, h¹_b,? ǫg_b¹˘`

t, x,?^y ǫ

˘ı, p^apt, x, yq “p⁰pt, x, yq `?

ǫp¹pt, x, yq `ǫp¹_bpt, x,?^y ǫq,

(2. 1)

where the functions with subscriptb denote the boundary layer profile. In the next six subsections, we will give the construction of the profiles in the above approximation (2. 1).

Keep in mind that the fast variable η “ ^?y_ǫ, and in the following derivation we assume first that for i“0,1,

ηÑ`8lim puⁱ_b, vⁱ_b, hⁱ_b, gⁱ_bqpt, x, ηq “0, lim

ηÑ`8p¹_bpt, x, ηq “0, which means the boundary layer profiles decay to zero away from the boundary.

2.1. Zeroth-order inner flow. From the arguments in the previous section, we know that the leading order inner flow pu⁰,H⁰, p⁰qpt, x, yqin the ansatz (2. 1) satisfies the following initial-boundary value problem for incompressible ideal MHD equations,

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Btu⁰` pu<sup>0</sup>¨∇qu<sup>0</sup>`∇p⁰´ pH⁰¨∇qH⁰“0, BtH⁰` pu⁰¨∇qH⁰´ pH⁰¨∇qu⁰“0,

∇¨u⁰“0, ∇¨H⁰“0,

pv⁰, g⁰q|y“0“0, pu⁰,H⁰q|t“0“ pu0,H0qpx, yq

(2. 2)

in the domain tpt, x, yq|t ą 0, x P T, y P R<sub>`</sub>u, where the velocity field u<sup>0</sup> “ pu<sup>0</sup>, v<sup>0</sup>q and magnetic field H<sup>0</sup> “ ph<sup>0</sup>, g<sup>0</sup>q. Note that the equations of (2. 2) can also be obtained by putting the ansatz (2. 1) into (1. 1), setting the terms of orderǫ<sup>0</sup>equal to zero and letting the fast variableηÑ `8.

The well-posedness of problem (2. 2) is guaranteed by the results in [55, 59] that can be stated as follows.

Proposition 2.1. Let mą1 be a integer, and let the initial datapu0,H0qpx, yq PH^mpTˆR_`qsatisfy that u0 and H0 are divergence free vector fields tangent to the boundary.Then there exists a timeT0ą0 and a smooth solution pu⁰,H⁰, p⁰qpt, x, yqto (2. 2) satisfying

pu⁰,H⁰,∇p⁰qpt, x, yq P čm j“0

C^j`

r0, T0s;H^m´jpTˆR_`q˘ ,

Remark 2.1. When the initial data of (2. 2)satisfy the assumption of initial data in Theorem 1.1, we can gain more information on the trace of the tangential componentsu⁰ andh⁰, which will be used to ensure the compatibility conditions of the problems investigated in the following subsections. More precisely,

u⁰pt, x,0q|t“0“u0px,0q “0, h⁰pt, x,0q|t“0“h0px,0q ěδ0ą0 (2. 3) and

Btⁱu⁰pt, x,0q|t“0“0, Bt^i´1Byh⁰pt, x,0q|t“0“0, 1ďiď rm

2s ´3. (2. 4)

Then, by the properties of the solutionpu⁰,H⁰, p⁰qpt, x, yqestablished in Proposition 2.1, it is not hard to see that there exists a time T1ďT0, such that the boundary value h⁰pt, x,0q ě ^δ2⁰ for all tP r0, T1s. Moreover, the trace theorem yields

sup

0ďtďT1

mÿ´1 i“0

››Btⁱpu⁰, h⁰,Bxp⁰qpt, x,0q››

H^m´1´ipTxqă `8. (2. 5) After establishing the leading order inner profilepu⁰,H⁰, p⁰qpt, x, yq, we now turn to construct the leading order MHD boundary layer profile.

2.2. Zero-order boundary layer. As in [43], we know that the zero-order MHD boundary layer profile pu⁰_b, v_b⁰, h⁰_b, g_b⁰qpt, x, ηqis given by

#pu⁰_b, h⁰_bqpt, x, ηq :“ pu^p, h^pqpt, x, ηq ´ pu⁰, h⁰qpt, x,0q, v_b⁰pt, x, ηq :“ ş₈

η Bxu⁰_bpt, x,η˜qd˜η, g⁰_bpt, x, ηq :“ ş₈

η Bxh⁰_bpt, x,η˜qd˜η, (2. 6) andpu^p, h^pqpt, x, ηqcan be solved by the following boundary layer system:

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Btu^p` pu<sup>p</sup>Bx`v^pBηqu^p´ ph^pBx`g<sup>p</sup>Bηqh<sup>p</sup>“µBη<sup>2</sup>u<sup>p</sup>´ Bxp<sup>0</sup>pt, x,0q, Bth<sup>p</sup>` pu^pBx`v<sup>p</sup>Bηqh<sup>p</sup>´ ph<sup>p</sup>Bx`g^pBηqu^p“κB²ηh^p,

Bxu^p` Bηv<sup>p</sup>“0, Bxh<sup>p</sup>` Bηg^p“0, pu^p, v^p,Bηh^p, g^pq|η“0“0, lim

ηÑ`8pu<sup>p</sup>, h<sup>p</sup>qpt, x, ηq “ pu<sup>0</sup>, h<sup>0</sup>qpt, x,0q, pu<sup>p</sup>, h<sup>p</sup>q|t“0“ pu<sup>0</sup>, h<sup>0</sup>qp0, x,0q “`

0, h0px,0q˘

(2. 7)

in the domain tpt, x, ηq|t P r0, T0s, x P T, η P R_`u, where we have used (2. 3) in the above initial data.

Moreover, it follows that from (2. 3),

h^pp0, x, ηq ě δ0 ą 0. (2. 8)

By the main theorem in [43], we have the local well-posedness theory of solutions to the initial-boundary value problem (2. 7). Before we state the well-posedness theorem, let us introduce some weighted Sobolev spaces used in this subsection. Denote by

Ω :“ px, ηq:xPT, ηPR_`( .

For anylPR,denote byL²_lpΩqthe weighted Lebesgue space with respect to the spatial variables:

L²_lpΩq:“!

fpx, ηq: ΩÑR, }f}L²_lpΩq:“´ ż

Ω

xηy^2l|fpx, ηq|²dxdη¯¹2

ă `8)

, xηy “1`η, and then, for any given mPN,denote byH_l^mpΩqthe weighted Sobolev spaces:

H_l^mpΩq :“ !

fpx, ηq: ΩÑR, }f}H^m_l pΩq:“´ ÿ

m1`m2ďm

}xηy^l`m2Bx^m1Bη^m2f}²L²pΩq

¯¹2

ă `8) . Combining Remark 2.1 with the condition (2. 8), we have the following result by the main theorem in [43].

Proposition 2.2. Letpu⁰,H⁰, p⁰qpt, x, yqbe the leading order inner flow with the initial datapu0,H0qwhich satisfy the assumptions of Theorem 1.1. Then, there exist a positive time 0ăT2ďT1 and a unique solution pu^p, v^p, h^p, g^pqpt, x, ηqto the initial boundary value problem (2. 7), such that

h^ppt, x, ηq ě δ0

2, @ pt, x, ηq P r0, T2s ˆΩ (2. 9) with the constantδ0ą0given in (2. 8), and for anylě0,

`u^ppt, x, ηq ´u⁰pt, x,0q, h^ppt, x, ηq ´h⁰pt, x,0q˘ P

rm{č2s´1 i“0

W^i,8

´

0, T2;H_l^rm{2s´1´ipΩq¯

. (2. 10)

Moreover, it holds that for the profile pu⁰_b, v_b⁰, h⁰_b, g_b⁰qpt, x, ηqdefined by (2. 6), pu⁰_b, h⁰_bqpt, x, ηq P

rmč{2s´1 i“0

W^i,8´

0, T2;H_l^rm{2s´1´ipΩq¯

, (2. 11)

pv⁰_b, g⁰_bqpt, x, ηq, pBηv_b⁰,Bηg⁰_bqpt, x, ηq P

rmč{2s´2 i“0

W^i,8`

0, T2;H_l^rm{2s´2´ipΩq˘

. (2. 12)

Proof. First of all, by using (2. 3) and (2. 20) it can be deduced that, the initial-boundary values of (2. 7) satisfy the compatibility conditions up to order rm{2s ´3. Then, the local well-posedness theory of the solution pu^p, v^p, h^p, g^pqpt, x, ηq to problem (2. 7), and the relation (2. 10) has been obtained in [43]. Note that the initial data of pu^p, h^pq, given in (2. 7), is independent of normal variable η, therefore the index l of weight with respect to η in (2. 10) can be arbitrary large. Moreover, (2. 11) follows automatically by combining (2. 6)₁ with (2. 10). Therefore, we only need to show (2. 12).

From (2. 6)₂ we obtainpBηv⁰_b,Bηg_b⁰q “ ´pBxu⁰_b,Bxh⁰_bq,and forαPN², lě0, ˇˇxηy^lBtx^αv⁰_bpt, x, ηqˇˇ“ˇˇxηy^l

ż₈

η

Btx^αBxu⁰_bpt, x,η˜qd˜ηˇˇď ż₈

η

ˇˇx˜ηy^lBtx^αBxu⁰_bpt, x,η˜qˇˇd˜η À xηy^12´l0››xηy^l`l0B^αtxBxu⁰_bpt, x, ηq››

L²_η

providedl0ą ¹2, which implies that

››B^αtxv⁰_bpt,¨q››

L²_lpΩqÀ››xηy^12´l0››

L²_稛›B^αtxBxu⁰_bpt,¨q››

L²_l`l0pΩqÀ››B^αtxBxu⁰_bpt,¨q››

L²_l`l0pΩq

providedl0ą1.Similarly, we have that for l0ą1,

››Btx^αg_b⁰pt,¨q››

L²_lpΩqÀ››B^αtxBxh⁰_bpt,¨q››

L²_l`l0pΩq.

By using the above two inequalities and combining with (2. 11), we get (2. 12) immediately.

From (2. 6) and the divergence free conditions in (2. 7) we have another expression forpu⁰_b, v⁰_b, h⁰_b, g⁰_bq:

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pu⁰_b, h⁰_bqpt, x, ηq “ pu^p, h^pqpt, x, ηq ´ pu⁰, h⁰qpt, x,0q, pv_b⁰, g_b⁰qpt, x, ηq “ pv^p, g^pqpt, x, ηq `ηpBxu⁰,Bxh⁰qpt, x,0q

`ş₈

0

´

Bxu^ppt, x, ηq ´ Bxu⁰pt, x,0q,Bxh^ppt, x, ηq ´ Bxh⁰pt, x,0q¯ dη,

(2. 13)

which implies that by virtue of the boundary conditionspv^p, g^pq|η“0“0 in (2. 7), pv_b⁰, g⁰_bqpt, x,0q “

ż₈

0

´

Bxu^ppt, x, ηq ´ Bxu⁰pt, x,0q,Bxh^ppt, x, ηq ´ Bxh⁰pt, x,0q¯

dη. (2. 14)

Then we can derive the problem of pu⁰_b, v⁰_b, h⁰_b, g⁰_bqpt, x, ηq. Indeed, from (2. 13) and (2. 14) it holds

#pu^p, h^pqpt, x, ηq “ pu⁰_b, h⁰_bqpt, x, ηq ` pu⁰, h⁰qpt, x,0q,

pv^p, g^pqpt, x, ηq “ pv_b⁰, g_b⁰qpt, x, ηq ´ pv_b⁰, g⁰_bqpt, x,0q ´ηpBxu⁰,Bxh⁰qpt, x,0q.

By using the notation of ¯fpt, xq to stand for the trace of function fpt, x, yq on the boundary ty “ 0u, we substitute the above expression into the problem (2. 7) to obtain that

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Btu⁰_b ` pu<sup>0</sup>`u⁰_bqBxu⁰_b` pv<sub>b</sub><sup>0</sup>´v<sup>0</sup><sub>b</sub> ´ηBxu<sup>0</sup>qBηu<sup>0</sup><sub>b</sub>´ ph<sup>0</sup>`h⁰_bqBxh⁰_b´ pg_b⁰´g⁰_b ´ηBxh⁰qBηh⁰_b

`Bxu⁰ u⁰_b´ Bxh⁰ h⁰_b “µBη²u⁰_b,

Bth⁰_b ` pu<sup>0</sup>`u⁰_bqBxh⁰_b` pv<sub>b</sub><sup>0</sup>´v<sub>b</sub><sup>0</sup>´ηBxu<sup>0</sup>qBηh<sup>0</sup><sub>b</sub>´ ph<sup>0</sup>`h⁰_bqBxu⁰_b´ pg⁰_b´g_b⁰´ηBxh⁰qBηu⁰_b

`Bxu<sup>0</sup> h<sup>0</sup><sub>b</sub>´ Bxh<sup>0</sup> u<sup>0</sup><sub>b</sub> “κB<sup>2</sup>ηh<sup>0</sup><sub>b</sub>, Bxu<sup>0</sup><sub>b</sub>` Bηv_b⁰“0, Bxh⁰_b` Bηg<sup>0</sup><sub>b</sub> “0, pu<sup>0</sup><sub>b</sub>,Bηh<sup>0</sup><sub>b</sub>q|η“0“ ´`

u⁰pt, xq,0˘

, lim

ηÑ`8pu⁰_b, h⁰_bq “0, pu⁰_b, h⁰_bq|t“0“0,

(2. 15)

where we have used the equations ofpu⁰, h⁰qon the boundaryty“0ufrom the problem (2. 2)-(??). Moreover, from (2. 15) we know thatg⁰_b satisfies the following equation:

Btg0^b` pu<sup>0</sup>`u⁰_bqBxg_b⁰` pv<sup>0</sup><sub>b</sub>´v<sup>0</sup><sub>b</sub>´ηBxu<sup>0</sup>qBηg<sup>0</sup><sub>b</sub>´ ph<sup>0</sup>`h⁰_bqBxv⁰_b´ pg_b⁰´g_b⁰´ηBxh⁰qBηv_b⁰

´ Bx

`g<sub>b</sub><sup>0</sup>`ηBxh⁰˘

u⁰_b´ Bxh⁰v⁰_b` Bx

`v<sub>b</sub><sup>0</sup>`ηBxu⁰˘

h⁰_b` Bxu⁰g_b⁰“κBη²g⁰_b. (2. 16) After constructing the leading order inner flowpu⁰,H⁰, p⁰qand boundary layer profilepu⁰_b, v⁰_b, h⁰_b, g⁰_bq, we proceed to construct the next order inner MHD flow.

2.3. First-order inner flow. Put the ansatz (2. 1) into (1. 1) and set the terms of orderǫ^1{2equal to zero, then letting η Ñ `8 yields the first order inner flow pu<sup>1</sup>,H<sup>1</sup>, p<sup>1</sup>qpt, x, yq satisfies the following linearized ideal MHD equations in the regiontpt, x, yq|tP r0, T2s, xPT, yPR<sub>`</sub>u:

$&

%

Btu¹` pu<sup>0</sup>¨∇qu<sup>1</sup>`∇p¹´ pH⁰¨∇qH¹` pu<sup>1</sup>¨∇qu<sup>0</sup>´ pH<sup>1</sup>¨∇qH<sup>0</sup>“0, BtH<sup>1</sup>` pu⁰¨∇qH¹´ pH⁰¨∇qu¹` pu¹¨∇qH⁰´ pH¹¨∇qu⁰“0,

∇¨u¹“0, ∇¨H¹“0,

(2. 17)

where u¹“ pu¹, v¹qand H¹“ ph¹, g¹q. The initial data is chosen to be zero:

pu¹,H¹q|t“0“0, (2. 18)

and the boundary conditions ofpv¹_e, g_e¹qin (2. 17) are imposed by pv_e¹, g_e¹qpt, x,0q “ ´pv_b⁰, g_b⁰qpt, x,0q “

ˆ

´ ż₈

0 Bxu⁰_bpt, x,η˜qd˜η,´ ż₈

0 Bxh⁰_bpt, x,η˜qd˜η

˙

, (2. 19)

to homogenous the boundary conditions of oder ǫ^1{2 for the normal components of pu^ǫ,H^ǫq. It is noted that the boundary condition (2. 19) is sufficient to solve the initial-boundary value problem (2. 17)-(2. 19).

Moreover, from (2. 11) we know that pv_e¹, g_e¹qpt, x,0q P

rm{2s´3

č

i“0

W^i,8´

0, T2;H^rm{2s´2´ipT_xq¯

. (2. 20)

By a similar argument as for the Proposition 2.1 for initial-boundary value problem of the linearized ideal MHD equations (2. 17)-(2. 19), or as a direct consequence of the main results in [48], we have

Proposition 2.3. Letpu⁰,H⁰, p⁰qpt, x, yqbe the leading order inner flow with the initial datapu0,H0qwhich satisfy the assumptions of Theorem 1.1. Let the boundary condition (2. 20)hold. Then there exists a smooth solution pu¹,H¹, p¹qpt, x, yqto problem (2. 17)-(2. 19) in the time interval r0, T3s, such that

pu¹,H¹,∇p¹qpt, x, yq P

rm{č2s´2 j“0

C^j´

r0, T3s;H^rm{2s´2´j`

TˆR_`˘¯

, where0ăT3ďT2 is the local lifespan of solution pu¹,H¹, p¹qpt, x, yq.

Now we consider the following approximation solutions to the problem (1. 1)-(1. 3):

# pu^a0,H^a0qpt, x, yq “ pu⁰,H⁰qpt, x, yq `` u⁰_b,?

ǫv⁰_b, h⁰_b,? ǫg_b⁰˘`

t, x,?^y ǫ

˘`?

ǫpu¹,H¹qpt, x, yq, p^a0pt, x, yq “p⁰pt, x, yq `?

ǫp¹pt, x, yq. (2. 21)

From the above construction ofpuⁱ,Hⁱ, pⁱqpi“0,1qandpu⁰_b, v_b⁰, h⁰_b, g_b⁰q, a direct calculation reads

$’

’’

’’

&

’’

’’

’%

Btu^a0` pu<sup>a0</sup>¨∇qu<sup>a0</sup>`∇p^a0´ pH^a0¨∇qH^a0´µǫ△u^a0“R^a0u , BtH^a0` pu^a0¨∇qH^a0´ pH^a0¨∇qu^a0´κǫ△H^a0“R^a0H,

∇¨u^a0“0, ∇¨H^a0 “0,

`u^a0,H^a0˘

|t“0“ pu0,H0qpx, yq, u^a0|y“0“`?

ǫu¹pt, x,0q,0˘

, Byh^a0|y“0“`

Byh⁰`? ǫByh¹˘

pt, x,0q, g^a0|y“0“0,

(2. 22)

where the remainder termsR^a0u “ pR^a0₁ , R^a0₂ qandR^a0H “ pR^a0₃ , R₄^a0qare summarized as follows,

$’

’’

’’

’’

’’

’’

’’

’’

’’

’’

’’

’’

’’

&

’’

’’

’’

’’

’’

’’

’’

’’

’’

’’

’’

’’

’%

R₁^a0“ `

u⁰´u⁰`? ǫu¹˘

Bxu⁰_b`“

v⁰´yByv⁰`? ǫ`

v¹´v¹˘‰

Byu⁰_b´`

h⁰´h⁰`? ǫh¹˘

Bxh⁰_b

´“

g⁰´yByg⁰`? ǫ`

g¹´g¹˘‰

Byh⁰_b`u<sup>0</sup><sub>b</sub>`

Bxu⁰´ Bxu⁰`? ǫBxu¹˘

`? ǫv_b⁰Byu⁰

´h⁰_b`

Bxh⁰´ Bxh⁰`? ǫBxh¹˘

´?

ǫg_b⁰Byh⁰`R^high₁ , R₂^a0“ ?

ǫ”

Btv⁰_b` pu<sup>0</sup>`u⁰_bqBxv⁰_b`` v⁰`?

ǫv¹`? ǫv⁰_b˘

Byv⁰_b´ ph⁰`h⁰_bqBxg⁰_b

´` g<sup>0</sup>`?

ǫg¹`? ǫg_b⁰˘

Byg⁰_b`v^b₀Byv⁰´g^b₀Byg⁰´µǫB²yv_b⁰ı

`u<sup>0</sup><sub>b</sub>`

Bxv⁰`? ǫBxv¹˘

´h⁰_b`

Bxg⁰`? ǫBxg¹˘

`R<sup>high</sup><sub>2</sub> , R<sub>3</sub><sup>a0</sup>“ `

u⁰´u⁰`? ǫu¹˘

Bxh⁰_b`“

v⁰´yByv⁰`? ǫ`

v¹´v¹˘‰

Byh⁰_b´`

h⁰´h⁰`? ǫh¹˘

Bxu⁰_b

´“

g⁰´yByg⁰`? ǫ`

g¹´g¹˘‰

Byu⁰_b`u<sup>0</sup><sub>b</sub>`

Bxh⁰´ Bxh⁰`? ǫBxh¹˘

`? ǫv⁰_bByh⁰

´h⁰_b`

Bxu⁰´ Bxu⁰`? ǫBxu¹˘

´?

ǫg_b⁰Byu⁰`R^high₃ , R₄^a0“ ?

ǫ”

Btg⁰_b` pu<sup>0</sup>`u⁰_bqBxg⁰_b`` v⁰`?

ǫv¹`? ǫv⁰_b˘

Byg⁰_b´ ph⁰`h⁰_bqBxv_b⁰

´` g<sup>0</sup>`?

ǫg¹`? ǫg_b⁰˘

Byv⁰_b`v^b₀Byg⁰´g^b₀Byv⁰´κǫB²yg⁰_b ı

`u<sup>0</sup><sub>b</sub>`

Bxg⁰`? ǫBxg¹˘

´h⁰_b`

Bxv⁰`? ǫBxv¹˘

`R^high₄ ,

(2. 23)

with$

’’

’’

’’

&

’’

’’

’’

%

R^high₁ “ǫ

!

u¹Bxu¹` pv<sup>1</sup>`v_b⁰qByu¹´h¹Bxh¹´ pg¹`g<sup>0</sup><sub>b</sub>qByh<sup>1</sup>´µ△` u⁰`?

ǫu¹˘

´µB²xu⁰_b )

, R^high₂ “ǫ

!

u¹Bxpv¹`v<sup>0</sup><sub>b</sub>q ` pv¹`v<sup>0</sup><sub>b</sub>qByv<sup>1</sup>´h<sup>1</sup>Bxpg<sup>1</sup>`g⁰_bq ´ pg¹`g<sup>0</sup><sub>b</sub>qByg<sup>1</sup>´µ△` v⁰`?

ǫv¹˘

´µ? ǫB²xv_b⁰

) , R^high₃ “ǫ!

u¹Bxh¹` pv<sup>1</sup>`v_b⁰qByh¹´h¹Bxu¹´ pg¹`g<sup>0</sup><sub>b</sub>qByu<sup>1</sup>´κ△` h⁰`?

ǫh¹˘

´κB²xh⁰_b) , R₄^high“ǫ!

u¹Bxpg¹`g<sup>0</sup><sub>b</sub>q ` pv¹`v<sub>b</sub><sup>0</sup>qByg<sup>1</sup>´h<sup>1</sup>Bxpv<sup>1</sup>`v⁰_bq ´ pg¹`g<sup>0</sup><sub>b</sub>qByv<sup>1</sup>´κ△` g⁰`?

ǫg¹˘

´κ? ǫB²xg_b⁰)

. (2. 24) The leading order terms of the remainders R^a0_i ,1ďiď4 mainly exist in the boundary layer, i.e., in the neighborhood of size of?

ǫnear the boundary, and it is easy to check that R_i^high “ Opǫq, i“1„4.

Thanks to the equation (2. 16) forg⁰_b we find that the remainderR^a0₄ is actually of orderǫ. Indeed, by virtue of (2. 16) we can rewriteR^a0₄ as

R₄^a0“? ǫ!`

u⁰´u⁰˘

Bxg_b⁰`“

v⁰´yByv⁰`? ǫ`

v¹´v¹˘‰

Byg_b⁰´`

h⁰´h⁰˘ Bxv_b⁰

´“

g⁰´yByg⁰`? ǫ`

g¹´g¹˘‰

Byv_b⁰`v<sub>b</sub><sup>0</sup>`

Byg⁰´ Byg⁰˘

´g_b⁰`

Byv⁰´ Byv⁰˘)

`u⁰_b“

Bxg⁰´yB²xyg⁰`? ǫ`

Bxg¹´ Bxg¹˘‰

´h⁰_b“

Bxv⁰´yB²xyv⁰`? ǫ`

Bxv¹´ Bxv¹˘‰

`R₄^high.

(2. 25)

Note that the major items of the remainders R^a0₁ , R₂^a0 and R₃^a0 with respect to ǫ are in fact of order ? ǫ.

Thus as shown in the ansatz (2. 1), we proceed in the next subsection to construct the boundary layer profile ǫp¹_bpt, x, ηqfor pressure to cover the major items inǫofR^a0₂ . And in subsection 2.5, we will construct boundary layer profile?

ǫpu¹_b,?

ǫv¹_b, h¹_b,?

ǫg¹_bqpt, x, ηqto cover the major items inǫofR^a0₁ andR^a0₃ , and homogenize the boundary conditions of (2. 22) as well.

2.4. Leading order boundary layer of pressure. In order to eliminate the major items inǫofR^a0₂ given in (2. 23), that is, the terms of order ?

ǫ produced bypu⁰_b, v_b⁰, h⁰_b, g⁰_bq, we define the boundary layer profile ǫp¹_bpt, x, ηqfor pressure in the following way,

Bηp¹_b “ ´ Btv_b⁰´`

u⁰_b`u⁰˘

Bxv_b⁰´`

v⁰_b`v<sup>1</sup>`ηByv⁰˘

Bηv⁰_b``

h⁰_b`h⁰˘

Bxg⁰_b``

g_b⁰`g<sup>1</sup>`ηByg⁰˘

Bηg⁰_b`µB²ηv_b⁰

´`

Bxv¹`ηBxy² v⁰˘

u⁰_b´ Byv⁰ v_b⁰``

Bxg¹`ηBxy² g⁰˘

h⁰_b` Byg⁰ g_b⁰, or

p¹_bpt, x, ηq “ ż₈

η

!

B^tv⁰_b``

u⁰_b`u⁰˘

B^xv⁰_b´`

h⁰_b`h⁰˘

B^xg⁰_b``

B^xv¹`η˜Bxy<sup>2</sup> v<sup>0</sup>˘ u<sup>0</sup><sub>b</sub>´`

B^xg¹`η˜B²xyg⁰˘ h⁰_b)

pt, x,η˜qd˜η

`”

´`v_b⁰

2 `v<sup>1</sup>`ηByv⁰˘

v_b⁰``g⁰_b

2 `g<sup>1</sup>`ηByg⁰˘

g_b⁰`µBηv_b⁰ı pt, x, ηq.

(2. 26) By using the above expression (2. 26) and combining with (2. 11)-(2. 12), we can obtain that

p¹_bpt, x, ηq, Bηp¹_bpt, x, ηq P

rm{č2s´3 i“0

W^i,8`

0, T2;H_l^rm{2s´3´ipΩq˘

, @lě0. (2. 27)

2.5. First-order boundary layer. In this subsection, we construct the first-oder boundary layer profiles pu¹_b,?

ǫv¹_b, h¹_b,?

ǫg_b¹qpt, x, ηq of the ansatz (2. 1) to cover the terms of order Op?

ǫq in remainders R^a0₁ and R^a0₃ given in (2. 23). To this end, applying (2. 1) into the equations p1.1q¹ and (1. 1)₃ and considering the terms of order ?

ǫ, it leads to the following equations for tangential components u¹_bpt, x, ηq and h¹_bpt, x, ηq respectively:

Btu¹_b``

u⁰_b`u⁰˘

Bxu¹_b``

v_b⁰`v<sup>1</sup>`ηByv⁰˘

Bηu¹_b´`

h⁰_b`h⁰˘

Bxh¹_b´`

g_b⁰`g<sup>1</sup>`ηByg⁰˘ Bηh¹_b

``

Bxu⁰_b` Bxu⁰˘

u¹_b` Bηu<sup>0</sup><sub>b</sub> v<sub>b</sub><sup>1</sup>´`

Bxh⁰_b` Bxh⁰˘

h¹_b ´ Bηh⁰_b g_b¹´µBη²u¹_b

“ ´`

u¹`ηByu⁰˘

Bxu⁰_b´`

ηByv¹`η² 2 By²v⁰˘

Bηu⁰_b``

h¹`ηByh⁰˘

Bxh⁰_b``

ηByg¹`η² 2 B²yg⁰˘

Bηh⁰_b

´`

Bxu¹`ηB²xyu⁰˘

u⁰_b´ Byu⁰ v_b⁰``

Bxh¹`ηBxy² h⁰˘

h⁰_b` Byh⁰ g_b⁰,

(2. 28)