MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory

Without Monotonicity I: Well-Posedness Theory

CHENG-JIE LIU Shanghai Jiao Tong University

FENG XIE

Shanghai Jiao Tong University AND

TONG YANG City University of Hong Kong

Abstract

We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl-type equations that are derived from the incompressible MHD system with non-slip boundary condition on the veloc- ity and perfectly conducting condition on the magnetic field. Under the assump- tion that the initial tangential magnetic field is not zero, we establish the local-in- time existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the well-posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for the MHD bound- ary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. © 2018 Wi- ley Periodicals, Inc.

Contents

1. Introduction and Main Result 64

2. Preliminaries 69

3. A Priori Estimates 73

4. Local-in-Time Existence and Uniqueness 108

5. A Coordinate Transformation 116

Appendix: Some Inequalities 117

Bibliography 119

Communications on Pure and Applied Mathematics, Vol. LXXII, 0063–0121 (2019)

© 2018 Wiley Periodicals, Inc.

1 Introduction and Main Result

One important problem about magnetohydrodynamics (MHD) is to understand the high Reynolds number limits in a domain with boundary. In this paper, we con- sider the following initial boundary value problem for the two-dimensional (2D) viscous MHD equations (cf. [4, 5, 7, 37]) in a periodic domain f.t; x; y/ W t 2 Œ0; T ; x 2T; y 2R_Cg W

8 ˆ<

ˆ:

@tuC.u r/u .H r/HC rpD4u;

@tH r .uH/D4H; r u D0; r HD0:

(1.1)

Here, we assume the viscosity and resistivity coefficients have the same order of a small parameter. u D .u₁; u₂/ denotes the velocity vector,H D .h₁; h₂/ denotes the magnetic field, andp D zpC jHj²=2denotes the total pressure with

z

p the pressure of the fluid. On the boundary, the non-slip boundary condition is imposed on velocity field

uˇ ˇ_y

D0D0;

(1.2)

and the perfectly conducting boundary condition on magnetic field h₂ˇ

ˇ_y

D0D@yh₁ˇ ˇ_y

D0 D0:

(1.3)

The formal limiting system of (1.1) yields the ideal MHD equations whentends to0. However, there is a mismatch in the tangential velocity between the equations (1.1) and the limiting equations on the boundaryy D 0. This is why a boundary layer forms in the vanishing viscosity and resistivity limit process. To find out the terms in (1.1) whose contribution is essential for the boundary layer, we use the same scaling as the one used in [32],

t Dt; xDx; yzD ¹²y;

then, set

(u1.t; x;y/z Du₁.t; x; y/;

u2.t; x;y/z D ¹²u₂.t; x; y/;

(h1.t; x;y/z Dh₁.t; x; y/;

h2.t; x;y/z D ¹²h₂.t; x; y/;

and

p.t; x;y/z Dp.t; x; y/:

Then by taking the leading order, the equations (1.1) are reduced to 8

ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

@tu1Cu1@xu1Cu2@yu1 h1@xh1 h2@yh1C@xpD@_y²u1;

@ypD0;

@th1C@y.u2h1 u1h2/D@_y²h1;

@th2 @x.u2h1 u1h2/D@_y²h2;

@xu1C@yu2D0; @xh1C@yh2D0;

(1.4)

inft > 0; x 2 T; y 2 R^Cg, where we have replaced yz by y for simplicity of notation.

The second equation of (1.4) implies that the leading order of boundary layers for the total pressurep.t; x; y/is invariant across the boundary layer and should be matched to the outflow pressureP .t; x/on top of the boundary layer, that is, the trace of pressure of ideal MHD flow. Consequently, we have

p.t; x; y/P .t; x/:

It is worth noting that the pressurepz of the fluid may have the leading order of boundary layers because of the appearance of the boundary layer for the magnetic field. It is different from the general fluid in the absence of the magnetic field, for which the leading boundary layer for the pressure of the fluid always vanishes.

The tangential component u1.t; x; y/ of the velocity field and h1.t; x; y/ of magnetic filed, respectively, should match the outflow tangential velocityU.t; x/

and the outflow tangential magnetic fieldH.t; x/on the top of the boundary layer, that is,

(1.5) u₁.t; x; y/ ! U.t; x/; h₁.t; x; y/ ! H.t; x/; asy ! C1; where U.t; x/and H.t; x/ are the trace of the tangential velocity and magnetic field, respectively. Therefore, we have the following “matching” condition:

Ut CU Ux HHx CPx D0; Ht CUHx H Ux D0;

(1.6)

which shows that (1.5) is consistent with the first and third equations of (1.4).

Moreover, on the boundaryfy D0g, the boundary conditions (1.2) and (1.3) give u1

ˇ ˇ_y

D0Du2

ˇ ˇ_y

D0D@yh1

ˇ ˇ_y

D0Dh2

ˇ ˇ_y

D0D0:

(1.7)

On the other hand, it is noted that equation (1.4)₄ is a direct consequence of equations (1.4)3, @xh1C@yh2 D 0in (1.4)5, and the boundary condition (1.7).

Hence, we only need to study the following initial boundary value problem of the MHD boundary layer equations inft 2Œ0; T ; x2T; y2R^Cg,

8 ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

@tu1Cu1@xu1Cu2@yu1 h1@xh1 h2@yh1D@_y²u1 Px;

@th1C@y.u2h1 u1h2/D@_y²h1;

@xu1C@yu2 D0; @xh1C@yh2D0;

u1ˇ ˇ_t

D0Du10.x; y/; h1ˇ ˇ_t

D0 Dh10.x; y/;

.u1; u2; @yh1; h2/ˇ ˇ_y

D0D0; lim

y!C1.u1; h1/D.U; H /.t; x/:

(1.8)

The aim of this paper is to show the local well-posedness of the system (1.8) with nonzero tangential component of that magnetic field, that is, without loss of generality, by assuming

h1.t; x; y/ > 0:

Let us first introduce some weighted Sobolev spaces for later use. Denote

WD f.x; y/W x 2T; y2R_Cg:

For anyl 2R;denote byL²_l./the weighted Lebesgue space with respect to the spatial variables:

L²_l./WD

f .x; y/W !R;

kfk_L²

l./WD Z

hyi^2ljf .x; y/j²dx dy¹₂

<C1

withhyi D 1Cy:Then, for any givenm 2 N;denote byH_l^m./the weighted Sobolev spaces,

H_l^m./WD ff .x; y/W !R; kfkH_l^m./<C1g (1.9)

with the norm

kfkH_l^m./WD X

m₁Cm₂m

hyi^lCm2@^m_x¹@_y^m2f

2 L²./

¹₂ : Now, we can state the main result as follows.

THEOREM1.1. Letm5be an integer andl 0a real number. Assume that the outer flow.U; H; Px/.t; x/satisfies that for someT > 0;

(1.10) M0WD

2mC2

X

iD0

sup

0tTk@ⁱ_t.U; H; P /.t;/kH^{2mC2 i}.Tx/

<C1: Also, we assume the initial data.u10; h10/.x; y/satisfies

(1.11) .u10.x; y/ U.0; x/; h10.x; y/ H.0; x//2H_l^3mC2./;

and the compatibility conditions up tom^th order. Moreover, there exists a suffi- ciently small constantı0> 0such that

(1.12) ˇ

ˇhyi^lC1@_yⁱ.u₁₀; h₁₀/.x; y/ˇ

ˇ.2ı₀/ ¹ fori D1; 2; .x; y/2;

h10.x; y/2ı0:

Then, there exist a positive time0 < TT and a unique solution.u1; u2; h1; h2/ to the initial boundary value problem(1.8)such that

.u1 U; h1 H / 2

m

\

iD0

W^i;1 0; TIH_l^{m i}./

; and

.u2CUxy; h2CHxy/2

m 1

\

iD0

W^i;1 0; TIH^{m 1 i}₁ ./

;

.@yu2CUx; @yh2CHx/2

m 1

\

iD0

W^i;1 0; TIH_l^{m 1 i}./

:

Moreover, ifl > ¹₂;

.u2CUxy; h2CHxy/ 2

m 1

\

iD0

W^i;1 0; TIL¹ Ry;CIH^{m 1 i}.Tx/ : Remark1.2. Note that the regularity assumption on the outflow.U; H; P /and the initial data.u10; h10/is not optimal. Here, we need the regularity to simplify the construction of approximate solution; cf. Section 4. One may relax the regularity requirement by using other approximations.

Let us briefly explain the main idea of the proof. Similar to the Prandtl equa- tions, the main difficulty in the analysis on the system (1.8) in the Sobolev frame- work is the loss ofx-derivatives in the vertical componentsu2 D @x@_y¹u1and h2D @x@_y¹h1with@_y¹f WDRy

0 f dyzappearing in the termsu2@yu1 h2@yh1

andu2@yh1 h2@yu1in the first and second equations of (1.8), respectively. This loss of x-derivative is coupled with the lack of any horizontal diffusion so that some kind of cancellation mechanism has to be used in the analysis. Our analysis is based on two new observations. One observation is that WD@_y¹h1satisfies

@t Cu2h1 u1h2D@_y² :

Another observation is that under the assumption on the nondegeneracy of h1, instead of estimating @^m_x.u1; h1/; m 2 N;we consider the following unknown functions to capture the cancellation:

umWD@^m_xu1

@yu1

h1

@^m_x ; hmWD@^m_xh1

@yh1

h1

@^m_x :

With the help of.um; hm/, the difficulty in the analysis on@^m_xu2@yu1 @^m_xh2@yh1

and@^m_xu2@yh1 @^m_xh2@yu1mentioned above can be overcome. Since@^m_x.u1; h1/ and.um; hm/ are equivalent in the Sobolev framework under the setting in this paper, the loss of thx-derivative can be avoided; see Section 3.2 for the detailed discussion. We also point out that in Section 5, a nonlinear coordinate transfor- mation in the spirit of the classical Crocco transformation to the system (1.8) is introduced, and it provides another approach to study the system with a similar well-posedness result.

We now review some related works to the problem studied in this paper. First of all, the study on fluid around a rigid body with high Reynolds numbers is an important problem in both physics and mathematics. The classical work can be traced back to Prandtl in 1904 about the derivation of the Prandtl equations for boundary layers from the incompressible Navier-Stokes equations with non-slip boundary condition; cf. [33]. About 60 years after its derivation, the first systematic work in rigorous mathematics was achieved by Ole˘ınik (cf. [31]), in which she showed that under the monotonicity condition on the tangential velocity field in the normal direction to the boundary, local-in-time well-posedness of the Prandtl system can be justified in 2D by using the Crocco transformation. This result

together with some extensions are presented in Oleinik-Samokhin’s classical book [32]. Recently, this well-posedness result was proved by simply using an energy method in the framework of Sobolev spaces in [1, 30] independently by taking care of the cancellation in the convection terms to overcome the loss of derivative in the tangential direction. Moreover, by imposing an additional favorable condition on the pressure, a global-in-time weak solution was obtained in [39]. Some three space dimensional cases were studied for both classical and weak solutions in [23, 25].

Since Oleinik’s classical work, the necessity of the monotonicity condition on the velocity field for well-posedness remained as a question until 1980s when Caflisch and Sammartino [35, 36] obtained the well-posedness in the framework of analytic functions without this condition (cf. [18–20, 28, 29, 41] and the references therein).

Recently the analyticity condition was further relaxed to Gevrey regularity; cf.

[10, 11, 21, 22].

When the monotonicity condition is violated, separation of the boundary layer is expected and observed for classical fluid. For this, E-Engquist constructed a finite-time blowup solution to the Prandtl equations in [8]. Recently, when the background shear flow has a nondegenerate critical point, some interesting ill- posedness (or instability) phenomena of solutions to both the linear and nonlinear Prandtl equations around the shear flow have been studied; cf. [9, 12, 14, 15, 24, 27]

and the references therein. All these results show that the monotonicity assumption on the tangential velocity is essential for the well-posedness except in the frame- work of analytic functions or Gevrey functions.

On the other hand, for electrically conducting fluid such as plasmas and liquid metals, the system of magnetohydrodynamics (denoted by MHD) is a fundamental system to describe the movement of fluid under the influence of an electromagnetic field. The study on the MHD was initiated by Alfvén [2], who showed that the mag- netic field can induce current in a moving conductive fluid with a new propagation mechanism along the magnetic field, called Alfvén waves.

For plasma, the boundary layer equations can be derived from the fundamen- tal MHD system, and they are more complicated than the classical Prandtl system because of the coupling of the magnetic field with the velocity field through the Maxwell equations. On the other hand, in physics, it is believed that the magnetic field has a stabilizing effect on the boundary layer that could provide a mecha- nism for containment of, for example, the high-temperature gas. If the magnetic field is transversal to the boundary, there are extensive discussions on the so-called Hartmann boundary layer; cf. [5, 16, 17]. In addition, there are works on the sta- bility of boundary layers with minimum Reynolds number for flow with different structures to reveal the difference from the classical boundary layers without an electromagnetic field; cf. [3, 6, 34].

In terms of mathematical derivation when the non-slip boundary condition for the velocity is present, the boundary layer systems that capture the leading order of fluid variables around the boundary depend on three physical parameters: the

magnetic Reynolds number, the Reynolds number, and their ratio, called the mag- netic Prandtl number. When the Reynolds number tends to infinity while the mag- netic Reynolds number is fixed, the derived boundary layer system is similar to the Prandtl system for classical fluid and its well-posedness was discussed in Ole˘ınik- Samokhin’s book [32], for which the monotonicity condition on the velocity field is needed. When the Reynolds number is fixed while the magnetic Reynolds num- ber tends to infinity, which corresponds to an infinite magnetic Prandtl number, the boundary layer system is similar to an inviscid Prandtl system, and the monotonic- ity condition on the velocity field is not needed for well-posedness. For the case with the finite magnetic Prandtl number, i.e., both the Reynolds number and the magnetic Reynolds number tend to infinity at the same rate, the boundary layers system is totally different from the classical Prandtl system, and this is the system to be discussed in this paper. Note that for this system, no mathematical well- posedness results have been obtained so far in the Sobolev spaces. Furthermore, we mention that in [38], the authors establish the vanishing viscosity limit for the MHD system in a bounded smooth domain ofR^d, d D 2; 3;with a slip bound- ary condition, while the leading order of boundary layers for both velocity and the magnetic field vanishes because of the slip boundary conditions.

To be precise, in this paper, to capture the stabilizing effect of the magnetic field, we establish the well-posedness theory for the problem (1.8) without any monotonicity assumption on the tangential velocity. The only essential condition is that the background tangential magnetic field have a lower positive bound. Hence, the result in this paper enriches the classical local well-posedness results of the classical Prandtl equations. At the same time, it is in agreement with the general physical understanding that the magnetic field stabilizes the boundary layer.

The rest of the paper is organized as follows. Some preliminaries are given in Section 2. In Section 3, we establish the a priori energy estimates for the nonlinear problem (1.8). The local-in-time existence and uniqueness of the solution to (1.8) in Sobolev space are given in Section 4. In Section 5, we introduce another method for the study of the well-posedness theory for (1.8) by using a nonlinear coordi- nate transformation in the spirit of Crocco transformation for the classical Prandtl system. Finally, the technical proof of a lemma is given in the Appendix.

2 Preliminaries

First, we introduce some notation. Use the tangential derivative operator

@^ˇ D@^ˇ_t¹@^ˇ_x² forˇD.ˇ1; ˇ2/2N²; jˇj Dˇ1Cˇ2; and then denote the derivative operator (in both time and space) by

D^˛D@^ˇ@_y^k for˛ D.ˇ1; ˇ2; k/2N³; j˛j D jˇj Ck:

Setei 2N²,i D1; 2;andEj 2N³,j D1; 2; 3;by

e1D.1; 0/2N²; e2D.0; 1/2N²;

and

E1D.1; 0; 0/2N³; E2D.0; 1; 0/2N³; E3D.0; 0; 1/2N³: Denote by @_y¹ the inverse of the derivative@y, i.e., .@_y¹f /.y/ WD Ry

0 f .´/d´:

Moreover, we use the notationŒ;to denote the commutator, and denote a non- decreasing polynomial function byP./, which may differ from line to line.

Form2N;define the function spacesH^m_l of measurable functionsf .t; x; y/W Œ0; T !Rsuch that for anyt 2Œ0; T ;

kf .t /kH^m_l WD X

j˛jm

khyi^lCkD^˛f .t;/k_L²²_./¹₂

<C1: (2.1)

The following inequalities will be used frequently in this paper, whose proofs are given in the Appendix.

LEMMA2.1. For proper functionsf; g; h, the following holds:

(i) Iflimy!C1.fg/.x; y/D0;then (2.2)

ˇ ˇ ˇ ˇ Z

Tx

.fg/ˇ ˇ_y

D0dx ˇ ˇ ˇ

ˇ k@yfkL²./kgkL²./C kfkL²./k@ygkL²./: In particular, iflimy!C1f .x; y/D0;then

(2.3)

fjyD0

_L2.Tx/p 2kfk

1 2

L²./k@yfk

1 2

L²./:

(ii) Forl 2Rand an integerm3;any˛D.ˇ; k/2N³,˛zD.ˇ;z k/z 2N³, withj˛j C jz˛j m,

(2.4) k.D^˛f D^˛zg/.t;/kL²

lCkC zk./Ckf .t /kH^m

l1kg.t /kH^m

l2; 8l1; l2 2R; l1Cl2 Dl:

(iii) For any > ¹₂,z > 0,

(2.5)

hyi .@_y¹f /.y/

_L2

y.R_C/ 2

2 1khyi¹ f .y/kLy².R_C/;

hyi ^z.@_y¹f /.y/

_L1

y .R_C/ 1

zkhyi^{1 z}f .y/kL¹_y .R_C/;

and then, forl 2R, an integerm3;and any˛ D.ˇ; k/2N³,ˇzD.ˇz1;ˇz2/2 N²withj˛j C j zˇj m,

D^˛g@^ˇz@_y¹h .t;/

_L2

lCk./Ckg.t /kH^m_l

Ckh.t /kH^m₁ : (2.6)

In particular, forD1;

(2.7)

hyi ¹ @_y¹f .y/

_L2

y.R_C/2kfk_Ly²_.RC/;

D^˛g@^ˇz@_y¹h .t;/

_L2

lCk./Ckg.t /kH^m_l

C1kh.t /kH^m₀:

For any > ¹₂,

@_y¹f .y/

_L1

y .R_C/CkfkL_y;² .R_C/; (2.8)

and then, forl 2R, an integerm2;and any˛D.ˇ; k/2N³;ˇzD.ˇz1;ˇz2/2 N²withj˛j C j zˇj m,

D^˛f @^ˇz@_y¹g .t;/

_L2

lCk./Ckf .t /kH^m_l kg.t /kH^m: (2.9)

To overcome the technical difficulty originating from the boundary terms at fy D C1g, we introduce an auxiliary function.y/2C¹.R_C/satisfying

.y/D

(y; y 2R0; 0; 0yR0; for some constantR₀ > 0. Then, set the new unknowns:

(2.10)

u.t; x; y/WDu1.t; x; y/ U.t; x/⁰.y/;

v.t; x; y/WDu2.t; x; y/CUx.t; x/.y/;

h.t; x; y/WDh1.t; x; y/ H.t; x/⁰.y/;

g.t; x; y/WDh2.t; x; y/CHx.t; x/.y/:

Choose the above construction for.u; v; h; g/to ensure the divergence-free condi- tions and homogenous boundary conditions, i.e.,

@xuC@yvD0; @xhC@ygD0;

.u; v; @yh; g/j^yD0D0; lim

y!C1.u; h/D0;

which implies thatvD @_y¹@xuandgD @_y¹@xh:It is easy to get that .u; h/.t; x; y/D u1.t; x; y/ U.t; x/; h1.t; x; y/ H.t; x/

C U.t; x/.1 ⁰.y//; H.t; x/.1 ⁰.y//

; which implies that by the construction of.y/,

(2.11) k.u; h/.t /kH^m_l CM0 k.u1 U; h1 H /.t /kH^m_l

k.u; h/.t /kH^m_l CCM0:

By using the new unknowns.u; v; h; g/ given by (2.10), we can reformulate the original problem (1.8) as follows:

8 ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ<

ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ:

@_tuCŒ.uCU⁰/@_xC.v U_x/@_yu Œ.hCH⁰/@_xC.g H_x/@_yh

@²_yuCUx⁰uCU⁰⁰v Hx⁰h H⁰⁰gDr1;

@_thCŒ.uCU⁰/@_xC.v U_x/@_yh Œ.hCH⁰/@_xC.g H_x/@_yu

@²_yhCHx⁰uCH⁰⁰v Ux⁰h U⁰⁰gDr2;

@_xuC@_yvD0; @_xhC@_ygD0;

.u; v; @_yh; g/jyD0D0;

.u; h/jtD0D u10.x; y/ U.0; x/⁰.y/; h10.x; y/ H.0; x/⁰.y/

,.u₀; h₀/.x; y/;

(2.12)

where

(r1DUtŒ.⁰/^{2 00 0}CPx

.⁰/^{2 00} 1

CU^.3/; r2DHtŒ.⁰/²C^{00 0}CH^.3/:

(2.13)

Note that we have used the divergence-free conditions in obtaining the equations of.u; h/ in (2.12), and the relations (1.6) in the calculation of (2.13). It is worth noting that by substituting (2.10) into the second equation of (1.8) directly, there is another equivalent form for the equation ofh, which may be convenient to use in some situations:

(2.14) @thC@yŒ.v Ux/.hCH⁰/ .uCU⁰/.g Hx/ @_y²hD Ht⁰CH^.3/: By the choice of.y/, it is easy to get that

r1.t; x; y/; r2.t; x; y/0; y 2R0; r1.t; x; y/ Px.t; x/; r2.t; x; y/0; 0y R0; (2.15)

and then for anyt 2Œ0; T ,0, andj˛j m, by virtue of (1.10),

hyi D^˛r1; D^˛r2

.t /

_L2./C X

jˇjj˛jC1

@^ˇ.U; H; Px/.t / _L2.Tx/

CM0: (2.16)

Furthermore, similarly to (2.11), we have that for the initial data k.u0; h0/kH_l^2m./ CM0

k.u10.x; y/ U.0; x/; h10 H.0; x//kH_l^2m./

k.u0; h0/kH_l^2m./CCM0: (2.17)

Finally, from the transformation (2.10) and the relations (2.11) and (2.17), it is easy to see that Theorem 1.1 is a corollary of the following result.

THEOREM2.2. Letm5be an integer,l 0a real number, and.U; H; Px/.t; x/

satisfy the hypotheses given in Theorem1.1. In addition, assume that for the prob- lem(2.12), the initial data .u0.x; y/; h0.x; y// 2 H_l^3mC2./and satisfies the compatibility conditions up tom^thorder. Moreover, there exists a sufficiently small constantı0 > 0such that for.x; y/2;

(2.18)

ˇ

ˇhyi^lC1@ⁱ_y.u0; h0/.x; y/ˇ

ˇ.2ı0/ ¹; i D1; 2;

h0.x; y/CH.0; x/⁰.y/2ı0;

Then, there exist a time0 < TT and a unique solution.u; v; h; g/to the initial boundary value problem(2.12)such that

.u; h/2

m

\

iD0

W^i;1.0; TIH_l^{m i}.//

(2.19) and

(2.20)

.v; g/2

m 1

\

iD0

W^i;1 0; TIH^{m 1 i}₁ ./

;

.@yv; @yg/2

m 1

\

iD0

W^i;1 0; TIH_l^{m 1 i}./

: Moreover, ifl > ¹₂;

.v; g/2

m 1

\

iD0

W^i;1 0; TIL¹ Ry;CIH^{m 1 i}.Tx/ : (2.21)

Therefore, our main task is to prove Theorem 2.2, and its proof will be given in the following two sections.

3 A Priori Estimates

In this section, we will establish a priori estimates for the nonlinear problem (2.12).

PROPOSITION3.1 (Weighted estimates forD^m.u; h/). Letm 5be an integer, l 0be a real number, and the hypotheses for.U; H; Px/.t; x/given in Theorem 1.1hold. Assume that.u; v; h; g/ is a classical solution to the problem(2.12)in Œ0; T satisfying that

.u; h/2L¹.0; TIH^m_l /; .@yu; @yh/2L².0; TIH_l^m/:

Moreover, for.t; x; y/2Œ0; T and sufficiently small constantı0> 0, (3.1)

h.t; x; y/CH.t; x/⁰.y/ı0;

hyi^lC1@_yⁱ.u; h/.t; x; y/ı₀¹; i D1; 2:

Then, it holds that for small time, sup

0stk.u; h/.s/kH^m_l

ı₀⁴ P.M0C k.u0; h0/kH_l^2m.//CCM₀⁶t¹₂ ˚

1 C ı₀²⁴ P M0C k.u0; h0/kH_l^2m./

CCM₀⁶t2

t

1 4: (3.2)

Also, we have that fori D1; 2;

hyi^lC1@_yⁱ.u; h/.t / _L1./

hyi^lC1@_yⁱ.u0; h0/ _L1./

CC ı₀⁴t P M0C k.u0; h0/kH_l^2m./

CCM₀⁶t¹₂ ˚

1 C ı₀²⁴ P M0C k.u0; h0/kH_l^2m./

CCM₀⁶t2

t

1 4

(3.3)

and

h.t; x; y/h0.x; y/ C ı₀⁴t P M0C k.u0; h0/kH_l^2m./

CCM₀⁶t¹₂ ˚

1 C ı₀²⁴ P M0C k.u0; h0/kH_l^2m./

CCM₀⁶t2

t

1 4: (3.4)

The proof of Proposition 3.1 will be given in the following two subsections.

Specifically, we will obtain the weighted estimates forD^˛.u; h/for˛ D.ˇ; k/D .ˇ1; ˇ2; k/satisfyingj˛j D jˇj Ckmandjˇj m 1, in the first subsection, and the weighted estimates for@^ˇ.u; h/withjˇj Dmin the second subsection.

3.1 WeightedH_l^m-Estimates with Normal Derivatives

The weighted estimates onD^˛.u; h/withj˛j D jˇj Ck mandjˇj m 1 can be obtained by the standard energy method because one order tangential regu- larity loss is allowed. That is, we have the following estimates:

PROPOSITION3.2 (Weighted estimates forD^˛.u; h/withj˛j m;jˇj m 1).

Letm 5be an integer, let l 0 be a real number, and let the hypotheses for .U; H; Px/.t; x/given in Theorem1.1hold. Assume that.u; v; h; g/is a classical solution to the problem(2.12)inŒ0; T and satisfies

.u; h/2L¹ 0; TIH^m_l

; .@yu; @yh/2L² 0; TIH^m_l :

Then, there exists a positive constantC, depending onm,l, and, such that for any small0 < ı1< 1;

(3.5)

X

j˛jm jˇjm 1

d

dtkD^˛.u; h/.t /k²_L²

lCk./CkD^˛@yu.t /k_L²²

lCk./

CkD^˛@yh.t /k²_L2 lCk./

ı1Ck.@yu; @yh/.t /k²_H^m

0 CC ı₁¹k.u; h/.t /k²_H^m

l 1C k.u; h/.t /k²_H^m

l

C X

j˛jm jˇjm 1

kD^˛.r1; r2/.t /k_L²²

lCk./CC X

jˇjmC2

@^ˇ.U; H; P /.t /

2 L².Tx/:

PROOF. Applying the operatorD^˛ D@^ˇ@^k_y for˛ D .ˇ; k/D .ˇ1; ˇ2; k/and satisfyingj˛j D jˇj Ck m; jˇj m 1in the first two equations of (2.12) yields

8 ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

@tD^˛uDD^˛r1C@²_yD^˛u D^˛˚

Œ.uCU⁰/@xC.v Ux/@yu Œ.hCH⁰/@xC.g Hx/@yh

CUx⁰uCU⁰⁰v Hx⁰h H⁰⁰g ;

@tD^˛hDD^˛r2C@_y²D^˛h D^˛˚

Œ.uCU⁰/@x C.v Ux/@yh Œ.hCH⁰/@xC.g Hx/@yu

CHx⁰uCH⁰⁰v Ux⁰h U⁰⁰g : (3.6)

Multiplying (3.6)₁ byhyi^2lC2kD^˛uand (3.6)₂byhyi^2lC2kD^˛hand integrating them overwith respect to the spatial variablesxandy, we obtain that

(3.7)

1 2

d

dtkhyi^lCkD^˛.u; h/.t /k_L²²_./

D Z



D^˛r₁ hyi^2lC2kD^˛uCD^˛r₂ hyi^2lC2kD^˛h dx dy C

Z



@²_yD^˛u hyi^2lC2kD^˛u dx dy C

Z



@_y²D^˛h hyi^2lC2kD^˛h dx dy Z



I1 hyi^2lC2kD^˛uCI2 hyi^2lC2kD^˛h dx dy;

where

8 ˆˆ ˆˆ ˆˆ ˆ<

ˆˆ ˆˆ ˆˆ ˆ:

I1DD^˛˚

Œ.uCU⁰/@x C.v Ux/@yu Œ.hCH⁰/@x C.g Hx/@yh CUx⁰uCU⁰⁰v Hx⁰h H⁰⁰g ; I2DD^˛˚

Œ.uCU⁰/@x C.v Ux/@yh Œ.hCH⁰/@x C.g Hx/@yu CHx⁰uCH⁰⁰v Ux⁰h U⁰⁰g : (3.8)

First of all, it is easy to see that by virtue of (2.16), (3.9)

Z



D^˛r1 hyi^2lC2kD^˛uCD^˛r2 hyi^2lC2kD^˛h

dx dy 1

2kD^˛.u; h/.t /k_L²²

lCk./C1

2kD^˛.r1; r2/.t /k²_L²

lCk./: Next, we assume that the following two estimates hold, which will be proved later:

for any small0 < ı1< 1;

(3.10)

Z



@_y²D^˛u hyi^2lC2kD^˛u dx dy C

Z



@_y²D^˛h hyi^2lC2kD^˛h dx dy

2kD^˛@yu.t /k²_L²

lCk./

2kD^˛@yh.t /k_L²²

lCk./

Cı₁k.@_yu; @_yh/.t /k²_H^m

0

CC ı₁¹k.u; h/.t /k²_H^m

l 1C k.u; h/.t /k²_H^m

l

CC X

jˇjm 1

@^ˇPx.t /

2 L².Tx/; and

(3.11) Z



I1 hyi^2lC2kD^˛uCI2 hyi^2lC2kD^˛h

dx dy

C X

jˇjmC2

@^ˇ.U; H /.t / _L2.T

x/C k.u; h/.t /kH^m_l

k.u; h/.t /k²H^m_l : At the moment, by plugging inequalities (3.9)–(3.11) into (3.7) and then summing over˛, we obtain that there exists a constantCm> 0, depending only onm;such that

X

j˛jm jˇjm 1

d dt

D^˛.u; h/.t /

2 L²_l

Ck./C

D^˛@yu.t /

2 L²_l

Ck./

CkD^˛@yh.t /k_L²²

lCk./

ı1Cmk.@yu; @yh/.t /k²_H^m

0 CC ı₁¹

.u; h/.t /k²_H^m

l 1C k.u; h/.t /

2 H^m_l

C X

j˛jm jˇjm 1

kD^˛.r1; r2/.t /k_L²²

lCk./

CC X

jˇjmC2

@^ˇ.U; H; P /.t /

2 L².Tx/;

which implies the estimate (3.5) immediately.

Now, it remains to show the estimates (3.10) and (3.11).

PROOF OF(3.10). In this part, we will first handle the term R

.@_y²D^˛u hyi^2lC2kD^˛u/dx dy; the term R

.@_y²D^˛h hyi^2lC2kD^˛h/dx dy can be esti- mated similarly. By integration by parts, we have

(3.12)

Z



@²_yD^˛u hyi^2lC2kD^˛u dx dy D

hyi^lCk@yD^˛u.t /

2 L²./

C2.lCk/

Z

 hyi^{2lC2k 1}@yD^˛uD^˛u dx dy C

Z

Tx

.@yD^˛uD^˛u/ˇ ˇ_y

D0dx:

By the Cauchy-Schwarz inequality, (3.13) 2.lCk/

Z

 hyi^{2lC2k 1}@yD^˛uD^˛u

dx dy

14

hyi^lCk@yD^˛u.t /

2

L²./C14.lCk/²khyi^lCkD^˛u.t /k²_L²_./; which implies that by plugging (3.13) into (3.12),

(3.14)

Z



@_y²D^˛u hyi^2lC2kD^˛u dx dy

13

14

hyi^lCkD^˛@yu.t /

2

L²./CCku.t /k²_H^m

l

C Z

Tx

.@yD^˛uD^˛u/ˇ ˇ_y

D0dx:

The last term in (3.14), that is, the boundary integralR

Tx.@yD^˛uD^˛u/ˇ ˇ_y

D0dx, is treated in the following two cases.

Case 1. j˛j m 1:By the inequality (2.2), we obtain that for any small 0 < ı1 < 1;

(3.15)

ˇ ˇ ˇ ˇ Z

Tx

.@yD^˛uD^˛u/ˇ ˇ_y

D0dx ˇ ˇ ˇ ˇ

@_y²D^˛u.t / _L2./

D^˛u.t / _L2./

C

@yD^˛u.t /

2 L²./ı1

@²_yD^˛u.t /

2 L²./

C ²

4ı1kD^˛u.t /k_L²²_./Ck@yD^˛u.t /k²_L²_./

ı1k@yu.t /kH^m₀ CC ı₁¹ku.t /k²H^m₀:

Case2. j˛j D jˇjCkDm. This equation implies thatk1fromjˇj m 1:

Then, if we denote ,˛ E3 D.ˇ; k 1/withjj D jˇj Ck 1Dm 1, the first equation in (2.12) reads that by virtue of@yD^˛D@_y²D;

@yD^˛uDD˚

@tuCŒ.uCU⁰/@xC.v Ux/@yu Œ.hCH⁰/@x C.g Hx/@yh

CUx⁰uCU⁰⁰v Hx⁰h H⁰⁰g r1 : Then, combining (2.15) with the fact 0foryR0, we get atyD0

(3.16)

@yD^˛uDDŒ@tuC.u@xCv@y/u .h@x Cg@y/hCPx DDPx CD^CE1uCD.u@xu h@xh/

CD.v@yu g@yh/:

It is easy to obtain from (2.3) that

(3.17) ˇ ˇ ˇ ˇ Z

Tx

.DPxD^˛u/ˇ ˇyD0dx

ˇ ˇ ˇ ˇ

kDPx.t /kL².Tx/kD^˛u.t /j^yD0kL².Tx/

p

2kDPx.t /kL².Tx/kD^˛u.t /k

1 2

L²./kD^˛@yu.t /k

1 2

L²./

14kD^˛@yu.t /k²_L²_./CCku.t /k²_H^m

0 CCkDPx.t /k_L²²_.Tx/; providedj˛j Dm:Also, by (2.2) andj CE₁j Dm,

(3.18) ˇ ˇ ˇ ˇ Z

Tx

.D^CE1uD^˛u/ˇ ˇ_y

D0dx ˇ ˇ ˇ ˇ

D^CE1@yu.t /

_L2./kD^˛u.t /kL²./

C kD^CE1u.t /kL²./kD^˛@yu.t /kL²./

ı1

3kD^CE1@yu.t /k²_L²_./C

14kD^˛@yu.t /k²_L²_./

CC ı₁¹ku.t /k²H^m₀: Next, as we know

D.u@xu DX

z

z

.D^zuD ^zCE2u/;