Acta Appl Math https://doi.org/10.1007/s10440-018-0197-7

Global Well-Posedness of an Initial-Boundary Value Problem of the 2-D Incompressible Navier-Stokes-Darcy System Pan Liu1 · Wenjuan Liu1

Received: 8 December 2017 / Accepted: 11 June 2018 © Springer Nature B.V. 2018

Abstract We investigate in the paper the initial boundary value problem of the two dimensional incompressible Navier-Stokes-Darcy system in a strip domain. It is shown that, without any small initial data assumption, the Navier-Stokes-Darcy equations have a unique global strong solution in the strip domain with a flat interface. The key is to establish the global-in-time regularity uniformly by pursuing the properties of Dirichlet-Neumann operator. Keywords Navier-Stokes-Darcy system · Global well-posedness · Dirichlet-Neumann operator Mathematics Subject Classification 35A07 · 74F10 · 76D03

1 Introduction We study herein the evolution of a two-dimensional coupling of the incompressible NavierStokes flow and Darcy flow separated by a flat interface where two kinds of fluid flow interact each other. Darcy equation formulates the Darcy’s law [13], which describes the motion of Darcy flow—flow of a fluid through a porous medium, where the fluid velocity u is characterized by u = −K∇p, here the matrix K denotes the permeability of porous medium and the scalar function p is the pressure. It was extensively applied in many physical models including porous media, such as the Hele-Shaw cell and laminar flow through sediments [10, 11]. For

B P. Liu

[email protected] W. Liu [email protected]

1

School of Mathematics, Northwest University, Xi’an, 710127, China

P. Liu, W. Liu

the incompressible fluid, the Darcy system reads in the following form [14, 15] ⎧ ⎪ ⎨− div(K∇p) = 0 in Ω(t), p = γκ on ∂Ω(t), ⎪ ⎩ −K∇p · n = v on ∂Ω(t), which has been developed by many physicists and mathematicians. H. Knüpfer and N. Masmoudi [14] proved the well-posedness and uniform bounds for Darcy’s flow on an infinite wedge. Furthermore, they showed the well-posedness of the Darcy’s flow with prescribed contract angle in [15]. Recently, more and more attentions have been devoted to the coupled fluid systems involving an initial-boundary value in a bounded open domain. In this tendency, the coupled two-dimensional incompressible Navier-Stokes-Darcy equations have already got preliminary development. Here the Navier-Stokes-Darcy system takes the form (see [2, 3, 8–10, 12, 13, 17, 19, 20, 22]): ⎧ ∂t u + u · ∇u − div(2μD(u) − p1 I ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ div u=0 ⎪ ⎪ ⎪ ⎪ ⎪− div(K∇p2 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨u · n = −K∇p2 · n u · τ = −2μG(D(u)n) · τ ⎪ ⎪ ⎪((−2μD(u) + p1 I )n) · n + 1 (u · u) = p2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ u = 0 ⎪ ⎪ ⎪ ⎪ ⎪p2 = 0 ⎪ ⎪ ⎩ u = u0

in (0, T ) × Ω1 , in (0, T ) × Ω1 , in (0, T ) × Ω2 , on (0, T ) × Γ, on (0, T ) × Γ, on (0, T ) × Γ, on (0, T ) × Γ1 , on (0, T ) × Γ2 , at {t = 0} × Ω1 ,

(1.1)

where Ωi ⊆ R2 (i = 1, 2) are two connected domain and the interface Γ = ∂Ω1 ∩ ∂Ω2 . u(t, x) = (u1 (t, x), u2 (t, x)) (x ∈ Ω1 ) denotes the velocity in Ω1 . pi = pi (t, x) (x ∈ Ωi ) denotes the pressure of the fluid in Ωi (i = 1, 2). D(u) = 12 [∇u + (∇u)T ] means the deformation tensor. The fluid viscosity μ is a positive constant, I is the 2 × 2 identity matrix, and the real constant symmetric matrix K is positive definite, which rely on the permeability of Ω2 . The domain Ω ⊂ R2 can be decomposed into two disjoint subdomains Ω1 and Ω2 separated by an interface Γ : Ω = Ω 1 ∪ Ω 2 , Ω1 ∩ Ω2 = ∅ and Ω 1 ∩ Ω 2 = Γ . We denote the boundary of Ωi by ∂Ωi with outer unit normal ni and Γi = ∂Ωi \ Γ for i = 1, 2. In addition, n is equal to n1 on Γ and τ is the tangential unit vector to Γ . (See Fig. 1.) Equation (1.1)1 is the balance of the momentum, (1.1)2,3 mean that the fluid is incompressible, and the condition (1.1)4 describes the continuity of the normal component of the velocity across the interface Γ . We denote (1.1)5 the Beaver-Joseph-Saffman law [4, 9, 21] with a positive constant G > 0, which usually obtained from experimental data. And the boundary value condition (1.1)6 means the balance of forces across the interface, which balance of forces includes the inertial force. In [9, 13], this interface condition was considered in the steady-state coupling of Navier-Stokes and Darcy equations. From the physical point of view, Γ is a surface separating the domain Ω1 filled by a fluid, and the domain Ω2 formed by a porous medium. We assume that the fluid contained in Ω1 has a fixed surface, namely, we do not consider the free surface fluid cases, and can filtrate through the adjacent porous medium. We also suppose that the fluid contained in Ω is incompressible. This model can be used to describe the filtration of blood through the

Global Well-Posedness of an Initial-Boundary Value Problem. . .

Fig. 1 Coupled domains Ω with interface Γ

arterial wall and also the simulation of transport of contaminants through rivers into the aquifers [3, 19]. We assume in the paper that Ω ⊆ R2 is a strip domain, Ω = E×(−a, b), Ω1 = E×(0, b), and Ω2 = E × (−a, 0) with E = T or E = R. Correspondingly, Γ = E × {y = 0}, Γ1 = E × {y = b}, and Γ2 = E × {y = −a} are all rigid and flat (see Fig. 1). Here and hereafter, we assume that a, b > 0 are two fixed and given constants. Remark 1.1 Without loss of generality, we may assume that μ = G = 1. To simplify subsequent calculations, it is convenient to assume that the boundary Γ = Ω 1 ∩ Ω 2 is flat. In fact, if Γ = E × {y = η(x)} is a small perturbation around the flat interface E × {y = 0}, then we can as usual straighten out the boundary to obtain the setting above by changing variables. Higher regularity for our solution requires compatibility conditions on the initial data. Throughout this paper, we assume that the following compatibility conditions hold   (1.2) ut |t=0 = div 2μD(u0 ) − p1 |t=0 I − u0 ∇u0 . Before presenting our results, we introduce the following functional space to describe the velocity and the pressure  HΓmi (Ω1 ) = f ∈ H m (Ω1 )|u = 0 on Γi , i = 1, 2, which is a Banach space equipped with the H m (Ωi ) norm. Our main result of this paper is to establish the global well-posedness of the NavierStokes-Darcy equations (1.1), which can be stated as follows. Theorem 1.1 Assume that the real symmetric constant matrix K is uniformly bounded and strictly elliptic, i.e., there are two positive constants Λ > λ > 0 such that λ|ξ |2  Kξ · ξ  Λ|ξ |2 ,

∀ξ ∈ R2 .

(1.3)

Suppose also that the initial data u0 ∈ HΓ21 (Ω1 ) is divergence free and satisfies the compatibility conditions (1.2). Then the Navier-Stokes-Darcy system (1.1) has a unique global solution (u, p2 ) on [0, +∞) such that       u ∈ C [0, +∞ ; HΓ21 (Ω1 )) ∩ C 1 [0, +∞ ; HΓ11 (Ω1 )) ∩ L2loc [0, +∞ ; HΓ31 (Ω1 ));     p2 ∈ L∞ [0, +∞ ; HΓ22 (Ω2 )) ∩ L2loc [0, +∞ ; HΓ32 (Ω2 )).

P. Liu, W. Liu

An approach to prove the Theorem 1.1 is to demonstrate the local well-possedness and the continuation criterion of the initial-boundary value problem (1.1). The main difficulty for the proof of global well-posedness lies in the uniform estimates for the normal derivatives of velocity and pressure on the interface, which are all unknown but we have get to establish estimates of the higher derivative. This difficulty will be overcome by a serious of careful energy estimates and delicate elliptic estimates, which will be elaborated detailedly via the using of the boundness of the Dirichlet-Neumann operator in Sect. 3. Structure of the Paper In Sect. 2, we present some basic inequalities in Sobolev space and the standard regularity theory of elliptic system. In Sect. 3, we establish the uniform a priori estimates of (1.1). The local well-posedness will be given in Sect. 4. Then the global well-posedness will be showed in Sect. 5. Notations Throughout this paper, · X denotes the norm of the Banach space X. (·, ·)Ω and (·, ·)Γ will mean the L2 -inner product in the domain Ω and the boundary Γ of some domain respectively. We apply the notation A : B to denote the scalar product of the matrix A = (aij ) and B = (bij ), meaning that A : B = aij bij (here and hereafter the Einstein sum1 mation convention is always employed), correspondingly, |A| = (A : A) 2 to be the norm of A. We also write AT and tr A to mean the transpose and trace of the matrix A, respec∂ ∂ , and ∂y = ∂2 = ∂y . We will also use ∇u tively. For simplicity, we set ∂t = ∂t∂ , ∂x = ∂1 = ∂x 2 and D u to denote the gradient and the Hessian matrix of the function u, respectively. The Fourier transform of the function f is defined by f (ξ ) = F [f ](ξ ) = RN f (x)e−2π ix·ξ dx (∀ξ ∈ RN ). We also apply the notation a  b to mean that a  Cb for a universal constant C > 0, which may be different on different lines.

2 Preliminaries In this preliminary section, we will give some useful lemmas and the standard regularity theory of elliptic equations which would be used in the next three sections. Firstly, let us recall some conclusions in Sobolev spaces. Lemma 2.1 (See [9, 10, 18]) Let Ω ⊂ R2 be an open, connected domain with C 1 boundary, as Fig. 1, and its boundary can be decomposed as ∂Ω = Γ ∪ Γ0 with Γ0 = ∅. Then there is a positive constant C > 0 which only depends on Ω such that for all u ∈ HΓ10 (Ω), (i) u L2 (Γ )  C ∇u L2 (Ω) , u L4 (Γ )  C ∇u L2 (Ω) ; (ii) u L2 (Ω)  C ∇u L2 (Ω) , u L4 (Ω)  C ∇u L2 (Ω) ; 1

1

3

1

(iii) u L∞ (Ω)  C ∇u L2 2 (Ω) ∂x u H2 1 (Ω) , u L4 (Ω)  C ∇u L4 2 (Ω) ∂x u L4 2 (Ω) . Lemma 2.2 (Korn’s inequality, [3, 7, 13]) Let Ω ⊂ RN (N  2) be an open, connected domain with Lipschitz boundary ∂Ω = Γ ∪ Γ0 . If Γ0 = ∅, then there is a positive constant C > 0 which only depends on Ω such that (2.1)

∇u L2 (Ω)  C D(u) L2 (Ω) for all u ∈ HΓ10 (Ω; RN ).

Global Well-Posedness of an Initial-Boundary Value Problem. . .

Remark 2.1 The reverse inequality of (2.1) also holds for all u ∈ H 1 (Ω; RN ). Secondly, let us review some fundamental Sobolev estimates for the elliptic system with the Dirichlet boundary value conditions. Lemma 2.3 (See [1]) Let k ∈ N be a nonnegative integer, and Ω ∈ RN (N  2) be an open 1 connected smooth domain. Assume that f ∈ H k−1 (Ω) and ϕ ∈ H k+ 2 (∂Ω). Then, there is a unique solution u ∈ H k+1 (Ω) to the Dirichlet problem

− u = f u=ϕ

in Ω, on ∂Ω,

satisfying 

u H k+1 (Ω)  C f H k−1 (Ω) + ϕ

 H

k+ 21

(∂Ω)

,

where the positive constant C > 0 only depends on Ω. Finally, let us conclude this section by introducing the Dirichlet-Neumann operator, which is one of the special Poincaré-Steklov operators, defined in an open, smooth, and connected domain Ω ⊂ RN (N  2). See [1, 5, 6, 15] for details. Let k ∈ N be a nonnegative integer and Ω ⊂ RN (N  2) be an open connected domain 1 1 with smooth boundary. Then the Dirichlet-Neumann operator B : H k+ 2 (∂Ω) → H k− 2 (∂Ω) is defined by Bϕ =

∂u μ∂ , ∂n

where u is the solution to the Dirichlet problem

− u = 0 u=ϕ

in Ω, on ∂Ω,

n is the exterior unit normal to the boundary ∂Ω, and μ∂ is the boundary volume form. Lemma 2.4 Let Ω be a strip domain defined as Ω1 . Assume that the Dirichlet-Neumann mapping B is defined as above. Then its inverse, the Neumann-Dirichlet operator 1

1

B−1 : H k− 2 (∂Ω) → H k+ 2 (∂Ω),

is well-defined. Furthermore, the Neumann-Dirichlet operator B−1 is bounded, i.e., there is 1 a positive constant M > 0 which only depends on Ω such that for all ϕ ∈ H k− 2 (∂Ω), −1 B ϕ

H

k+ 21

(∂Ω)

 M ϕ

H

k− 21

(∂Ω)

.

(2.2)

Remark 2.2 Equation (2.2) in the case of k = 0 has been proved by J. Behrndt and A.F.M. Terelst in [5]. In the following, we will immediately demonstrate (2.2) holds for each positive integer k ∈ N+ based on their result.

P. Liu, W. Liu

Proof By extension, in view of the definition of Dirichlet-Neumann operator and Poisson’s kernel of Laplace equations for R2+ , we obtain      ∂u ∂u  ∂  ϕ(y) x2 μ∂ = − Bϕ(x1 ) = =− dy ∂n ∂x2 x2 =0 ∂x2 x2 =0 π R |x − y|2   1 ∂  x2 =− ϕ(y) dy π ∂x2 x2 =0 R |x − y|2  1 ∂  =− (g ∗ ϕ)(x), π ∂x2 x2 =0 here g(x) =

x2 |x|2

=

we can find

x2 x12 +x22

(x1 ∈ R, x2 > 0). Therefore, by the properties of Fourier transform,

  1 ∂  1 ∂   g ∗ ϕ(ξ ) = − g(ξ, ˆ x2 )ϕ(ξ ˆ ) π ∂x2 x2 =0 π ∂x2 x2 =0  ∂  1 ˆ ) = − ϕ(ξ g(ξ, ˆ x2 ) π ∂x2 x2 =0   −2π |ξ |x  ∂  1 2 ˆ ) = − ϕ(ξ πe π ∂x2 x2 =0

 B ϕ(ξ ) = −

= 2π|ξ |ϕ(ξ ˆ ) 1

Notice that for each ψ ∈ H k− 2 (R), the Neumann problem

− u = 0 ∂u =ψ ∂n

in R2+ , on ∂R2+ ,

admits a unique solution u ∈ H k+1 (R2+ ). Therefore we restrict this solution u on ∂R2+ . By 1 the trace theorem, we know that there is a function ϕ ∈ H k+ 2 (R) such that ϕ = u|∂R2+ . 1

According to the definition of B , we have Bϕ = ψ . Thus, ∀ψ ∈ H k− 2 (R), we get −1 2 B ψ k+ 1 H

2 (R)

2 = B−1 Bϕ k+ 1 H

 =

R

2 (R)

= ϕ 2 k+ 1 H

2 (R)

2  k+ 1  ϕ (ξ ) dξ 1 + |ξ |2 2 

= ϕ 2 k− 1 H

2 (R)

+

1 4π 2

1 = ϕ 2 k− 3 + 4π 2 H 2 (R)





R



R



1 + |ξ |2

2 k− 12  B  ϕ(ξ ) dξ

1 + |ξ |2

2 k− 32  1 B  ϕ(ξ ) dξ +

ψ 2 k− 1 , 4π 2 H 2 (R)

Global Well-Posedness of an Initial-Boundary Value Problem. . .

which implies 1 

ψ 2 m− 1 2 (R) 2 2 (R) 4π 2 m=1 H H (R)   1 = k max M02 , 2 ψ 2 k− 1 . 4π H 2 (R)

−1 2 B ψ k+ 1 H

Hence (2.2) holds for M =

k

 M02 ψ 2 − 1

+

√ 1 k max{M0 , 2π }. Here the positive constant M0 satisfies −1 B ψ

1

H 2 (R)

 M0 ψ

1

− H 2 (R)

, 

which has be proved by J. Behrndt et al. [5] in 2015.

3 A Priori Estimates In this section, we establish the a priori estimates for the system (1.1) in Sobolev spaces H m for all integer m  2. For simplicity, we detailedly present a sketch for m = 2, and the general cases of m  3 will be briefly illustrated by induction arguments. Let (u, p2 ) be a strong solution to the initial boundary value problem (1.1) in [0, T ) with the regularity stated in Theorem 4.1. For the Navier-Stokes-Darcy equations (1.1), we will show the following estimates. Proposition 3.1 Assume that the inequalities (1.3) hold. Suppose also that (u, p2 ) is a smooth solution to the system (1.1) with u0 ∈ HΓ21 (Ω1 ) satisfying div u0 = 0 and the compatibility conditions (1.2). Then there is a constant T > 0, such that there exists a constant C = C(T ) > 0 satisfying  t   D(u) 2 2 + ∇p2 2H 2 (Ω ) dτ  C(T )

u 2H 2 (Ω ) + ut 2H 1 (Ω ) + p2 2H 2 (Ω ) + H (Ω ) 1

1

2

0

2

1

(3.1)

for all t ∈ [0, T ). Proof We will divide the proof of Proposition 3.1 into eight steps. We now start with the basic L2 -energy estimate. Step 1. L2 estimate of u. Taking the L2 inner product of the first equation in the system (1.1) with u over Ω1 yields     (∂t u, u)Ω1 + (u · ∇u, u)Ω1 − div 2μD(u) − p1 I , u Ω = 0. (3.2) 1

Applying Green’s formula to the third term, employing the orthogonal decomposition of the fluid velocity field u = (u · n)n + (u · τ )τ , using the boundary value conditions (1.1)5−7 , noticing that  2 D(u) : ∇u = D(u) : D(u) = D(u) ,

P. Liu, W. Liu

and noting I : ∇u = δij ∂j ui = ∂i ui = div u = 0, we obtain 2 1 d 1

u 2L2 (Ω ) + 2μ D(u) L2 (Ω ) + u · τ 2L2 (Γ ) + 1 1 2 dt G

 p2 (u · n)dS = 0.

(3.3)

Γ

Taking the L2 inner product of (1.1)3 with p2 over Ω2 , then employing the boundary value conditions (1.1)4,8 after integration by parts, we have 

 K∇p2 · ∇p2 dx −

Ω2

(u · n)p2 dS = 0.

(3.4)

Γ

Noticing (1.3), adding (3.4) to (3.3) yields 2 1 d

u 2L2 (Ω ) + 2μ D(u) L2 (Ω ) + λ ∇p2 2L2 (Ω )  0. 1 2 1 2 dt

(3.5)

Step 2. H 1 estimate of u. Taking L2 inner product of (1.1)1 with ut = ∂t u over Ω1 , applying Green’s formula, (1.1)2 , the orthogonal decomposition of the fluid velocity, and the boundary value conditions (1.1)5−7 , we obtain    2 1 d 1 2 + 2μ D(u) L2 (Ω ) + u · τ L2 (Γ ) + (ut · n)p2 dS 1 2 dt G Γ   1 (u · u)(ut · n) dS − (u · ∇u) · ut dx. = 2 Γ Ω1   |u|2 · ut dx − ∇ (u · ∇u) · ut dx. = 2 Ω1 Ω1

ut 2L2 (Ω ) 1

Thanks to Hölder’s inequality and Cauchy’s inequality, we get    2 1 d 1 2μ D(u) L2 (Ω ) + u · τ 2L2 (Γ ) + (ut · n)p2 dS 1 1 2 dt G Γ |u|2 

ut L2 (Ω1 ) ∇ 2 − u · ∇u 2 L (Ω1 ) 2 1 |u|2 − u · ∇u  η ut 2L2 (Ω ) + ∇ 2 1 4η 2 L (Ω1 )

ut 2L2 (Ω ) +

 η ut 2L2 (Ω ) + 1

1 |u||∇u| 2 2 , L (Ω1 ) 2η

where η > 0 is an arbitrary constant, which will be determined later. Taking η = 12 , using Minkowski’s inequality, Hölder’s inequality, Lemma 2.1, and Lemma 2.2, we arrive at

Global Well-Posedness of an Initial-Boundary Value Problem. . .

   2 1 d 1 1

ut 2L2 (Ω ) + 2μ D(u) L2 (Ω ) + u · τ 2L2 (Γ ) + (ut · n)p2 dS 1 1 2 2 dt G Γ 2  |u||∇u| L2 (Ω ) 1



u 2L∞ (Ω1 ) ∇u 2L2 (Ω ) 1

2  u H 1 (Ω1 ) ux H 1 (Ω1 ) D(u) L2 (Ω ) . 1

Furthermore, by Lemma 2.1, we have    2 1 d 1 1

ut 2L2 (Ω ) + 2μ D(u) L2 (Ω ) + u · τ 2L2 (Γ ) + (ut · n)p2 dS 1 1 2 2 dt G Γ 3  D(ux ) L2 (Ω ) D(u) L2 (Ω ) . 1

1

(3.6)

Applying ∂t to (1.1)3 and then taking the L2 inner product of the resulting equation with p2 over Ω2 , Green’s formula and the boundary value conditions (1.1)4,8 give that   1 d K∇p2 · ∇p2 dx − p2 (ut · n) dS = 0. (3.7) 2 dt Ω2 Γ Adding (3.6) and (3.7) together, we have    2 d 1 2 2 K∇p2 · ∇p2 dx

ut L2 (Ω ) + 2μ D(u) L2 (Ω ) + u · τ L2 (Γ ) + 1 1 dt G Ω2 3  D(ux ) L2 (Ω ) D(u) L2 (Ω ) . 1

1

(3.8)

Step 3. L2 estimate of ut . Differentiating (1.1)1 with respect to t and then taking the L2 inner product of the resulting equation with ut over Ω1 , employing Green’s formula, (1.1)2 , the orthogonal decomposition of the fluid velocity, the boundary value conditions (1.1)5−7 , Minkowski’s inequality, Hölder’s inequality, Lemma 2.1, and Lemma 2.2, we obtain  2 1 1 d

ut 2L2 (Ω ) + 2μ D(ut ) L2 (Ω ) + ut · τ 2L2 (Γ ) + (∂t p2 )(ut · n) dS 1 1 2 dt G Γ   2 |u|

ut L2 (Ω1 )  ∂t ∇ 2 − u · ∇u 2 L (Ω1 ) 3 1   D(u) H4 1 (Ω ) D(ux ) L4 2 (Ω ) D(ut ) L2 (Ω 1

1 + D(u) 2 2

L (Ω1

1

1)

1  D(ux ) 2 2 D(ut ) 2

ut L2 (Ω1 ) , L (Ω ) ) L (Ω ) 1

1

which along with Cauchy’s inequality gives  2 1 1 d 2 2

ut L2 (Ω ) + μ D(ut ) L2 (Ω ) + ut · τ L2 (Γ ) + (∂t p2 )(ut · n) dS 1 1 2 dt G Γ 32 12    D(u) H 1 (Ω ) D(ux ) L2 (Ω ) + D(u) L2 (Ω ) D(ux ) L2 (Ω ) ut 2L2 (Ω ) . 1

1

1

1

1

P. Liu, W. Liu

Applying ∂t to (1.1)3 and then taking L2 inner product of the resulting equation with ∂t p2 over Ω2 , the Green’s formula and the boundary value conditions (1.1)4,8 give that 

 K∇(∂t p2 ) · ∇(∂t p2 ) dx − Ω2

(∂t p2 )(ut · n) dS = 0.

(3.9)

Γ

Considering (1.3), we find 2 2 d 1

ut 2L2 (Ω ) + 2μ D(ut ) L2 (Ω ) + ut · τ 2L2 (Γ ) + λ ∇(∂t p2 ) L2 (Ω ) 1 1 2 dt G 32 12    D(u) H 1 (Ω ) D(ux ) L2 (Ω ) + D(u) L2 (Ω ) D(ux ) L2 (Ω ) ut 2L2 (Ω ) . 1

1

1

1

1

(3.10)

Step 4. L2 estimate of D 2 u. Applying ∂x to (1.1)1 and then taking the L2 inner product of the resulting equation with ut,x over Ω1 , employing Green’s formula, (1.1)2 , the orthogonal decomposition of the fluid velocity, the boundary value conditions (1.1)5−7 , Cauchy’s inequality, Minkowski’s inequality, Hölder’s inequality, Lemma 2.1, and Lemma 2.2, we know    2 1 d 1 2μ D(ux ) L2 (Ω ) + ux · τ 2L2 (Γ ) + (∂x p2 )(ut,x · n) dl 1 2 dt G Γ 32 34 54 12    C D(u) L2 (Ω ) D(ux ) L2 (Ω ) + D(u) H 1 (Ω ) D(ux ) L2 (Ω ) ut,x L2 (Ω1 )

ut,x L2 (Ω1 ) +

1

1  ut,x 2L2 (Ω ) 1 2

1

 + C D(u)

L2 (Ω1 )

1

D(ux ) 3 2

L (Ω1 )

1

3 + D(u) 2 1

H (Ω1 )

5  D(ux ) 2 2 . L (Ω ) 1

(3.11) Applying ∂t ∂x to (1.1)3 and then taking L2 inner product of the resulting equation with ∂x p2 over Ω2 , the Green’s formula and the boundary value conditions (1.1)4,8 give that 1 d 2 dt



 K∇(∂x p2 ) · ∇(∂x p2 ) dx − Ω2

(∂x p2 )(ut,x · n) dS = 0.

(3.12)

Γ

Adding (3.11) and (3.12) together, we get

ut,x 2L2 (Ω ) + 1

   2 d 1 K∇(∂x p2 ) · ∇(∂x p2 ) dx 2μ D(ux ) L2 (Ω ) + ux · τ 2L2 (Γ ) + 1 dt G Ω2

3 3 5  D(u) L2 (Ω ) D(ux ) L2 (Ω ) + D(u) H2 1 (Ω ) D(ux ) L2 2 (Ω ) . 1

1

1

1

(3.13)

Next, we estimate the pure normal derivatives of the second order via the system. (perhaps, we can find that ∂y u2 = −∂x u1 from (1.1)2 . Hence ∂y2 u2 = −∂y ∂x u1 . Therefore, only the second pure normal derivative of u1 needs to be estimated). For this end, we first rewrite (1.1)1 as μuyy = ut + u · ∇u + ∇p1 − μuxx . Thus, it can be derived from Lemma 2.1 that

(3.14)

Global Well-Posedness of an Initial-Boundary Value Problem. . .

uyy L2 (Ω1 )  ut L2 (Ω1 ) + u · ∇u L2 (Ω1 ) + ∇p1 L2 (Ω1 ) + uxx L2 (Ω1 ) 1 1  ut L2 (Ω1 ) + u H2 1 (Ω ) ux H2 1 (Ω ) D(u) L2 (Ω ) 1 1 1 + ∇p1 L2 (Ω1 ) + D(ux ) L2 (Ω ) 1 2 2 2  ut L2 (Ω ) + u 1 + ux 1 + D(u) 2

H (Ω1 )

1

H (Ω1 )

L (Ω1 )

+ ∇p1 L2 (Ω1 ) + D(ux ) L2 (Ω ) , 1

which implies 4

uyy 2L2 (Ω )  1 + ut 4L2 (Ω ) + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) + ∇p1 2L2 (Ω ) . 1

1

1

1

1

(3.15)

By the definition of D 2 u, combining (3.15) we have 4 2 2 D u 2  1 + ut 4L2 (Ω ) + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) + ∇p1 2L2 (Ω ) . L (Ω ) 1

1

1

1

1

(3.16)

Similarly, we can also obtain

uyy H 1 (Ω1 )  ut H 1 (Ω1 ) + u · ∇u H 1 (Ω1 ) + uxx H 1 (Ω1 ) + ∇p1 H 1 (Ω1 )  ut H 1 (Ω1 ) + u · ∇u L2 (Ω1 ) + ∇(u · ∇u) L2 (Ω ) + uxx H 1 (Ω1 ) + ∇p1 H 1 (Ω1 ) 1

3 1 1 3  ut H 1 (Ω1 ) + D(ux ) L2 2 (Ω ) D(u) L2 2 (Ω ) + D(ux ) L2 2 (Ω ) D 2 u L2 2 (Ω 1

1

1

1)

+ D(ux ) H 1 (Ω ) D(u) L2 (Ω ) D 2 u L2 (Ω ) + D(uxx ) L2 (Ω ) + ∇p1 H 1 (Ω1 ) , 1 2

1 2

1

1

1

1

which means 4 4

∇uyy 2L2 (Ω )  1 + ut 2H 1 (Ω ) + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) 1 1 1 1 1 2 2 2 + D u 2 + ∇p1 2 . (3.17) L (Ω1 )

L (Ω1 )

Moreover, considering the definition of D 3 u and (3.17), we have 4 4 3 2 D u 2  1 + ut 2H 1 (Ω ) + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) L (Ω1 ) 1 1 1 1 2 + ∇p1 2H 1 (Ω ) + D 2 u L2 (Ω ) 1 1 4 2 4  1 + ut H 1 (Ω ) + u H 1 (Ω ) + D(ux ) L2 (Ω ) 1 1 1 4 + D(uxx ) L2 (Ω ) + ∇p1 2H 1 (Ω ) 1 1 4  2 4 4 + C 1 + ut L2 (Ω ) + u H 1 (Ω ) + D(ux ) L2 (Ω ) + ∇p1 2L1 (Ω ) 1 1 1 1 8  1 + ut 8 1 + u 8 1 + D(ux ) 2 H (Ω1 )

H (Ω1 )

8 + D(uxx ) L2 (Ω ) + ∇p1 4H 1 (Ω 1

L (Ω1 )

1)

(3.18)

P. Liu, W. Liu

Now, we calculate the estimates on the terms ∇p1 , D(ux ), and D(uxx ). Step 4.1. L2 estimate of ∇p1 . Applying the divergence operator to (1.1)1 and using (1.1)2 , we can find − p1 = div(u · ∇u) = (∇u)T : ∇u = tr(∇u)2 .

(3.19)

According to (1.1)1−2 and the boundary value conditions (1.1)6−7 , we know that the pressure p1 satisfies the follow boundary conditions:

on Γ1 × (0, T ), p1 = 0, (3.20) 1 1 p1 = p2 − 2 (u · u) − ∂x u , on Γ × (0, T ). Along with (3.19) and (3.20), it then follows from the Lemma 2.3 that

∇p1 L2 (Ω1 )  (∇u)T : ∇u H −1 (Ω ) + p2 1 + |u|2 1 + ∂x u1 1

H 2 (Γ )

H 2 (Γ )

1

H 2 (Γ )

 div(u · ∇u) H −1 (Ω ) + p2 H 1 (Ω2 ) + |u|2 H 1 (Ω ) + ∂x u1 H 1 (Ω ) 1 1 1  1  2  u · ∇u L2 (Ω1 ) + ∇p2 L2 (Ω2 ) + ∇|u| L2 (Ω ) + D ux L2 (Ω ) , 1

1

hence, we know 3 1

∇p1 L2 (Ω1 )  D(u) L2 2 (Ω ) D(ux ) L2 2 (Ω ) + ∇p2 L2 (Ω2 ) + D(ux ) L2 (Ω ) 1 1 1 2  1 + u 2H 1 (Ω ) + D(ux ) L2 (Ω ) + ∇p2 2L2 (Ω ) . (3.21) 1

2

1

And

∇p1 H 1 (Ω1 )  (∇u)T : ∇u L2 (Ω ) + p2

1

3

H 2 (Γ )

+ |u|2

3

H 2 (Γ )

+ ∂x u1

3

H 2 (Γ )

 |∇u|2 L2 (Ω ) + p2 H 2 (Ω2 ) + |u|2 H 2 (Ω ) + ∂x u1 H 2 (Ω ) 1 1 1  ∇u L4 (Ω ) + ∇p2 H 1 (Ω ) + ∇|u|2 2 + D 2 |u|2 2 1

2

L (Ω1 )

L (Ω1 )

+ D(ux ) H 1 (Ω

1)

2 2  1 + u 2H 1 (Ω ) + D 2 u L2 (Ω ) + D(uxx ) L2 (Ω ) + uxyy 2L2 (Ω ) + ∇p2 H 1 (Ω2 ) 1 1 1 1 (3.22) Step 4.2. L2 estimate of uxyy . Differentiating equations (3.14) with respect to x yields μuxyy = ut,x + ∂x (u · ∇u) + ∇(∂x p1 ) − μuxxx , which implies

uxyy 2L2 (Ω

1)

2 2  ut,x 2L2 (Ω ) + ∂x (u · ∇u) L2 (Ω ) + ∇(∂x p1 ) L2 (Ω ) + uxxx 2L2 (Ω ) 1 1 1 1 2 2 2 2 2 2  1 + ut,x L2 (Ω ) + u H 1 (Ω ) + D u L2 (Ω ) + D(uxx ) L2 (Ω ) + ∇(∂x p1 ) L2 (Ω 1

1

1

1

1)

Global Well-Posedness of an Initial-Boundary Value Problem. . .

2 2 2  1 + ut 2H 1 (Ω ) + u 2H 1 (Ω ) + D 2 u L2 (Ω ) + D(uxx ) L2 (Ω ) + ∇(∂x p1 ) L2 (Ω ) 1 1 1 1 1 (3.23) Step 4.3. L2 estimate of ∇(∂x p1 ). Applying ∂x to equations (3.19) and (3.20) obtains

− (∂x p1 ) = div(∂x (u · ∇u)) ∂x p1 = ∂x p2 − (u · ux ) − ∂x2 u1

in (0, T ) × Ω1 , in (0, T ) × Γ.

Thanks to Lemma 2.3, we arrive at ∇(∂x p1 ) 2 2

L (Ω1 )

 2  2  div ∂x (u · ∇u) H −1 (Ω ) + ∇(∂x p2 )

1 H 2 (Γ )

1

+ u · ux 2

1 H 2 (Γ )

2 + ∂x2 u1

 2  2  ∂x (u · ∇u) L2 (Ω ) + u · ux 2L2 (Ω ) + ∇(u · ux ) L2 (Ω ) + uxx 2 1

1

2 + ∇(∂x p2 ) L2 (Ω ) 2 D(uxx )  D(ux ) 2 L (Ω1 )

3 + D(u) 2

L (Ω1 )

L2 (Ω1 )

D(ux )

L2 (Ω1

1

D(u) 2 2

L (Ω1 )

1

H 2 (Γ )

1

H 2 (Γ )

3 + D(u) L2 (Ω ) D(ux ) L2 (Ω 1

2 + D(uxx ) L2 (Ω ) + ∇p2 2H 1 (Ω ) . ) 1

1)

2

Furthermore, we have 4 4 ∇(∂x p1 ) 2 2  1 + u 4H 1 (Ω ) + D 2 u L2 (Ω ) + D(uxx ) L2 (Ω ) + ∇p2 4H 1 (Ω ) L (Ω1 ) 1 2 1 1 (3.24) Step 4.4. H 2 estimate of p2 . Consider that (1.1)3 and the boundary value conditions (1.1)4,8 compose an elliptic equation with the Neumann boundary conditions: ⎧ ⎪ ⎨− div(K∇p2 ) = 0 in Ω2 , K∇p2 · n = −u · n on Γ, ⎪ ⎩ p2 = 0 on Γ2 . It follows from Lemma 2.3, Lemma 2.4 and the trace theorem that  (u · n) 1  u 1  u H 1 (Ω1 )

∇p2 H 1 (Ω2 )  B−1 (u · n) 3 H 2 (Γ ) H 2 (Γ ) H 2 (Γ )  D(u) L2 (Ω ) , (3.25) 1

and  (u · n) 3  u 3  u H 2 (Ω1 )

∇p2 H 2 (Ω2 )  B−1 (u · n) 5 H 2 (Γ ) H 2 (Γ ) H 2 (Γ )  D(u) H 1 (Ω ) . (3.26) 1

P. Liu, W. Liu

Step 5. L2 estimate of D(ux ). Applying ∂x to (1.1)1 and then taking the L2 inner product of the resulting equation with ux . Meanwhile, applying ∂x to (1.1)3 and then taking L2 the inner product of the resulting equation with ∂x p2 . Adding those two resulting estimate equations, Green’s formula gives 2 2 1 1 d

ux 2L2 (Ω ) + 2μ D(ux ) L2 (Ω ) + ux · τ L2 (Γ ) + λ ∇(∂x p2 ) 2L2 (Ω ) 1 2 1 2 dt G 1 3 3 5    C u H2 1 (Ω ) D(ux ) L2 2 (Ω ) + D 2 u L4 2 (Ω ) D(ux ) L4 2 (Ω ) ux L2 (Ω1 ) 1

1

1

1

8 2 2    μ D(ux ) L2 (Ω ) + C u 2H 1 (Ω ) ux 4L2 (Ω ) + D 2 u L2 (Ω ) ux L3 2 (Ω ) , 1

1

1

1

1

which implies 2 2 1 d

ux 2L2 (Ω ) + 2μ D(ux ) L2 (Ω ) + ux · τ 2L2 (Γ ) + λ ∇(∂x p2 ) L2 (Ω ) 1 1 2 dt G 8 2  u 2 1

ux 4 2 + D 2 u 2

ux 3 2 H (Ω1 )

L (Ω1 )

L (Ω1 )

L (Ω1 )

3  u 4H 1 (Ω ) + ux 8L2 (Ω ) + D 2 u L2 (Ω ) + ux 8L2 (Ω ) 1 1 1 1 2 3 4 8  u 1 + ux 2 + D u 2 . H (Ω1 )

L (Ω1 )

(3.27)

L (Ω1 )

Step 6. L2 estimate of D(uxx ). Applying ∂x2 to (1.1)1 and then taking the L2 inner product of the resulting equation with ut,xx . Meanwhile, applying ∂x2 to (1.1)3 and then taking L2 the inner product of the resulting equation with ∂x2 p2 . Adding those two resulting estimate equations, it follows from Green’s formula and Young’s inequality that  2 1 d 1 2μ D(uxx ) L2 (Ω ) + uxx · τ 2L2 (Γ ) 1 1 2 dt G       + K∇ ∂x2 p2 · ∇ ∂x2 p2 dx

ut,xx 2L2 (Ω ) +

Ω2

3 1 2  1  ut,xx 2L2 (Ω ) + C D 2 u L2 2 (Ω ) D(ux ) L2 2 (Ω ) D(uxx ) L2 (Ω ) 1 1 1 1 2 3 2  + D(ux ) L2 (Ω ) D(uxx ) L2 (Ω ) + u H 1 (Ω1 ) D(ux ) L2 (Ω ) D(uxx ) L2 (Ω ) . 1

1

1

1

Therefore, we arrive at   2   d 1

ut,xx 2L2 (Ω ) + 2μ D(uxx ) L2 (Ω ) + uxx · τ 2L2 (Γ ) + λ ∇ ∂x2 p2 L2 (Ω ) 1 1 2 dt G 2 2 10 10 4  D u L2 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) 1 1 1 1 1 4 + D(uxx ) 2 L (Ω1 )

10 10 2  1 + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) + D 2 u L2 (Ω ) . 1

1

1

1

(3.28)

Global Well-Posedness of an Initial-Boundary Value Problem. . .

Step 7. L2 estimate of D(ut ). Applying ∂t to the first equation in the system (1.1) and then taking the L2 inner product of the resulting equation with utt . Meanwhile, applying ∂t2 to the third equation in the system (1.1) and then taking L2 the inner product of the resulting equation with ∂t p2 . Adding those two resulting estimate equations, it follows from Green’s formula and Young’s inequality that   2 2 1 d 1 2μ D(ut ) L2 (Ω ) + ut · τ L2 (Γ ) + λ ∇(∂t p2 ) L2 (Ω )

utt 2L2 (Ω ) + 1 1 2 2 dt G 3 1   1  utt 2L2 (Ω ) + C D 2 u L2 2 (Ω ) D(ux ) L2 2 (Ω ) + u H 1 (Ω1 ) D(ux ) L2 (Ω ) 1 1 1 1 2 2 × D(ut ) L2 (Ω ) . 1

Thus, we have   2 2 d 1 2μ D(ut ) L2 (Ω ) + ut · τ L2 (Γ ) + λ ∇(∂t p2 ) L2 (Ω ) 1 1 2 dt G 2 2 10 10 4  D u L2 (Ω ) + D(ux ) L2 (Ω ) + D(ut ) L2 (Ω ) + u 4H 1 (Ω ) + D(ux ) L2 (Ω ) 1 1 1 1 1 8 + D(ut ) L2 (Ω ) . (3.29)

utt 2L2 (Ω ) +

1

Step 8. Closing of the a priori estimates. Along with (3.5), (3.8), (3.10), (3.13), (3.16), (3.18), (3.21)–(3.29), we have 2 2 d

u 2H 1 (Ω ) + ut 2H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) + ∇p2 2L2 (Ω ) 1 1 2 1 1 dt 2  2  2 2   + ∇(∂x p2 ) L2 (Ω ) + ∇ ∂x p2 L2 (Ω ) + C0 D(u) H 2 (Ω ) 2 2 1  2 2 + ut H 1 (Ω ) + ∇p2 H 2 (Ω ) 1

2

3 1     C ut L2 (Ω1 ) D 2 u L2 2 (Ω ) D(ux ) L2 2 (Ω ) + u H 1 (Ω1 ) D(ux ) L2 (Ω ) 1

1

1

3 5 3 + u 3H 1 (Ω ) D(ux ) L2 (Ω ) + D(ux ) L2 (Ω ) u H 1 (Ω1 ) + D(ux ) L2 2 (Ω ) D 2 u L2 2 (Ω ) 1 1 1 1 1 4 2 4 4 + 1 + ut 2 + u 1 + D(ux ) 2 + D(ux ) 2 L (Ω1 )

H (Ω1 )

L (Ω1 )

L (Ω1 )

8 8 10 + ut 8H 1 (Ω ) + u 8H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) + D(ux ) L2 (Ω ) 1 1 1 1 1 10 2 2 10 8 12 + D(uxx ) 2 + D u 2 + D(ut ) 2 + ux 2 + ut 2 L (Ω1 )

L (Ω1 )

L (Ω1 )

12  + D(ux ) L2 (Ω ) + u 12 , H 1 (Ω )

L (Ω1 )

1

1

where C0 > 0 is a constant. Therefore 2 2 d

u 2H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) + ut 2H 1 (Ω ) 1 1 1 1 dt    2 2 + ∇p2 2L2 (Ω ) + ∇(∂x p2 ) L2 (Ω ) + ∇ ∂x2 p2 L2 (Ω ) 2

2

2

L (Ω1 )

P. Liu, W. Liu

2   + C0 D(u) H 2 (Ω ) + ut 2H 1 (Ω ) + ∇p2 2H 2 (Ω ) 1 2 1 12 12    C 1 + ut 12 + u 12 + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) . H 1 (Ω ) H 1 (Ω ) 1

1

1

(3.30)

1

Hence, by the bootstrap argument, there is a positive constant C = C(T ) > 0 such that 2 2

u 2H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω ) + ut 2H 1 (Ω ) 1 1 1 1 2  2  2 2 + ∇p2 L2 (Ω ) + ∇(∂x p2 ) L2 (Ω ) + ∇ ∂x p2 L2 (Ω )  C, 2

1

(3.31)

1

which together with (3.16), (3.18), (3.25) and (3.26) yields the uniform estimate 

u 2H 2 (Ω ) + ut 2H 1 (Ω ) + p2 2H 2 (Ω ) + 1

1

2

t

0

  D(u) 2 2 + ∇p2 2H 2 (Ω ) dτ  C, H (Ω ) 2

1

(3.32) 

for all t ∈ [0, T ). This completes the proof of Proposition 3.1.

Moreover, we can build the following H m energy estimates for the general cases of m  3. Proposition 3.2 Let m  2 be an integer. Suppose that the assumptions of Proposition 3.1 are satisfied. If (u, p2 ) is an smooth solution to system (1.1) with u0 ∈ HΓm1 (Ω1 ) being divergence free and satisfying the compatibility conditions (1.2), then there is a positive constant T > 0, such that there exists a positive constant C = C(T ) > 0 such that

u 2H m (Ω1 ) + p2 2H m (Ω2 )  C(T ) for all t ∈ [0, T ). Induction Firstly, applying Lemma 2.3 and Lemma 2.4 to equations ⎧ ⎪ ⎨− div(K∇p2 ) = 0 −K∇p2 · n = u · n ⎪ ⎩ p2 = 0

in Ω2 on Γ on Γ2 ,

(3.33)

we arrive at 2

p2 2H m (Ω2 )  B−1 (u · n) m− 1

2 (Γ )

H

 u · n 2 m− 3 H

2 (Γ )

 u 2H m−1 (Ω

1)

for all integer m  3. Therefore, we only need to calculate the H m estimates of u. In Proposition 3.1, we have obtained the H 2 -estimates of u: 2

u 2H 2 (Ω ) + D(uxx ) L2 (Ω ) + ut 2H 1 (Ω ) + p2 2H 2 (Ω )  C. 1

1

1

2

By (3.18), (3.22)–(3.25), thus we know 8 8 3 2 D u 2  1 + ut 8H 1 (Ω ) + u 8H 1 (Ω ) + D(ux ) L2 (Ω ) + D(uxx ) L2 (Ω )  C. L (Ω ) 1

1

1

1

1

Global Well-Posedness of an Initial-Boundary Value Problem. . .

Now, we assume that m−1 2   2 D u L2 (Ω ) + D ∂xm−2 u L2 (Ω )  C 1

1

holds for all m − 1  3. Next, we detailedly present a sketch of L2 estimates of D m u for integer m  4. Firstly, by the first equation in the system (1.1), we have μuyy = ut + u · ∇u − μuxx + ∇p1 . Then

uyy 2H m−2 (Ω )  ut 2H m−2 (Ω ) + u · ∇u 2H m−2 (Ω ) + uxx 2H m−2 (Ω ) + ∇p1 2H m−2 (Ω ) . 1

1

1

1

2

Due to D m u = {D m−2 uyy } ∪ {D(∂xm−1 u)}, we get m 2 D u 2  ut 2H m−2 (Ω ) + u · ∇u 2H m−2 (Ω ) + uxx 2H m−2 (Ω ) L (Ω1 ) 1 1 1  m−1  2 2 + D ∂x u H m−2 (Ω ) + ∇p1 H m−2 (Ω ) 2

1

 1 + ut 2H m−2 (Ω ) + u 4H m−3 (Ω 1

1

m−1  k 4 D u 2 + ) L (Ω

1)

k=2 m−4  2+i m−2−i 2   4 ∂ ∂ + D ∂xm−1 u L2 (Ω ) + u L2 (Ω x y 1

1)

i=0

+ ∇p1 2H m−2 (Ω ) . 2

Next, we estimate ut H m−2 (Ω1 ) , D(∂xm−1 u) 4L2 (Ω ) , ∂x2+i ∂ym−2−i u H m−2 (Ω1 ) 1

∇p1 2H m−2 (Ω ) .

(3.34) and

2

Step 1. Applying ∂xm−1 to the first equation of system (1.1) and then taking the L2 inner product of the resulting equation with ∂xm−1 ut over Ω1 . Meanwhile, employing ∂xm−1 to the third equation of system (1.1) then taking L2 inner product of the resulting equation with ∂xm−1 p2 over Ω2 . And then adding those two resulting estimate equations, it follows from the results of H m−1 -estimates and Young’s inequality    2  m−1  2  m−1   m−1  1 d 1 m−1 ∂ u · τ L2 (Γ ) + K ∇∂x p2 · ∇∂x p2 dx 2μ D ∂x u L2 (Ω ) + 1 2 dt G x Ω2 + ∂xm−1 ut 2L2 (Ω

1)

2   2 1  ∂xm−1 ut L2 (Ω ) + C D ∂xm−1 u L2 (Ω ) + C1 . 1 1 2 In view of Grönwall’s inequality, it yields

P. Liu, W. Liu

2    2  2 2μ D ∂xm−1 u L2 (Ω ) + λ ∇∂xm−1 p2 L2 (Ω )  1 + D ∂xm−1 u0 L2 (Ω 1

1



1)

1 + u0 2H m (Ω1 )

 1.

(3.35)

Step 2. Applying ∂t to the first equation in the system (1.1) obtains μ ut = utt + ∂t (u · ∇u) + ∇(∂t p1 ),

(3.36)

which implies 2 2

ut 2H m (Ω1 )  utt 2H m−2 (Ω ) + ∂t (u · ∇u) H m−2 (Ω ) + ∇(∂t p1 ) H m−2 (Ω ) 1 1 1 2 2 2  1 + utt m−2 + ut m−1 + ∇(∂t p1 ) m−2 . H

H

(Ω1 )

(Ω1 )

Applying Lemma 2.3 to the equations

− p1 = tr(∇u)2 p1 = p2 − 12 (u · u) − u1x obtains ∇(∂t p1 ) 2 m−2 H (Ω1 )   2  ∂t tr(∇u)2 H m−4 (Ω ) + ∂t p2 2 m− 5 1

2 (Γ )

H

H

(Ω1 )

(3.37)

in (0, T ) × Ω1 on (0, T ) × Γ

+ u · ut 2 m− 5 H

2 (Γ )

+ u1t,x 2 m− 5 H

2 (Γ )

  2  ∂t ∇u : (∇u)T H m−4 (Ω ) + ∂t p2 2H m−2 (Ω ) + u · ut 2H m−2 (Ω ) + u1t,x 2H m−2 (Ω 2

1



1 + ut 4H m−3 (Ω ) 1

+ ut 4H m−2 (Ω ) 1

1

+ ut 4H m−1 (Ω ) 1

1)

+ ∂t p2 2H m−2 (Ω ) . 2

Hence, we know ∇(∂t p1 ) 2 m−2  1 + ut 4H m−1 (Ω ) + ∂t p2 2H m−2 (Ω )  1, (Ω ) H 1

1

2

here we used the fact that

∂t p2 2H m−2 (Ω )  ut 2H m−3 (Ω ) , 2

1

which follows from (3.33). Applying ∂t2 to the first equation of system (1.1) and then taking the L2 inner product of the resulting equation with utt over Ω1 . Meanwhile, applying ∂t2 to the third equation of system (1.1) then taking L2 inner product of the resulting equation with ∂t2 p2 over Ω2 . And then adding those two resulting estimate equations, it follows from Green’s formula and Young’s inequality 2 2 1 1 d

utt 2L2 (Ω ) + 2μ D(utt ) L2 (Ω ) + utt · τ 2L2 (Γ ) + λ ∂t2 p2 L2 (Ω )  1 + utt 2L2 (Ω ) , 1 1 1 2 2 dt G which implies 

utt 2L2 (Ω ) + 1

t 0



2  2  2  2μ D(utt ) L2 (Ω ) + λ ∇ ∂t2 p2 L2 (Ω ) dτ  1 + utt (0) L2 (Ω ) . 1

1

1

Global Well-Posedness of an Initial-Boundary Value Problem. . .

Hence, we arrive at 

utt 2L2 (Ω ) + 1

t 0

2    2  2μ D(utt ) L2 (Ω ) + λ ∇ ∂t2 p2 L2 (Ω ) dτ  C. 1

(3.38)

1

On the other hand,

uttt 2L2 (Ω ) + 1

   2     1 d 1 K∇ ∂t2 p2 · ∇ ∂t2 p2 dx 2μ D(utt ) L2 (Ω ) + utt · τ 2L2 (Γ ) + 1 2 dt G Ω2

2 1  uttt 2L2 (Ω ) + C D(utt ) L2 (Ω ) + C1 . 1 1 2 Thus D(utt ) 2 2

L (Ω1 )

  2 + ∇ ∂t2 p2 L2 (Ω )  C. 1

Hence, combining (3.38) yields 2

utt 2H 1 (Ω )  utt 2L2 (Ω ) + D(utt ) H 1 (Ω )  1. 1

1

(3.39)

1

Furthermore m 2 ∂ u 1  C. t H (Ω ) 1

Therefore

ut 2H m−2 (Ω )  1

⎧ m ⎨1 + ∂t 2 u 2 2 ⎩1 + ∂

t

m−1 2

m is even,

L (Ω1 )

u 2H 1 (Ω ) 1

(3.40)

m is odd.

Step 3. Applying ∂x2+i to (3.36), we get m−4−i  2+i  2 D ∂x uyy L2 (Ω ) 1 2+i 2 2 2  ∂x ut H m−4−i (Ω ) + ∂x2+i (u · ∇u) H m−4−i (Ω ) + ∂x4+i u H m−4−i (Ω ) 1 1 1 2+i 2 + ∂x (∇p1 ) H m−4−i (Ω ) 1 2 2 2  ut m−2 + u m−1 + D(∂ m−1 u 2 + ∇p1 2 m−2 H



(Ω1 )

H

x

(Ω1 )

L (Ω1 )

1 + ∇p1 2H m−2 (Ω ) . 1 1

Step 4. Applying Lemma 2.3 to the equations

we obtain

(Ω1 )

(3.41)

Hence, it is only needed to estimate ∇p1 2H m−2 (Ω ) .

− p1 = tr(∇u)2 p1 = p2 − 12 (u · u) − u1x

H

in (0, T ) × Ω1 , on (0, T ) × Γ,

P. Liu, W. Liu

∇p1 2H m−2 (Ω ) 1 2  tr(∇u)2 H m−4 (Ω ) + p2 2 m− 5 1

H

2 (Γ )

+ u · u 2 m− 5 H

2 (Γ )

+ u1x 2 m− 5 H

2 (Γ )

2  ∇u : (∇u)T H m−4 (Ω ) + p2 2H m−2 (Ω ) + u · u 2H m−2 (Ω ) + u1x 2H m−2 (Ω 2

1



1 + u 4H m−2 (Ω ) 1

+ u 4H m−3 (Ω ) 1

+ u 4H m−1 (Ω ) 1

1

1)

 1.

(3.42)

Substituting (3.35), (3.40), (3.41) and (3.42) into (3.34) yields m 2 D u 2  C, L (Ω ) 1

where C is a positive constant which only depends on T and u0 . Therefore, the proof of the Proposition 3.2 is completed. 

4 Local Well-Posedness This section is devoted to the proof of local well-posedness for (1.1). With the a priori estimate in hand, we are now ready to complete the proof of the following Theorem. Theorem 4.1 Let m  2 be an integer. Assume that the inequalities (1.3) hold. Suppose also that u0 ∈ HΓm1 (Ω1 ) is an initial divergence free velocity field satisfying the compatibility conditions (1.2). Then there is a positive constant T > 0, such that the Navier-Stokes-Darcy system (1.1) admits a unique solution (u, p2 ) on [0, T ) satisfying        (Ω1 ) ∩ L2loc [0, T ); HΓm+1 (Ω1 ) (u, p2 ) ∈ C [0, T ); HΓm1 (Ω1 ) ∩ C 1 [0, T ); HΓm−1 1 1      (Ω2 ) . × L∞ [0, T ); HΓm2 (Ω2 ) ∩ L2loc [0, T ); HΓm+1 2

4.1 Existence Part Now we are in a position to complete the proof of the existence part of Theorem 4.1. Proof of Theorem 4.1 (Existence Part). We will first construct an iteration scheme for (1.1) to obtain the approximate solution and then derive uniform bounds to pass the limit. This procedure is more or less standard and thus we just give a sketch for completeness. By setting u(0) = u0 , we can construct the iterative sequence {(u(k) , p2(k) )}∞ k=1 as follows ⎧ (k) (k−1) ⎪ · ∇u(k) − div(2μD(u(k) ) − p1(k) I ) = 0 in Ω1 ; ⎨∂t u + u (4.1) div u(k) = 0 in Ω1 ; ⎪ ⎩ (k) − div(K∇p2 ) = 0 in Ω2 . Analogously, the initial-boundary value conditions of the iterative sequence as follows. ⎧ (k) at t = 0; u = u0 ⎪ ⎪ ⎪ ⎪ (k) ⎪ u = 0 on Γ1 ; ⎪ ⎪ ⎪ ⎨p (k) = 0 on Γ2 ; 2 (4.2) (k) (k) ⎪ u · n = −K∇p · n on Γ ; ⎪ 2 ⎪ ⎪ ⎪ ⎪u(k) · τ = −2μG(D(u(k) )n) · τ on Γ ; ⎪ ⎪ ⎩ ((−2μD(u(k) ) + p1(k) I )n) · n + 12 (u(k−1) · u(k) ) = p2(k) on Γ.

Global Well-Posedness of an Initial-Boundary Value Problem. . .

By the theories of linear evolution equations and second order elliptic equations, we may get there is a unique smooth solution (u(k) , p2(k) ) to (4.1) and (4.2) satisfies 

        u(k) , p2(k) ∈ C [0, T ); HΓm1 (Ω1 ) ∩ C 1 [0, T ); HΓm−1 (Ω1 ) ∩ L2loc [0, T ); HΓm+1 (Ω1 ) 1 1      (Ω2 ) , × L∞ [0, T ); HΓm2 (Ω2 ) ∩ L2loc [0, T ); HΓm+1 2

for every divergence free initial data u0 ∈ HΓm1 (Ω1 ) (m  2 is an integer). Next, we can get    (k) (k)  ∞ u , p2 k=1 is a Cauchy sequence in L∞ (0, T ); L2 (Ω1 ) , which together with the Sobolev embedding theorems and the regularity theories implies that {(u(k) , p2(k) )}∞ k=1 may converge to some limit function        (Ω1 ) ∩ L2loc [0, T ); HΓm+1 (Ω1 ) (u, p2 ) ∈ C [0, T ); HΓm1 (Ω1 ) ∩ C 1 [0, T ); HΓm−1 1 1      (Ω2 ) . × L∞ [0, T ); HΓm2 (Ω2 ) ∩ L2loc [0, T ); HΓm+1 2 Then we can naturally deduce that this limit function (u, p2 ) solves the system (1.1) by the standard argument [16]. This ends the proof of the existence part of the Theorem 4.1. 

4.2 Uniqueness Part The rest of this section, we only need to prove the uniqueness of the solution to (1.1). Proof of Theorem 4.1 (Uniqueness Part). If (u, p1 , p2 ) and (v, q1 , q2 ) were two solutions to

2 = p2 − q2 . Then (

u, p 1 , p 2 ) (1.1). For simplicity, we denote

u = u − v, p

1 = p1 − q1 and p would satisfy ⎧ ⎪ u +

u · ∇u + v · ∇

u − div(2μD(

u) − p

1 I ) = 0 in Ω1 × (0, T ); ⎨ ∂t

(4.3) div

u=0 in Ω1 × (0, T ); ⎪ ⎩ in Ω2 × (0, T ); − div(K∇ p 2 ) = 0 with ⎧

u0 = 0 ⎪ ⎪ ⎪ ⎪ ⎪

u=0 ⎪ ⎪ ⎪ ⎨p

2 = 0 ⎪

u · n = −K∇ p

2 · n ⎪ ⎪ ⎪ ⎪ ⎪

u · τ = −2μG(D(

u)n) · τ ⎪ ⎪ ⎩ ((−2μD(

u) + p

1 I )n) · n + 12 (u, u) − 12 (v, v) = p 2

on Ω1 × {t = 0}; on Γ1 × (0, T ); on Γ2 × (0, T ); on Γ × (0, T ); on Γ × (0, T ); on Γ × (0, T ).

Firstly, taking L2 -energy estimate on the first equation of the system (4.3), we get  2 1 1 d

u 2L2 (Ω ) + 2μ D(

u · τ 2L2 (Γ ) + p u) L2 (Ω ) +

2 (

u · n) dS 1 1 2 dt G Γ   1 1 = (u · u)(

u · n) dS − (v · v)(

u · n) dS 2 Γ 2 Γ

(4.4)

P. Liu, W. Liu



 v · ∇

u ·

u dx .



−

u · ∇u ·

u dx + Ω1

(4.5)

Ω1

Taking L2 -energy estimate on the third equation of (4.3), we find   K∇ p

2 · ∇ p

2 dx − p

2 (

u · n) dS = 0. Ω2

(4.6)

Γ

Adding (4.5) and (4.6) together yields 2 1 d 1 u) L2 (Ω ) +

2 2L2 (Ω )

u 2L2 (Ω ) + 2μ D(

u · τ 2L2 (Γ ) + λ ∇ p 1 2 1 2 dt G   1 1  (u · u)(

u · n) dS − (v · v)(

u · n) dS 2 Γ 2 Γ   

u · ∇u ·

u dx + v · ∇

u ·

u dx . − Ω1

(4.7)

Ω1

Applying Green’s formula to the right hand side of (4.7), we know   1 1 (u · u)(

u · n) dS − (v · v)(

u · n) dS 2 Γ 2 Γ   =

u · ∇u · u dx −

u · ∇v · v dx 

Ω1

=



Ω1

u · ∇u ·

u dx + Ω1

u · ∇

u · v dx

(4.8)

Ω1

Substituting (4.8) into (4.7) obtains 2 1 d 1

u 2L2 (Ω ) + 2μ D(

u · τ 2L2 (Γ ) + λ ∇ p u) L2 (Ω ) +

2 2L2 (Ω ) 1 2 1 2 dt G   

u · ∇

u · v dx − v · ∇

u ·

u dx Ω1

(4.9)

Ω1

Thanks to Young’s inequality and Lemma 2.1, we get       

u · ∇

u · v dx − v · ∇

u ·

u dx   Ω1

Ω1

u L2 (Ω1 )

u L2 (Ω1 )  2 v L∞ (Ω1 ) ∇

  2 C η 2 D(

u) L2 (Ω ) +

u L2 (Ω )  2 v H 2 (Ω1 ) 1 1 2 η 2  η D(

u) L2 (Ω ) + C(η)

u 2L2 (Ω ) , 1

1

(4.10)

where we take η sufficiently small. Then substituting (4.10) into (4.9) obtains 2 2 d

u 2L2 (Ω ) + 2μ D(

u) L2 (Ω ) + λ ∇ p

2 L2 (Ω ) 

u 2L2 (Ω ) . 1 1 1 2 dt

(4.11)

Thus, it follows from Grönwall’s inequality that

u = 0 and p

2 = 0 for all t ∈ [0, T ). The proof of Theorem 4.1 is completed. 

Global Well-Posedness of an Initial-Boundary Value Problem. . .

5 Global Well-Posedness In this section, we will establish the global well-posedness for (1.1), namely, we are going to prove the Theorem 1.1. First of all, Theorem 4.1 provides us a local strong solution. Let T ∗ > 0 be the maximal existence time of the solution in Theorem 4.1. It suffices to prove T ∗ = +∞. For this end, we first prove the following continuation criterion.

5.1 Continuation Criterion Theorem 5.1 Continuation criterion Let K and u0 satisfy the assumptions of Theorem 1.1. Assume that (u, p2 ) solves equations (1.1) with initial data u0 on [0, T ∗ ) with the maximal existence time T ∗ < ∞. Suppose also that (u, p2 ) is a local-in-time smooth solution on [0, T ∗ ) in Theorem 4.1, that is,        (u, p2 ) ∈ C [0, T ∗ ); HΓ21 (Ω1 ) ∩ C 1 [0, T ∗ ); HΓ11 (Ω1 ) ∩ L2loc [0, T ∗ ); HΓ31 (Ω1 )      × L∞ [0, T ∗ ); HΓ22 (Ω2 ) ∩ L2loc [0, T ∗ ); HΓ32 (Ω2 ) . Then the solution (u, p2 ) may be continued beyond T ∗ into a smooth solution of (1.1) provided  T∗

∇u L∞ (Ω1 ) dt < +∞. 0

Remark 5.1 For the local solution (u, p2 ) ∈ C ([0, T ∗ ); HΓ21 (Ω1 )) × L2 ([0, T ∗ ); HΓ22 (Ω2 )) in Theorem 4.1, if T ∗ < +∞ is its maximal existence time, then 

T∗

∇u L∞ (Ω1 ) dt = +∞. 0

We now start to prove Theorem 5.1. Proof of Theorem 5.1 Taking L2 -estimate to the first equation in the system (1.1) over Ω1 and using div u = 0 and boundary conditions, meanwhile, taking L2 inner product of the third equation in the system (1.1) with p2 , then adding those two resulting estimate equations, we arrive at 2 1 1 d

u 2L2 (Ω ) + 2μ D(u) L2 (Ω ) + u · τ 2L2 (Γ ) + λ ∇p2 L2 (Ω2 )  0. 1 1 2 dt G

(5.1)

Integrating (5.1) with respect to the time t , we get for t ∈ [0, T ∗ ), u(t) 2 2 + L (Ω ) 1



t 0



2   2  2μ D u(s) L2 (Ω ) + λ ∇p2 (s) L2 (Ω ) ds  u0 2L2 (Ω ) . 1

2

1

(5.2)

Taking L2 inner product of the first equation in the system (1.1) with ut over Ω1 , meanwhile, applying ∂t to the third equation in the system (1.1) and then taking L2 (Ω2 ) inner product of the resulting equation with p2 , then adding those two resulting estimate equations, we have

P. Liu, W. Liu

ut 2L2 (Ω ) + 1

  2 1 d 1 2μ D(u) L2 (Ω ) + u · τ 2L2 (Γ ) + λ ∇p2 2L2 (Ω ) 2 1 2 dt G

1  ut 2L2 (Ω ) + u 2L∞ (Ω1 ) D(u) 2L2 (Ω ) . 1 1 2

(5.3)

Integrating both sides with respect to t it yields 2 2μ D(u) L2 (Ω ) + λ ∇p2 2L2 (Ω ) +



2

1

t

ut (s) 2 2 ds L (Ω ) 1

0

2 2  2μ D(u0 ) L2 (Ω ) + λ ∇p2 (0) L2 (Ω ) + 1



2

  D u(s) 2 2 u(s) 2 ∞ ds L (Ω ) L (Ω )

t

1

0

1

where trace theorem H 1 (Ω1 ) → L2 (Γ ) has been applied. Thanks to Lemma 2.3 and Lemma 2.4, from equations ⎧ ⎪ ⎨− div(K∇p2 ) = 0 −K∇p2 · n = u · n ⎪ ⎩ p2 = 0

in Ω2 on Γ on Γ2

it can be derived that

∇p2 H 1 (Ω2 )  C u

1 H 2 (Γ )

 C u H 1 (Ω1 )  C D(u) L2 (Ω ) , 1

(5.4)

and

∇p2 H 2 (Ω2 )  C u

3

H 2 (Γ )

 C u H 2 (Ω1 ) .

(5.5)

Therefore, we arrive at 2 2μ D(u) L2 (Ω ) + λ ∇p2 2L2 (Ω ) + 2

1

2  C D(u0 ) L2 (Ω ) +



1

t



ut (s) 2 2

t

L (Ω1 )

0

  D u(s) 2 2

L (Ω1 )

0

dτ

u(s) 2 ∞ ds. L (Ω ) 1

Now applying Grönwall’s inequality to (5.6) gives 2   2 2μ D u(t) L2 (Ω ) + λ ∇p2 (t) L2 (Ω ) + 1



2

2  C D(u0 ) L2 (Ω ) exp



1

t 0

t

1

0

u(s) 2 ∞ L (Ω

1

ut (s) 2 2 ds L (Ω )

 ds . )

Noticing that u 2L∞ (Ω1 )  C u0 L2 (Ω1 ) ∇u L∞ (Ω1 ) , we have 2   2 2μ D u(t) L2 (Ω ) + λ ∇p2 (t) L2 (Ω ) + 1

2



2  C D(u0 ) L2 (Ω ) exp u0 L2 (Ω1 ) 1

 0

t



t 0

ut (s) 2 2 ds L (Ω ) 1

∇u(s)

L∞ (Ω1 )

 ds

(5.6)

Global Well-Posedness of an Initial-Boundary Value Problem. . .

   t ∇u(s) ∞  C u0 2H 1 (Ω ) exp u0 L2 (Ω1 ) ds . L (Ω ) 1

(5.7)

1

0

Differentiating (1.1)1 with respect to t and then taking the L2 inner product of the resulting equation with ut over Ω1 . Meanwhile, applying ∂t to (1.1)3 and then taking L2 inner product of the resulting equation with ∂t p2 over Ω2 , then adding those two resulting estimate equations, we get 2 2 1 1 d

ut 2L2 (Ω ) + 2μ D(ut ) L2 (Ω ) + ut · τ 2L2 (Γ ) + λ ∇(∂t p2 ) L2 (Ω ) 1 1 2 2 dt G  ∇u L∞ (Ω1 ) ut 2L2 (Ω ) + u L∞ (Ω1 ) ut L2 (Ω1 ) ∇ut L2 (Ω1 ) 1



∇u L∞ (Ω1 ) ut 2L2 (Ω ) 1

+ C u 2L∞ (Ω1 ) ut 2L2 (Ω ) + μ ∇ut 2L2 (Ω ) . 1

1

(5.8)

Applying the Grönwall’s inequality, it yields  t 2 2   2μ D(ut ) L2 (Ω ) + λ ∇(∂t p2 ) L2 (Ω ) dτ

ut 2L2 (Ω ) + 1

1

0



 u0 2H 2 (Ω ) exp



1

here we used the fact that

2

C u0 2L2 (Ω ) + 1 1





t



∇u L∞ (Ω1 ) dτ ,

(5.9)

0

ut (0) 2 2  C u0 2H 2 (Ω ) , L (Ω ) 1

1

which follows from the compatibility conditions (1.2)   ut (0) = div 2μD(u0 ) − p1 (0)I − u0 · ∇u0 , where the pressure p1 (0) is determined by

− p1 (0) = div(u0 · ∇u0 ) p1 (0) = p2 (0) − 12 |u0 |2 − ∂x u10

in Ω1 , on Γ1 .

Applying ∂x to (1.1)1 and then taking the L2 inner product of the resulting equation with ut,x over Ω1 and employing ∂t ∂x to (1.1)3 and then taking L2 inner product of the resulting equation with ∂x p2 over Ω2 , then adding those two resulting estimate equations, we know   2 2 1 d 2 2μ D(ux ) L2 (Ω ) + ux · τ L2 (Γ ) + λ ∇(∂x p2 ) L2 (Ω ) + ut,x 2L2 (Ω ) 1 1 2 dt G 2 2  ∇u L∞ (Ω1 ) D(ux ) L2 (Ω ) D(u) L2 (Ω ) + u 2L∞ (Ω1 ) D(ux ) L2 (Ω ) . (5.10) 1

1

1

Applying Grönwall’s inequality to (5.10) obtains  t 2 2

ut,x 2L2 (Ω ) dτ 2μ D(ux ) L2 (Ω ) + λ ∇(∂x p2 ) L2 (Ω ) + 1

  2  D ux (0) L2 (Ω

2

1

0

1

   t ∞ exp C u0 L2 + sup D(u) L2 (Ω )

∇u L (Ω1 ) dτ . ) 0
1

0

(5.11)

P. Liu, W. Liu

Now, we estimate uyy . For this end, we first rewrite the first equation in the system (1.1) as μuyy = ut + u · ∇u − μuxx + ∇p1 .

(5.12)

Hence, we have  

uyy L2 (Ω1 )  C ut L2 (Ω1 ) + u · ∇u L2 (Ω1 ) + ∇p1 L2 (Ω1 ) + uxx L2 (Ω1 ) .

(5.13)

For the nonlinear term on right hand side, we have 3 1

u · ∇u L2 (Ω1 )  C u L∞ (Ω1 ) D(u) L2 (Ω )  C D(ux ) L2 2 (Ω ) D(u) L2 2 (Ω ) . 1

1

Next we estimate ∇p1 . From system ⎧ 2 ⎪ ⎨− p1 = tr(∇u) p1 = p2 − 12 |u|2 − u1x ⎪ ⎩ p1 = 0

1

(5.14)

in Ω1 , on Γ, on Γ1 ,

it can be derived that 

∇p1 L2 (Ω1 )  C |∇u|2 H −1 (Ω ) + p2

1

1

H 2 (Γ )

+ |u|2

1

H 2 (Γ )

+ u1x

 1

H 2 (Γ )

   C u · ∇u L2 (Ω1 ) + p2 H 1 (Ω2 ) + |u|2 H 1 (Ω ) + ux H 1 (Ω1 ) 1

3 1    C D(ux ) L2 2 (Ω ) D(u) L2 2 (Ω ) + D(u) L2 (Ω ) + D(ux ) L2 (Ω ) , 1 1 1 1 (5.15) where inequality (5.4) has been used. Substituting (5.14) and (5.15) into (5.13) and then adding D(ux ) L2 (Ω1 ) , we have 2 D u

L2 (Ω1 )

3 1    C ut L2 (Ω1 ) + D(ux ) L2 2 (Ω ) D(u) L2 2 (Ω ) + D(u) L2 (Ω ) + D(ux ) L2 (Ω ) 1 1 1 1 2 2    C 1 + ut 2 2 + D(u) 2 + D(ux ) 2 . (5.16) L (Ω1 )

L (Ω1 )

L (Ω1 )

Combining the estimates (5.2), (5.4), (5.5), (5.7), (5.9), (5.11) and (5.16) we arrive at  t

ut 2H 1 (Ω ) ds

u 2H 2 (Ω ) + p2 2H 3 (Ω ) + 1

2

0



1



   C 1 + u0 4H 2 (Ω ) exp C1 u0 L2 (Ω1 ) + u0 2H 1 (Ω 1

1)

   t   t × exp C2 u0 L2 (Ω1 )

∇u L∞ (Ω1 ) ds

∇u L∞ (Ω1 ) ds 0

(5.17)

0

where C, C1 and C2 are three nonnegative constants which all are independent of u and p2 . Based on the H 2 energy estimate the higher energy estimate can be obtained by bootstrap method. Thus the proof of Theorem 5.1 is completed. 

Global Well-Posedness of an Initial-Boundary Value Problem. . .

5.2 Proof of Theorem 1.1 Along with the local well-posedness (Theorem 4.1) and the continuation criterion (Theorem 5.1), we can prove our main result, i.e., Theorem 1.1. Proof of Theorem 1.1 Thanks to Theorem 4.1, we conclude that: given u0 ∈ HΓ21 (Ω2 ) satisfying div u0 = 0, (1.1) has a unique local solution (u, p2 ) satisfying        (u, p2 ) ∈ C [0, T ∗ ); HΓ21 (Ω1 ) ∩ C 1 [0, T ∗ ); HΓ11 (Ω1 ) ∩ L2loc [0, T ∗ ); HΓ31 (Ω1 )      × L∞ [0, T ∗ ); HΓ22 (Ω2 ) ∩ L2loc [0, T ∗ ); HΓ32 (Ω2 ) for some T ∗ > 0. Our aim of what fallows is to prove that this maximal existence time T ∗ = ∞. Thanks to the inequality (3.1) in Proposition 3.1, we have for all t ∈ [0, T ∗ ) 

u 2H 2 (Ω ) + ut 2H 1 (Ω ) + 1

1

t

 D(u) 2 2

H (Ω1 )

0

+ ∇p2 2H 2 (Ω

 2)

  dτ  C T ∗ ,

where C(T ∗ ) is continuously depends on T ∗ . Let t → T ∗ , we get 

T∗

D(u) 2 2

H (Ω1 )

0

  dt  C T ∗ < +∞,

(5.18)

which along with Korn’s inequality implies 

T∗

 

∇u 2H 2 (Ω ) dt  C T ∗ < +∞, 1

0

which implies that ∇u is bounded in L2 ([0, T ∗ ); H 2 (Ω1 )). Then, by the Sobolev embedding theorem H 2 (Ω1 ) → L∞ (Ω1 ), we have  0

T∗

 

∇u 2L∞ (Ω1 ) dt  C T ∗ < +∞,

which implies 

T∗

∇u L∞ (Ω1 ) dt  0

√

 T∗ 0

T∗

∇u 2L∞ (Ω1 ) dt

 12

   C T ∗ < +∞.

From this, by the Theorem 5.1 we may extend the solution (u, p2 ) beyond T ∗ into a smooth solution of (1.1), which contradicts the assumption of the maximal existence time T ∗ < +∞. This ends the proof of the Theorem 1.1.  Acknowledgements The authors would like to thank Professor Guilong Gui for helpful discussions. This work of the authors is partially supported by the National Natural Science Foundation of China under the grants 11571279 and 11331005. The authors would like to thank the referees for constructive suggestions and comments.

P. Liu, W. Liu

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