The combined inviscid and non-resistive limit for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions

Acta Mathematica Scientia 2018,38B(6):1655–1677 http://actams.wipm.ac.cn

THE COMBINED INVISCID AND NON-RESISTIVE LIMIT FOR THE NONHOMOGENEOUS INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS∗

ܓ*)

Zhipeng ZHANG (

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China; Department of Mathematics, Nanjing University, Nanjing 210093, China E-mail : [email protected] Abstract In this paper, we establish the existence of the global weak solutions for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions for the velocity field and the magnetic field in a bounded domain Ω ⊂ R3 . Furthermore, we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible magnetohydrodynamic equations in energy space. Key words

nonhomogeneous incompressible MHD equations; Navier boundary conditions; inviscid and non-resistive limit

2010 MR Subject Classification

1

35Q30; 76D03; 76D05; 76D07

Introduction

In this paper, we consider the nonhomogeneous incompressible magnetohydrodynamic (MHD) equations (for example, see [1])    ρt + u · ∇ρ = 0,     ρut + ρu · ∇u − H · ∇H + ∇π − ν∆u = 0, (1.1)    Ht + u · ∇H − H · ∇u − µ∆H = 0,    div u = div H = 0 in Q := (0, T ) × Ω, where Ω is a smooth bounded domain of R3 and T > 0. The unknowns are ρ, u = (u1 , u2 , u3 )t , H = (H1 , H2 , H3 )t and π, denoting the density, the velocity, the magnetic field and the pressure of the fluid, respectively, where (·, ·, ·)t represent the transpose of (·, ·, ·). The constant ν > 0 is the viscosity coefficient of the fluid, and µ > 0 is the resistivity constant acting as the magnetic diffusion coefficient of the magnetic field. ∗ Received April 24, 2017; revised October 16, 2017. This paper is supported by the National Natural Science Foundation of China (11671193) and the China Scholarship Council.

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We add to ρ, u and H the following initial-boundary value conditions:   ρ(0) = ρ0 ,   (ρu)(0) = v0 ,    H(0) = H0

in Ω, and

  u · n = 0,      (Su · n)τ = −αuτ ,

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(1.2)

(1.3)

 H · n = 0,      (SH · n)τ = −βHτ

on Σ, where Σ := (0, T ) × ∂Ω, and n is the outward unit normal vector to ∂Ω. ρ0 ≥ 0, and v0 has to be at least such that v0 (x) = 0 whenever ρ0 (x) = 0. Moreover, α and β are two given nonnegative constants, and the constant α measures the tendency of the fluid to slip on the boundary, S is the strain tensor defined by Sf =

1 (∇f + (∇f )t ), 2

where (∇f )t denotes the transpose of the matrix ∇f , and fτ stands for the tangential part of f on Σ, i.e., fτ = f − (f · n)n. The boundary condition (1.3)1 and (1.3)2 are the so-called Navier boundary conditions, which were introduced by Navier in [26] to show that the velocity is proportional to the tangential part of the stress. This kind boundary condition allows the fluid slip along the boundary and has important applications for problems with rough boundaries [2, 5]. For the magnetic field H, the Navier boundary conditions (1.3)3 and (1.3)4 can be rewritten in the form of the vorticity as H · n = 0,

n × (∇ × H) = (BH)τ

on Σ,

(1.4)

see [33] for details, where B is a smooth symmetric matrix. We remark that the perfectly conducting boundary conditions for the magnetic field take the form: H · n = 0,

n × (∇ × H) = 0 on Σ.

(1.5)

Therefore, the boundary condition (1.3)3 and (1.3)4 can be regarded as the generalization of (1.5). Additional, the Navier boundary conditions are adaptable to system (1.1), since it ensures the boundary balance of the quantities on the boundary. The interested reader can refer to [15, 34]. The problem (1.1)–(1.3) describes the macroscopic behavior of electrically conducting incompressible fluids in a magnetic field. Due to the significance of the physical background, the MHD equations were studied by many physicists and mathematicians on various topics, refer to [1, 3, 4, 7, 11, 12, 14, 16–18, 20–22, 24, 25, 27, 32, 35, 36]. Since the resistivity coefficient µ is inversely proportional to the electrical conductivity, it is reasonable to assume that there is no magnetic diffusion when the conducting fluid under consideration is of very high conductivity, we refer, for example, to [7, 27]. Consequently, for the extremely high electrical conductivity

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cases which occur frequently in many cosmical and geophysical problems, it is more rational to ignore the resistivity term µ∆H in (1.1)3 . Thus, it is very interesting to study the inviscid and non-resistive limits to the problem (1.1)–(1.3). The purpose of this paper is to establish the convergence of the weak solutions for (1.1)– (1.3), as ν and µ go to zero simultaneously, toward the strong one of the following ideal nonhomogeneous incompressible MHD equations   ρ0t + u0 · ∇ρ0 = 0,      ρ0 u0 + ρ0 u0 · ∇u0 − H 0 · ∇H 0 + ∇π 0 = 0, t  H 0 + u0 · ∇H 0 − H 0 · ∇u0 = 0,  t     div u0 = div H 0 = 0

(1.6)

with the following initial-boundary value conditions:   ρ0 (0) = ρ00       u0 (0) = u00   H 0 (0) = H00     u0 · n = 0      H0 · n = 0

in Ω, in Ω, in Ω,

(1.7)

on Σ, on Σ.

We aim at giving a rigorous justification of this formal procedure. Before we proceed our concerns, we first recall some results on the homogeneous or nonhomogeneous incompressible MHD equations with Dirichlet boundary condition. For the homogeneous incompressible MHD equations, Duvaut and Lions [4] constructed a class of global weak solution and a class of local strong solution to the initial-boundary value problem of the MHD equations, which is similar to the Leray-Hopf solutions to the Navier-Stokes equations. Sermange and Teman [14] mainly considered the large time behavior of the solutions to the homogeneous incompressible MHD equations. Additional, Ferreira and Villamizar-Roa [11] studied the stability of steady solutions for the homogeneous incompressible MHD equations in a bounded domain of R3 . They also proved the existence of fast decaying strong solutions for the non-steady MHD equations. Li and Cai [24] investigated the global L2 stability for large solutions to the homogeneous incompressible MHD equations in bounded or unbounded domains of R3 . Under suitable conditions for the large solutions, they showed that the large solutions are stable. For the nonhomogeneous incompressible MHD equations with the initial density away from vacuum, the existence and uniqueness are established for the local strong solution as well as for the global strong solution with small data by Li and Wang in [21]. We also refer to [12, 18, 22, 32] and the references therein for more results. However, there are only a few results on the inviscid and non-resistive limits for the incompressible MHD equations. In 2002, D´ıaz and Lerena [16] utilized the C 0 -semigroup technique developed in [19] to study the inviscid and non-resistive limits in the whole space R3 for the homogeneous incompressible MHD equations. Recently, Zhang in [36] extended the results in [16] to the nonhomogeneous case. Xiao, Xin and Wu [35] studied the inviscid limit for the

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homogeneous incompressible MHD equations with the following slip boundary conditions: ( u · n = 0, n × (∇ × u) = 0 on Σ, (1.8) H · n = 0, n × (∇ × H) = 0 on Σ, where they followed the approaches in [34] and formulated the boundary value in a suitable functional setting so that the Stokes operator is well behaved and the nonlinear terms fall into the desired functional spaces. Specially, the authors of this paper in [25] investigated the inviscid limit in a bounded domain of R3 for the homogeneous incompressible MHD equations with the boundary condition (1.3) in conormal Sobolev space. Note that, if we take H = 0 in (1.1), it is reduced to the nonhomogeneous incompressible Navier-Stokes equations. As for the research on the vanishing viscosity limit for the nonhomogeneous incompressible fluids, we can refer to [6, 23, 28, 31] and the references cited therein. In particular, the authors in [23] proved the convergence, as the viscosity coefficient goes to zero, of the weak solutions for the nonhomogeneous incompressible Navier-Stokes eqautions with Navier boundary conditions to the strong solution of the nonhomogeneous incompressible Euler equations. Now we give more detailed explanations on the purpose of this paper. The first goal is to prove the existence for a global weak solution of the nonhomogeneous incompressible MHD equations. We mainly follow and modify some ideas developed in [8, 9, 23, 29, 30]. We first introduce a regularized problem depending on a small positive parameter ǫ through a regularization of the problem (1.1)–(1.3) (see (2.8)–(2.10) below). In order to get the existence of a weak-strong solution (see Definition 2.5) to the regularized problem, we set up the approximate problem (2.17)–(2.19) of the regularized problem (2.8)–(2.10) by Galerkin approximate method. Next, we verify the existence of the solution for the approximate solution independent of the index m. Third, we prove more estimates for the approximate solution independent of the index m. By virtue of some compactness arguments, we give the existence of a weak-strong solution to the regularized problem. Finally, based on the lower semi-continuity of norms, we obtain the estimates for the solution of the regularized problem which is independent of ǫ. With the help of these estimates and some compactness arguments, we show the existence of a global weak solution for the problem (1.1)–(1.3). The second goal of this paper is using the energy method to prove the convergence of a global weak solution of the nonhomogeneous incompressible MHD equations with Navier boundary conditions, as ν and µ go to zero simultaneously, to the strong solution of the ideal nonhomogeneous incompressible MHD equations in energy space. In fact, due to the appearance of the magnetic field H in system (1.1), we need to overcome some new difficulties and to face more complicated energy estimates in the proof of the above two goals. Here, we list the main difficulties. On one hand, we need to use ω · ∇ρm , ρm ω · ∇um and ω · ∇H m to replace um · ∇ρm , ρm um · ∇um and um · ∇H m in (2.17) and (2.18), respectively, to get a nonlinear ordinary differential problem (2.23), where ω is a given function. Hence we can only give the local existence of the solution for problem (2.22) by using the classical theory of nonlinear ordinary equations directly, which is different from [23], where the ordinary differential equations are linear. In order to prove the global existence of the solution for problem (2.22), we need to do some energy estimates. Fortunately, we find that the local solution of (2.23) satisfies uniform estimate (2.32), such that we can extend the local solution of (2.22) to

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any time T > 0. On the other hand, we will need to estimate more terms which are caused by the terms which contain the magnetic field H, such as the nonlinear terms u · ∇H, H · ∇u and H · ∇H in system (1.1), see Step 3 in Section 2 for details. This paper is organized as follows. In the next Section, we give the precise definition of the weak solutions for the problem (1.1)–(1.3) and establish the existence of these weak solutions. In Section 3, we first introduce a result of local strong solution for the ideal nonhomogeneous incompressible MHD equations, and investigate the proof of the combined inviscid and nonresistive limit in energy space. Throughout the rest of the paper, we denote by D(Ω) and D′ (Ω) the space of C ∞ -functions with compact support and the space of distributions in Ω, respectively. Lp (Ω) and W m,p (Ω) stand for the usual Lebesgue and Sobolev spaces with norms k · kp and k · kW m,p (Ω) , respectively. In particular, W m,2 (Ω) is replaced by H m (Ω) with norm k · kH m (Ω) . For a Banach space X, we indicate by h·, ·iX ′ ,X the duality product between X ′ (the dual space of X) and X. We also denote by Hσ1 (Ω) the subspace of H 1 (Ω) in which the vector fields are divergence-free and tangent to the boundary.

2

The Weak Solutions

In this section, we establish the global existence of weak solutions to the nonhomogeneous incompressible MHD equations with Navier boundary conditions. Before stating the main result of this section, we first introduce the definition of global weak solutions to the problem (1.1)–(1.3). We assume that the initial density ρ0 ∈ Lp (Ω) (6 < p ≤ +∞), allowing vacuum, 2 2p |v0 |2 ∈ L1 (Ω) which imply v0 ∈ L p+1 (Ω), and H0 ∈ L2 (Ω). The initial data v0 and |vρ00| ρ0 correspond formally to the initial value for ρu and ρ|u|2 , respectively. In the case p = ∞, Lr -exponents depending on p should be understood in the natural way, that is, as the limit 2p when p → ∞. For instance, L p+1 becomes L2 in that case. Now we introduce the definition of a weak solution with finite energy for the problem (1.1)–(1.3). Definition 2.1 We call (ρ, u, H) a weak solution to the problem (1.1)–(1.3) if (ρ, u, H) satisfies   ρ ∈ C([0, T ]; W −1,p(Ω)) ∩ L∞ (0, T ; Lp (Ω)), ρ ≥ 0 a.e. in Q,     u ∈ L2 (0, T ; H 1 (Ω)), √ρu ∈ L∞ (0, T ; L2 (Ω)),   σ   ∞ H ∈ L (0, T ; L2(Ω)) ∩ L2 (0, T ; Hσ1 (Ω)), (2.1)   4p−6 4p−6  2 2  3p 3p ρui uj ∈ L (0, T ; L (Ω)), Hi Hj ∈ L (0, T ; L (Ω)),     3p  ui Hj ∈ L2 (0, T ; L 2p+2 (Ω)),

where i, j = 1, 2, 3, and 6 < p ≤ ∞ such that (1) The continuity equation holds in D′ (Q), the momentum equation and the magnetic field equation are verified in the following senses Z TZ Z TZ Z TZ − ρu · ∂t f dxdt − (ρu · ∇f ) · u dxdt + (H · ∇f ) · H dxdt 0

+ 2ν

Z

0

Ω T Z

Ω

0

Ω

Su : Sf dxdt + 2αν

Z

0

T

Z

∂Ω

0

u · f dσdt =

Z

Ω

Ω

v0 · f (0) dx,

(2.2)

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−

Z

T

0

+ 2µ

Z

Z

0

H · ∂t g dxdt −

Ω T Z

Z

0

T

Z

(u · ∇g) · H dxdt +

Ω

SH : Sg dxdt + 2βµ

Ω

Z

0

T

Z

T

0

Z

H · g dσdt =

∂Ω

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Z

Z

(H · ∇g) · u dxdt Ω

H0 · g(0) dx,

(2.3)

Ω

where (f, g) ∈ C 1 ([0, T ]; Hσ1 (Ω)) and (f, g)(T, x) = 0 a.e. in Ω. (2) The initial datum ρ(0) = ρ0 is verified in the following sense: Z ′ hρ(0), ψiW −1,p (Ω),W 1,p′ (Ω) = ρ0 ψ dx ∀ ψ ∈ W01,p (Ω). 0

(2.4)

Ω

(3) The energy inequality  Z t  Z t
1 √ 2 2 2 2 k( ρu)(t)k2 + kH(t)k2 + 2 ν kSuk2ds + µ kSHk2ds 2 0 0  Z tZ    Z tZ 1 v0 2 2 2 2 + 2 να |u| dσds + µβ |H| dσds ≤ k √ k2 + kH0 k2 2 ρ0 0 ∂Ω 0 ∂Ω

(2.5)

holds for a.e. t ∈ (0, T ). Remark 2.2 Note that div u = div H = 0 in Q and u · n = H · n = 0 on Σ are given by the choice of the space Hσ1 (Ω). The weak formulation (2.2) and (2.3) also contain the boundary condition (1.3)2 and (1.3)4 , respectively, in the sense that if (u, H) ∈ Hσ1 (Ω) ∩ H 2 (Ω), then (1.3)2 and (1.3)4 can be recovered, see [23] for the details. Remark 2.3 We also note that by taking test functions in D([0, T ) × Ω) in the form ϕ = ψ(x)θh (z) in (2.2) and (2.3), where ψ belongs to D(Ω) and is divergence-free, and θh (z) ∈ D([0, T )) such that θh (z) = 1 for z ≤ t and θh (z) = 0 for z ≥ t + h, and taking the limit as h → 0, we obtain that Z tZ Z Z tZ (H · ∇ψ) · H dxds (ρu · ∇ψ) · u dxds + (ρu)(t) · ψ dx − 0 Ω 0 Ω Ω Z tZ Z tZ Z + 2ν Su : Sψ dxds + 2αν u · ψ dσds = v0 · ψ dx, (2.6) 0

Z

Ω

0

∂Ω

Z tZ

Ω

Z tZ

H(t) · ψ dx − (u · ∇ψ) · H dxds + (H · ∇ψ) · u dxds 0 Ω 0 Ω Z tZ Z tZ Z + 2µ SH : Sψ dxds + 2βµ H · ψ dσds = H0 · ψ dx, Ω

0

Ω

0

∂Ω

(2.7)

Ω

which imply that ((ρu)(t), H(t)) converges weakly to (v0 , H0 ) as t → 0+ . The main result in this section reads 2

Theorem 2.4 Let 6 < p ≤ ∞, ρ0 ∈ Lp (Ω), ρ0 ≥ 0 a.e. in Ω, |vρ00| ∈ L1 (Ω), and H0 ∈ L2 (Ω). Then there exists a weak solution (ρ, u, H) of the problem (1.1)–(1.3) in the sense of Definition 2.1. In order to prove the global existence of a weak solution, we first introduce a regularized problem depending on a small positive parameter ǫ through a regularization of the problem (1.1)–(1.3). Specifically, fixed ǫ > 0, we need to find the solution (ρǫ , uǫ , H ǫ ) of the following

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regularized problem related to (1.1)–(1.3)   ρǫt + uǫ · ∇ρǫ = ǫ∆ρǫ ,       ρǫ uǫt + ρǫ uǫ · ∇uǫ − H ǫ · ∇H ǫ + ∇π ǫ − ν∆uǫ = − ǫ (∆ρǫ )uǫ , 2  ǫ ǫ ǫ ǫ ǫ ǫ  H + u · ∇H − H · ∇u − µ∆H = 0,  t     div uǫ = div H ǫ = 0

in Q, with the following initial-boundary value conditions   ρǫ (0) = ρǫ0 ,   (ρǫ uǫ )(0) = v0ǫ ,    ǫ H (0) = H0ǫ in Ω, and

(2.8)

(2.9)

 ǫ ∂ρ   = 0,   ∂n    ǫ    u · n = 0,

(2.10)

(Suǫ · n)τ = −αuǫτ ,       H ǫ · n = 0,     (SH ǫ · n)τ = −βHτǫ

on Σ, where ρǫ0 belongs to C 2,r (Ω), r ∈ (0, 1) (see [9]) with  ǫ ∂ρ0   = 0 on ∂Ω, 0 < ǫ ≤ ρǫ0 in Ω,   ∂n      x ∈ Ω : ρǫ0 (x) < ρ0 (x) → 0 as ǫ → 0,     ρǫ0 (x) → ρ0 in Lp (Ω) as ǫ → 0, 6 < p < ∞,     ǫ ρ0 (x) ⇀ ρ0 weakly − ∗ in L∞ (Ω) as ǫ → 0, and the initial datum (v0ǫ , H0ǫ ) is defined as ( ( v0 if ρǫ0 (x) ≥ ρ0 (x), H0 ǫ ǫ v0 = and H0 = ǫ 0 if ρ0 (x) < ρ0 (x) 0

1661

(2.11) p = +∞,

if ρǫ0 (x) ≥ ρ0 (x), if

ρǫ0 (x) < ρ0 (x).

(2.12)

We introduce the definition of a weak-strong solution to the above regularized problem. Definition 2.5 We call (ρǫ , uǫ , H ǫ ) a weak-strong solution of the problem (2.8)–(2.10), if (ρǫ , uǫ , H ǫ ) satisfies   ρǫ ∈ C([0, T ]; W −1,p (Ω)) ∩ L∞ (0, T ; Lp(Ω)),        ∇ρǫ ∈ L2 (0, T ; L2 (Ω)), ρǫ > 0 a.e. in Q,      √ ǫ   ∞ 2 ǫ 2 1   u ∈ L (0, T ; Hσ (Ω)), ρǫ u ∈ L (0, T ; L (Ω)), (2.13) ǫ ∞ 2 2 1    H ∈ L (0, T ; L (Ω)) ∩ L (0, T ; Hσ (Ω)),     4p−6 4p−6    ρǫ uǫi uǫj ∈ L 3p (0, T ; L2(Ω)), Hiǫ Hjǫ ∈ L 3p (0, T ; L2 (Ω)),       3p  ǫ ǫ ui Hj ∈ L2 (0, T ; L 2p+2 (Ω)),

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where i, j = 1, 2, 3 and 6 < p ≤ +∞, such that (1) Equation (2.8)1 holds a.e. in Q. The boundary condition (2.10)1 holds a.e. on Σ, and the initial condition (2.9)1 holds a.e. in Ω. (2) The momentum equation (2.8)2 is verified in the sense −

Z

Z

T

0

+

Z

ρ u · ∂t f dxdt −

Ω

Z

T

0

=−

ǫ ǫ

ǫ 2

Z

T

0

ǫ

Z

(ρǫ uǫ · ∇f ) · uǫ dxdt Ω

ǫ

(H · ∇f ) · H dxdt + 2ν

Ω Z T 0

Z

Z

T

0

∇ρǫ · ∇(uǫ · f ) dxdt +

Ω

Z

Z

Ω

ǫ

Su : Sf dxdt + 2αν

Ω

Z

0

T

Z

uǫ · f dσdt

∂Ω

v0ǫ · f (0) dx.

(2.14)

(3) The magnetic field equation (2.8)3 is verified in the sense −

Z

T 0

+ 2µ

Z

Z

0

H ǫ · ∂t g dxdt −

Ω T Z

Z

T

0

Z

(uǫ · ∇g) · H ǫ dxdt +

Ω

SH ǫ : Sg dxdt + 2βµ

Z

T

0

Ω

Z

Z

0

H ǫ · g dσdt =

∂Ω

T

Z

Z

Ω

(H ǫ · ∇g) · uǫ dxdt

Ω

H0ǫ · g(0) dx,

(2.15)

where (f, g) ∈ C 1 ([0, T ]; Hσ1 (Ω)) with (f, g)(T, x) = 0 a.e. in Ω. Now, we give the existence of a weak-strong solution to the regularized problem. Lemma 2.6 Let 6 < p ≤ ∞, ρǫ0 be as in (2.11), and (u0 , H0 ) as in Theorem 2.4. Then there exists a weak-strong solution (ρǫ , uǫ , H ǫ ) of the problem (2.8)–(2.10) in the sense of 6p Definition 2.5. Moreover, (ρǫ , uǫ ) satisfies ρǫ uǫ ∈ L2 (0, T ; L p+6 (Ω)), ρǫ ∈ Lς (0, T ; W 2,ς (Ω)), and ∂t ρǫ ∈ Lς (0, T ; Lς (Ω)) for some ς > 32 . Since the proof is complicated and lengthy, we divide it into five steps.  Step 1 Approximate problem Let ω k k∈N be a smooth basis of Hσ1 , orthonormal  in L2 (Ω). We define Ym := span ω 1 , · · · , ω m . For fixed ǫ > 0, we will define a sequence of approximated solutions (ρm,ǫ , um,ǫ , H m,ǫ ), m ∈ N, which converges to a weak-strong solution of the problem (2.8)–(2.10) as m → ∞. For presentation convenience, we shall omit the parameter ǫ, and write (ρm , um , H m ) in place of (ρm,ǫ , um,ǫ , H m,ǫ ), Proof

m

u (t) =

m X

j

φj (t)ω ,

m

H (t) =

j=1

m X

ψj (t)ω j .

(2.16)

j=1

For each m ∈ N, we will find ρm ∈ C([0, T ]; C 2 (Ω)) and (um , H m ) ∈ C 1 ([0, T ]; Ym ) satisfying  m ρ + um · ∇ρm = ǫ∆ρm    t    m ∂ρ =0  ∂n      m ρ (0) = ρǫ0

in Q, on Σ, in Ω

(2.17)

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and

Z  m m m m  ρm u m · ∇f ) · H m  t · f + (ρ u · ∇u ) · f + (H   Ω   Z    m  + 2νSu : Sf dx + 2αν um · f dσ    ∂Ω   Z     = ǫ ∇ρm · ∇(um · f ) dx, ∀ f ∈ Ym , 2 Ω Z   m    Ht · g − (um · ∇g) · H m + (H m · ∇g) · um    Ω  Z     m  + 2µSH : Sg dx + 2βµ H m · g dσ = 0, ∀ g ∈ Ym ,    ∂Ω    m u (0) = um H m (0) = H0m , 0 ,

where (um (0), H m (0)) ∈ Ym is uniquely determined by Z Z Z Z ǫ m ρǫ0 um · φ dx = v · φ dx, H · φ dx = H0ǫ · φ dx, 0 0 0 Ω

Ω

Ω

(2.18)

∀ φ ∈ Ym .

(2.19)

Ω

Note that (um (0), H m (0)) is well-defined because the matrices with coefficients R and Ω ω i · ω j dx are invertible, respectively.

R

Ω

ρǫ0 ω i · ω j dx

Step 2 Existence of the solution to the approximate problem For fixed m, by virtue of the Schauder’s fixed point theorem, some estimates independent m and compactness arguments, we prove the existence of (ρm , um , H m ). We first pay attention to the following linearized problem  ∂ ρm + ω · ∇ρm = ǫ∆ρm in Q,    t    m ∂ρ (2.20) =0 on Σ,  ∂n      m ρ (0) = ρǫ0 in Ω,

where ω ∈ C([0, T ]; Ym ).

From the following lemma, we get the existence of ρm ∈ C([0, T ]; C 2(Ω)) for fixed m ∈ N and ω. Lemma 2.7 (see Lemma 3.1 in [10]) Let ω ∈ C([0, T ]; Ym) be a given vector field. Suppose ∂ρǫ that ρǫ0 ∈ C 2,r (Ω), r ∈ (0, 1), inf x∈Ω ρǫ0 (x) > 0, and satisfies the compatibility condition ∂n0 = 0 on ∂Ω. Then (2.20) possesses a unique classical solution  ρm = ρm (ω) ∈ W = ρm ∈ C([0, T ]; C 2,r (Ω)), ∂t ρm ∈ C([0, T ]; C 0,r (Ω)) .

Moreover, the mapping ω → ρm (ω) maps bounded sets in C([0, T ]; Ym) into bounded sets in W and it is continuous with values in C 1 ([0, T ] × Ω). Finally, as div ω = 0, it holds that inf ρǫ0 (x) ≤ ρm (t, x) ≤ sup ρǫ0 (x),

x∈Ω

(t, x) ∈ [0, T ] × Ω.

(2.21)

x∈Ω

Next, given ω ∈ C([0, T ]; Ym) and ρm ∈ C([0, T ]; C 2(Ω)), we need to deal with the following

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problem: Z  m m m   ρm u m · ∇f ) · H m + 2νSum : Sf dx  t · f + (ρ ω · ∇u ) · f + (H   Ω  Z Z   ǫ  m   + 2αν u · f dσ = ∇ρm · ∇(um · f ) dx ∀ f ∈ Ym ,   2 Ω  ∂Ω Z  m Ht · g − (ω · ∇g) · H m + (H m · ∇g) · um + 2µSH m : Sg dx   Ω   Z      + 2βµ H m · g dσ = 0 ∀ g ∈ Ym ,    ∂Ω    m u (0) = um H m (0) = H0m . 0 ,

(2.22)

 m Putting (2.16) into (2.22), we obtain the following ordinary differential equations for φj j=1  m and ψj j=1 :  m m m X X X  1 dφj 2   a + a φ + a3ijk ψj ψk = 0, ij ij j   dt  j=1 j=1  j,k=1    m m m  X  X 1 dψj X 2 bij + bij ψj + b3ijk ψk φj = 0, (2.23) dt  j=1 j=1 j,k=1     m    φj (0) j=1 = components of um  0 ,      ψ (0) m = components of H m , j 0 j=1 where

a1ij = a2ij

Z Z

ρm ω i · ω j dx, Ω j

Ω

i

Z

i

Z

(ρ ω · ∇ω ) · ω dx + 2ν Sω : Sω dx + 2αν ω i · ω j dσ Ω ∂Ω Z ǫ ∇ρm · ∇(ω i · ω j ) dx, − 2 Ω Z a3ijk = (ω j · ∇ω i ) · ω k dx, ZΩ b1ij = ω i · ω j dx, Ω Z Z Z b2ij = − (ω · ∇ω i ) · ω j dx + 2µ Sω i : Sω j dx + 2βµ ω i · ω j dσ, Ω Ω ∂Ω Z b3ijk = (ω k · ∇ω i ) · ω j dx. =

m

j

Ω

 Due to the orthogonality of {ω k }k∈N in L2 (Ω), we easily find that B := b1ij = Id. Additional, since ρm ≥ ǫ, it holds that X 2 Z m m X T m i ξ Aξ = ρ (x, t) ξi ω (x) dx ≥ ǫ |ξi |2 , Ω

i=1

 where A := a1ij . We obtain that the following matrix

i=1

A

0

!

is symmetric, positive definite 0 B and invertible. Based on the classical theory of ordinary differential equations, problem (2.23)  m  m has a local unique solution φj j=1 ∈ (C 1 ([0, T1 ]))m and ψj j=1 ∈ (C 1 ([0, T1 ]))m , where,

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without loss of generality, we assume 0 < T1 < T . Then the local solvability of problem (2.22) is proved. Now, we aim at extending the solution (um , H m ) to the time T . Multiplying (2.20)1 by 1 m 2 m 2 |u | , integrating on Ω, and adding this result to (2.22)1 with f = u , we have Z Z 1 d √ m m 2 k ρ u k2 + 2νkSumk22 + 2να |um |2 dσ = (H m · ∇H m ) · um dx. (2.24) 2 dt ∂Ω Ω Taking g = H m in (2.22)2 , we obtain that Z Z 1 d kH m k22 + 2µkSH m k22 + 2µβ |H m |2 dσ = − (H m · ∇H m ) · um dx. 2 dt ∂Ω Ω

(2.25)

From (2.24) and (2.25), we arrive at

1 d √ m m 2 k ρ u k2 + kH m k22 + 2 νkSum k22 + µkSH m k22 2 dt Z  Z |um |2 dσ + µβ

+ 2 να

|H m |2 dσ

∂Ω

= 0.

(2.26)

∂Ω

Integrating (2.26) on (0, t] ⊂ [0, T1 ], we can obtain that p √ 2 m 2 k( ρm um )(t)k22 + kH m (t)k22 ≤ k ρǫ0 um 0 k2 + kH0 k2 .

(2.27)

As for the initial data, we have the following observations Z Z Z Z |v0 |2 |v0ǫ |2 ǫ m 2 dx ≤ dx ≤ C, v0ǫ · um dx = ρ |u | dx ≤ 0 0 0 ǫ Ω ρ0 Ω Ω Ω ρ0 Z Z Z |H0m |2 dx ≤ |H0ǫ |2 dx ≤ |H0 |2 dx ≤ C. Ω

Ω

(2.28) (2.29)

Ω

Here and in what follows, C denotes a generic constant independent of m and ǫ, and may change from line to line. Hence, we conclude that Z Z p ǫ m 2 √ m m |v0ǫ |2 2 m 2 m 2 |H0ǫ |2 dx. (2.30) k( ρ u )(t)k2 + kH (t)k2 ≤ k ρ0 u0 k2 + kH0 k2 ≤ ǫ dx + Ω Ω ρ0 By virtue of (2.21), for small ǫ, we obtain that R |v0ǫ |2 R p ǫ m 2 ǫ 2 ρ0 u0 k2 + kH0m k22 Ω ρǫ0 dx + Ω |H0 | dx ku ≤ . + kH ≤ ǫ ǫ m m P P Thanks to |φj (t)|2 = kum (t)k22 and |ψj (t)|2 = kH m (t)k22 , we get m

(t)k22

m

(t)k22

k

j=1

j=1

R p ǫ m 2 ρ0 u0 k2 + kH0m k22 Ω ≤ (|φj (t)| + |ψj (t)| ) ≤ ǫ j=1

m X

(2.31)

2

2

k

|v0ǫ |2 ρǫ0

dx + ǫ

R

Ω

|H0ǫ |2 dx

.

(2.32)

m Hence, using the uniform estimate (2.32), and the continuity of {φj (t)}m j=1 and {ψj (t)}j=1 with m respect to time, and regarding {φj (T1 )}m j=1 and {ψj (T1 )}j=1 as new initial data, we can easily m m extend the solution {φj (t)}j=1 and {ψj (t)}j=1 to the time T by repeating the previous processes, which implies that (um , H m ) exists on [0, T ]. Moreover, from (2.28)–(2.30), the solution also satisfies Z Z √ |v0 |2 dx + |H0 |2 dx, ∀ t ∈ [0, T ]. (2.33) k ρm um (t)k22 + kH m (t)k22 ≤ Ω ρ0 Ω

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Furthermore, by virtue of (2.21) and (2.33), we also conclude that (um , H m ) is bounded m P in C([0, T ], L2(Ω)) which is independent of m and ω. Using the equalities |φj (t)|2 = kum k22 and

m P

j=1

2

|ψj (t)| =

kH m k22



again, we obtain that φj m

m

m

j=1

j=1

 m and ψj j=1 are bounded in C([0, T ]),

which further imply that (u , H ) is bounded in C([0, T ], Ym ) (independent of ω). Additional, if ω is bounded in C([0, T ]; Ym), from (2.23) and the properties of the matrix M , we obtain that  m  m ∂t φj j=1 and ∂t ψj j=1 are bounded in C([0, T ]), which imply that (∂t um , ∂t H m ) is bounded in C([0, T ], Ym ). Thus, we obtain that (um , H m ) is bounded in C 1 ([0, T ], Ym ) provided that ω is bounded in C([0, T ], Ym ). It is precise that, given ω ∈ C([0, T ]; Ym ), based on Lemma 2.7, there exists a unique solution ρm ∈ C([0, T ]; C 2,r (Ω)) of (2.20). With the help of the known functions ρm and ω, we can verify that there exists a unique solution (um , H m ) ∈ C 1 ([0, T ]; Ym ) of (2.22), which is bounded in C 1 ([0, T ]; Ym ) provided that ω is bounded in C([0, T ]; Ym). Thus, there are two positive constants M1 and M2 such that kum kC 1 ([0,T ];Ym ) ≤ M2 if kωkC([0,T ];Ym) ≤ M1 . Denote by B1 the closed ball in C([0, T ]; Ym) of radius M1 and B2 the closed ball in C 1 ([0, T ]; Ym ) of radius M2 . Then, the mapping T : B1 → B2 , ω → um

is continuous. Furthermore, by the Arzel` a-Ascoli’s theorem, we know that B2 is compact in C([0, T ]; Ym). Therefore, the mapping T is continuous and compact from B1 to B1 . Then, in view of the Schauder’s fixed point theorem, we obtain that the existence of a fixed point um for given time T > 0. Taking ρm the corresponding solution of (2.20), we obtain the existence of an approximate solution (ρm , um , H m ) of (2.17)–(2.19). Step 3 Estimates for (ρm , um , H m ) We will show several estimates for (ρm , um , H m ) which are independent of m and ǫ. (1) From Lemma 2.7, we obtain that 0 < inf ρǫ0 ≤ ρm (x, t), x∈Ω

(t, x) ∈ [0, T ] × Ω.

(2.34)

Multiplying (2.17)1 by ρm |ρm |p−2 , integrating by parts and using the incompressible condition of um , we have Z p−2 1 d m p kρ kp + ǫ(p − 1) (2.35) ||∇ρm ||ρm | 2 |2 dx = 0. p dt Ω Hence, based on (2.11)3 and (2.11)4 , we obtain that kρm (t)kp ≤ kρǫ0 kp ≤ C,

∀ t ∈ [0, T ].

(2.36)

(2) We multiply (2.17)1 by ρm , and use the incompressible condition of um and integration by parts to obtain that Z t (2.37) kρm (t)k22 + 2ǫ k∇ρm (s)k22 ds = kρǫ0 k22 . 0

Thus √ √ k ǫ∇ρm kL2 (0,T ;L2 (Ω)) + k ǫ∆ρm kL2 (0,T ;W −1,2 (Ω)) ≤ C.

(2.38)

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m (3) Integrating (2.17)1 on Ω and using ∂ρ ∂n Σ = 0, we obtain the total mass conversation Z Z ρm (t) dx = ρǫ0 dx, t ∈ [0, T ]. (2.39)

Ω

Ω

(4) Similar to (2.26), taking f = um and g = H m in (2.18), we can easily get that

1 d √ m m 2 k ρ u k2 + kH m k22 + 2 νkSum k22 + µkSH m k22 2 dt Z  Z |um |2 dσ + µβ

+ 2 να

∂Ω

|H m |2 dσ

= 0.

(2.40)

∂Ω

Furthermore, integrating (2.40) on (0, t] ⊂ [0, T ], we obtain that  Z t  Z t
1 √ m m 2 m 2 m 2 m 2 k( ρ u )(t)k2 + kH (t)k2 + 2 ν kSu k2 ds + µ kSH k2 ds 2 0 0  Z tZ  Z tZ + 2 να |um |2 dσds + µβ |H m |2 dσds 0

∂Ω

0

∂Ω


1 p 2 m 2 (2.41) ≤ k ρǫ0 um 0 k2 + kH0 k2 . 2 Using (2.28) and (2.29), we arrive at √ k( ρm um )(t)k22 + kH m (t)k22 ≤ C, ∀ t ∈ [0, T ]. (2.42)  Additional, (Sum , SH m ) m∈N+ is bounded in L2 (0, T ; L2(Ω)). In order to prove that  m m (u , H ) m∈N+ is bounded in L2 (0, T ; H 1 (Ω)), we shall apply the following generalized Korn’s inequality (see [10])
kvk2H 1 (Ω) ≤ C(K, M, p) kSvk22 + kRvk21
≤ C(K, M, p) kSvk22 + kRk1 kRv 2 k1 (2.43) R for v ∈ Hσ1 (Ω) and any function R ≥ 0 satisfying 0 < M ≤ Ω R dx, kRkp ≤ K, for some p > 1.  In view of the generalized Korn inequality, we can easily get that H m m∈N+ is bounded  in L2 (0, T ; H 1(Ω)) ⊂ L2 (0, T ; L6(Ω)), independent of ǫ. As for um m∈N+ , without loss of generality, we can assume that ρ0 ≥ 0 and ρ0 6≡ 0. It follows from (2.39) and the fact that ρǫ0 → ρ0 in Lp (Ω), 6 < p < ∞, that Z Z 1 m M := kρ0 k1 ≤ ρ dx = ρǫ0 dx ≤ 2kρ0 k1 , for small ǫ > 0 and any m. (2.44) 2 Ω Ω  Hence, taking R = ρm in (2.43) and using (2.44), we infer that um m∈N+ is bounded in L2 (0, T ; H 1(Ω)) ⊂ L2 (0, T ; L6(Ω)), independent of ǫ. For the case p = ∞, since the weak-∗ R R convergence ρǫ0 ⇀ ρ0 in L∞ implies that Ω ρǫ0 dx → Ω ρ0 dx, we can also obtain the same  m conclusion for u m∈N+ by choosing the same M .   √ ρm um m∈N+ and um m∈N+ , we deduce the (5) Using the bounds for ρm m∈N+ ,  following estimates for ρm um m∈N+ : kρm um k

kρm um k

2p L∞ (0,T ;L p+1

(Ω))

6p

L2 (0,T ;L p+6 (Ω))

1 √ ≤ kρm kL2 ∞ (0,T ;Lp (Ω)) k ρm um kL∞ (0,T ;L2 (Ω)) ≤ C,

≤ kρm kL∞ (0,T ;Lp (Ω)) kum kL2 (0,T ;L6 (Ω)) ≤ C.

(2.45)

(2.46)

Furthermore, in view of interpolation inequality, we have kρm um k2 ≤ kρm um kθ

2p

L∞ (0,T ;L p+1 (Ω))

kρm um k1−θ 6p , p+6

(2.47)

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< 1. Thus kρm um kLq (0,T ;L2 (Ω)) ≤ C with q =

2(2p − 3) > 6. 3

(6) Using H¨older’s inequality, we have 1 √ 12p ≤ kρm kL2 ∞ (0,T ;Lp (Ω)) kum kL2 (0,T ;L6 (Ω)) ≤ C. k ρm u m k 2 6+2p (Ω))

L (0,T ;L

Since p > 6, we have

12p 6+2p

(2.48)

(2.49)

> 4. It follows from interpolation inequality that

p−6 3p √ √ √ k ρm um k4 ≤ k ρm um k24p−6 k ρm um k 4p−6 12p .

(2.50)

6+2p

Furthermore, taking ζ =

2(4p−6) , 3p

we obtain ζ > 2 and

2(p−6) √ √ √ k ρm um kζ4 ≤ k ρm um k2 3p k ρm um k212p . 6+2p √ m ζ 4 Consequently, ρm u m∈N+ is bounded in L (0, T ; L (Ω)), which implies that

m kρm um i uj k

ζ

L 2 (0,T ;L2 (Ω))

≤ C.

(2.51)

(2.52)

  (7) Since H m m∈N+ is bounded in L2 (0, T ; H 1 (Ω)), we can easily deduce that H m m∈N+ is bounded in the following spaces 6p

12p

L2 (0, T ; L p+6 (Ω)), L2 (0, T ; L 6+2p (Ω)), Lζ (0, T ; L4 (Ω)).

(2.53)

Thus, we also obtain that kHim Hjm k

ζ

L 2 (0,T ;L2 (Ω))

≤ C.

(2.54)

Additional, from kum kL2 (0,T ;H 1 (Ω)) ≤ C and kH m kL∞ (0,T ;L2 (Ω)) ≤ C, we obtain that m kum i Hj k

3p 2p+2

≤ kum i k

6p p+1

kHjm k

2p p+1

,

(2.55)

 3p m is bounded in L2 (0, T ; L 2p+2 (Ω)). which implies um i Hj m∈N+   (8) Finally, we estimate the temporal derivative of ρm m∈N+ , ρm um m∈N+ and  m 2p H m∈N+ . First, since ∂t ρm = −div (ρm um ) + ǫ∆ρm and W −1,2 (Ω) ⊂ W −1, p+1 (Ω), by using (2.38) and (2.45), we conclude that
√ √ 2p 2p k∂t ρm k 2 ≤ C kρm um k ∞ + ǫk ǫ∆ρm kL2 (0,T ;W −1,2 (Ω)) −1, p+1 p+1 L (0,T ;W

(Ω))

L

≤C

(0,T ;L

(Ω))

(2.56)

for small ǫ > 0.  Next, for ρm um m∈N+ , in view of (2.18)1 , H¨older’s inequality, and Sobolev’s embedding inequality, we obtain, for all f ∈ D(Ω) and s > 3, that d Z ρm um f dx ≤ kρm um ⊗ um k2 k∇f k2 + 2νkSum k2 kSf k2 dt Ω ǫ + 2ανkum kL2 (∂Ω) kf kL2(∂Ω) + k∇ρm k2 kum k6 k∇f k3 2 ǫ + k∇ρm k2 k∇um k2 kf k∞ + kH m k2 k∇H m k2 kf k∞ 2 ≤ C kρm um ⊗ um k2 + kum kH 1 (Ω) + ǫk∇ρm k2 kum kH 1 (Ω)

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+ kH m k2 k∇H m k2 kf kW 1,s (Ω) ,

1669 (2.57)

 m where um ⊗ um = um i uj 3×3 . Hence, based on (2.38) and (2.52), we deduce that d Z ρm um f dx ≤ fm kf kW 1,s (Ω) , ∀ f ∈ D(Ω) dt Ω

(2.58)

for some fm ∈ L1 (0, T ), which implies that

k∂t (ρm um )kL1 (0,T ;W −1,s′ (Ω)) ≤ C,

1 1 + = 1. s s′

(2.59)

 Finally, as for H m m∈N+ , in view of (2.18)2 , the uniform estimates in (4), H¨older’s inequality, and Sobolev embedding inequality, we have, for s > 3, d Z
H m g dx ≤ kum k2 k∇H m k2 + k∇um k2 kH m k2 kgk∞ dt Ω + 2µkSH m k2 kSgk2 + 2βµkH m kL2 (∂Ω) kgkL2(∂Ω)

≤C kum k2 k∇H m k2 + k∇um k2 kH m k2
+ kH m kH 1 (Ω) kgkW 1,s (Ω) ≤ gm kgkW 1,s (Ω) ,

∀ g ∈ D(Ω)

(2.60)

for some gm ∈ L1 (0, T ), which implies that k∂t H m kL1 (0,T ;W −1,s′ (Ω)) ≤ C,

1 1 + ′ = 1. s s

(2.61)

Step 4 Convergence properties We will use the uniform estimates obtained in Step 3 to infer some convergence properties of the approximate solution. We first pay attention to 2p ρm . Since H 1 (Ω) ֒→֒→ L2 (Ω) ֒→ W −1, p+1 (Ω), kρm kL2 (0,T ;H 1 (Ω)) ≤ C(ǫ) and k∂t ρm k

L2 (0,T ;W

−1,

2p p+1

(Ω))

≤ C.

(2.62)

Here and in what follows, C(ǫ) indicates that the constant C depends on ǫ. By using Lemma 4 in [29], we obtain that  m ρ m∈N+ is relatively compact in L2 (0, T ; L2(Ω)).

 2p Similarly, since Lp (Ω) ֒→֒→ W −1,p (Ω) ֒→ W −1, p+1 (Ω) and ρm m∈N+ is bounded in L∞ (0, T ; Lp(Ω)) , we deduce that  m ρ m∈N+ is relatively compact in C([0, T ]; W −1,p(Ω)). 6p

6p

′

6p For ρm um , as s > 3 ≥ 5p−6 , L p+6 (Ω) ֒→֒→ W −1, p+6 (Ω) ֒→ W −1,s (Ω), by using (2.46) and (2.59), we conclude that  m m 6p ρ u m∈N+ is relatively compact in L2 (0, T ; W −1, p+6 (Ω)).  Similar to ρm um m∈N+ , we can easily obtain that



Hm



m∈N+

6p

is relatively compact in L2 (0, T ; W −1, p+6 (Ω)).

Additional, in view of the uniform bounds obtained in Step 3, we deduce that the se m m m m m quence (ρm , um , H m , ρm um , ρm um i uj , Hi Hj , ui Hj ) m∈N+ converges (up to subsequences)

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to (ρǫ , uǫ , H ǫ , ξ1ǫ , (ξ2ǫ )ij , (ξ3ǫ )ij , (ξ4ǫ )ij ) as m → ∞, in the following senses   strongly in L2 (0, T ; L2 (Ω)) and a.e. in Q,   ρm → ρǫ weakly − ∗ in L∞ (0, T ; Lp(Ω)),    strongly in C([0, T ]; W −1,p(Ω)), um → uǫ weakly in L2 (0, T ; Hσ1 (Ω)),  weakly in L2 (0, T ; Hσ1 (Ω)),     6p   weakly in L2 (0, T ; L 6+p (Ω)), m ǫ H →H ∞  weakly − ∗ in L (0, T ; L2(Ω)),     6p  strongly in L2 (0, T ; W −1, p+6 (Ω)),  6p  weakly in L2 (0, T ; L p+6 (Ω)),    2p ρm um → ξ1ǫ weakly − ∗ in L∞ (0, T ; L p+1 (Ω)),   6p   strongly in L2 (0, T ; W −1, p+6 (Ω)), ζ

2 m ǫ 2 ρm u m i uj → (ξ2 )ij weakly in L (0, T ; L (Ω)), ζ 2

Him Hjm → (ξ3ǫ )ij weakly in L (0, T ; L2(Ω)), m ǫ 2 um i Hj → (ξ4 )ij weakly in L (0, T ; L

3p 2p+2

(2.63)

(2.64)

(2.65)

(2.66)

(2.67) (2.68)

(Ω)).

(2.69)

Finally, we identify the limits ξ1ǫ , ξ2ǫ , ξ3ǫ and ξ4ǫ . Note that the product mapping from 6p H 1 (Ω) × W −1,p (Ω) to W −1, p+1 (Ω) is continuous (see Lemma 3 in [29]). Thus, from (2.63)3 and (2.64), we can obtain that 6p

ρm um → ρǫ uǫ weakly in L2 (0, T ; W −1, p+1 (Ω)),

(2.70) 6p

which implies that ξ1ǫ = ρǫ uǫ . Also, notice that the product mapping from H 1 (Ω)×W −1, p+1 (Ω) 3p to W −1, p+3 (Ω) is continuous (see also Lemma 3 in [29]). Therefore, based on (2.64), (2.65)1 , (2.65)4 , (2.66)3 and (2.70), we can infer that 3p

−1, p+3 m ǫ ǫ ǫ 1 ρm u m (Ω)), i uj → ρ ui uj weakly in L (0, T ; W

Him Hjm → Hiǫ Hjǫ weakly in L1 (0, T ; W m um i Hj

→ uǫi Hjǫ

1

weakly in L (0, T ; W

3p −1, p+3

3p −1, p+3

(Ω)),

(Ω)).

(2.71) (2.72) (2.73)

The uniqueness of the limit in the sense of distributions implies that (ξ2ǫ )ij = ρǫ uǫi uǫj , (ξ3ǫ )ij = Hiǫ Hjǫ , (ξ4ǫ )ij = uǫi Hjǫ . Step 5 Passing to the limit as m → ∞ In this last step, we will show the existence of a weak-strong solution to the problem (2.8)–(2.10) by passing to the limit as m → ∞ in the approximate problem (2.17)–(2.19). First, using (2.48) and the maximal regularity for  parabolic equations, we obtain that ρm m∈N+ is bounded in Lq (0, T ; H 1(Ω)), which implies  that ∇ρm m∈N+ is bounded in Lq (0, T ; L2(Ω)). This bound together with the estimate for  m u m∈N+ in L2 (0, T ; L6 (Ω)) us to apply the classical Lς -theory of parabolic equations (see Theorem 10.22 in [10]) to conclude that k∂t ρm kLς (0,T ;Lς (Ω)) + kρm kLς (0,T ;W 2,ς (Ω)) ≤ C(ǫ),

(2.74)

No.6

where ς = Since

Z.P. Zhang: NONHOMOGENEOUS INCOMPRESSIBLE MHD EQUATIONS 2q q+2

1671

> 32 . ρm → ρǫ ,

ρm u m → ρ ǫ u ǫ ,

∆ρm → ∆ρǫ

in

D′ (Q),

by passing to the limit in (2.17)1 in the sense of distributions, we can conclude that ∂t ρǫ + uǫ · ∇ρǫ = ǫ∆ρǫ

in D′ (Q).

(2.75)

In view of (2.74) and the lower semi-continuity of norms, we further get that ∂t ρǫ + uǫ · ∇ρǫ = ǫ∆ρǫ

a.e. in Q.

(2.76)

As for the initial value of ρǫ , from (2.63)3 , we have ρm (0) → ρǫ (0) in W −1,p (Ω). In addition, the initial condition (2.17)3 shows that ρm (0) = ρǫ0 ∈ C 2,r (Ω), so we have ρǫ (0) = ρǫ0 . Moreover, it is easy to see that the boundary condition is also satisfied. Next, we pass to the limit in (2.18) as m → ∞. Using the energy identity (2.37) and (2.76) for ρm and ρǫ , respectively, we can deduce that ∇ρm converges strongly to ∇ρǫ in L2 (0, T1 ; L2 (Ω)) (see [8]). Hence, using the convergence obtained in Step 4 and passing to the limit in (2.18) as m → ∞, we can verify that (2.14) and (2.15) hold. We complete the proof of Lemma 2.6.  Proof of Theorem 2.4 Let (ρǫ , uǫ , H ǫ ) be the weak-strong solution of the problem (2.8)–(2.10) given by Lemma 2.6. Recall that the uniform estimates obtained in Step 3 of the proof of Lemma 2.6 are independent of ǫ > 0. So, similar to the Step 4 in the proof of Lemma 2.6,  we can conclude that there exists a subsequence of (ρǫ , uǫ , H ǫ , ρǫ uǫ , ρǫ uǫi uǫj , Hiǫ Hjǫ , uǫi Hjǫ ) ǫ>0 converging to (ρ, u, H, β1 , (β2 )ij , (β3 )ij , (β4 )ij ) as ǫ → 0 in the following senses ( weakly − ∗ in L∞ (0, T ; Lp(Ω)), ǫ (2.77) ρ →ρ strongly in C([0, T ]; W −1,p(Ω)), uǫ → u weakly in L2 (0, T ; Hσ1 (Ω)),  2 1   weakly in L (0, T ; Hσ (Ω)),   6p   weakly in L2 (0, T ; L 6+p (Ω)), ǫ H →H ∞  weakly − ∗ in L (0, T ; L2(Ω)),     6p  strongly in L2 (0, T ; W −1, p+6 (Ω)),  6p 2   weakly in L (0, T ; L p+6 (Ω)),   2p ρǫ uǫ → β1 weakly − ∗ in L∞ (0, T ; L p+1 (Ω)),   6p   strongly in L2 (0, T ; W −1, p+6 (Ω)), ζ

ρǫ uǫi uǫj → (β2 )ij weakly in L 2 (0, T ; L2(Ω)), ζ 2

Hiǫ Hjǫ → (β3 )ij weakly in L (0, T ; L2(Ω)), uǫi Hjǫ

2

→ (β4 )ij weakly in L (0, T ; L

3p 2p+2

(Ω)).

(2.78)

(2.79)

(2.80)

(2.81) (2.82) (2.83)

Moreover, using (2.38) and the continuity of the product mapping from H 1 (Ω) × W −1,2 (Ω) to 3 W −1, 2 , we obtain ǫ∇ρǫ → 0 strongly in L2 (0, T ; L2(Ω)),

(2.84)

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3

ǫuǫ ∆ρǫ → 0 weakly in L1 (0, T ; W −1, 2 (Ω)).

(2.85)

Similar to the compactness arguments of Step 4 in the proof of Lemma 2.6, we can verify that β1 = ρu, (β2 )ij = ρui uj , (β3 )ij = Hi Hj , (β4 )ij = ui Hj . From the above convergence properties, analogous to Step 5, we can pass to the limit in the problem (2.8)–(2.10). The only difference is that the two terms involving the parameters ǫ vanish. Then, we obtain that the mass equation is proved in D′ (Q). From (2.77)2 , we have that ρǫ0 = ρǫ (0) → ρ(0) in W −1,p (Ω). However, as ρǫ0 → ρ0 in Lp (Ω), when 6 < p < ∞, and ρǫ0 ⇀ ρ0 weakly-∗ in L∞ (Ω), when p = ∞, then ρ(0) = ρ0 at least in W −1,p (Ω). Finally, we verify the energy inequality (2.5). Multiplying (2.41) by φ ∈ D(0, T ), φ > 0, integrating over (0, T ), using the convergence properties (2.63)–(2.69), we arrive at   Z t  Z T  Z t √ ǫ ǫ 1 2 ǫ 2 ǫ 2 ǫ 2 k( ρ u )(t)k2 + kH (t)k2 + 2 ν kSu k2 ds + µ kSH k2 ds 2 0 0 0  Z tZ  Z tZ + 2 να |uǫ |2 dσds + µβ |H ǫ |2 dσds φ(t)dt 0

∂Ω

0

∂Ω

 Z T |v0,ǫ | 2 1 k√ k2 + kH0ǫ k22 ≤ φ(t)dt. 2 ρ0,ǫ 0

(2.86)

As ǫ → 0, we have Z

Ω

|v0,ǫ |2 dx → ρ0,ǫ

Z

Ω

|v0 |2 dx, ρ0

Z

Ω

|H0ǫ |2 dx →

Z

|H0 |2 dx.

Ω

Furthermore, in view of the convergence properties (2.77)–(2.83), we can easily get   Z t  Z T  Z t 1 √ 2 2 2 2 k( ρu)(t)k2 + kH(t)k2 + 2 ν kSuk2ds + µ kSHk2ds 2 0 0 0  Z tZ  Z tZ + 2 να |u|2 dσds + µβ |H|2 dσds φ(t)dt 0

∂Ω

0

 Z T |v0 | 1 k √ k22 + kH0 k22 φ(t)dt ≤ 2 ρ0 0

∂Ω

(2.87)

for any φ ∈ D(0, T ), φ > 0. This yields the energy inequality (2.5). Consequently, we obtain the existence of a weak solution in the sense of Definition 2.1. 

3

Combined Inviscid and Non-resistive Limit to the MHD Equations

In this section, we prove the convergence of the weak solutions obtained in Section 2 to the strong solution of the ideal nonhomogeneous incompressible MHD equations as ν and µ go to zero simultaneously. To this end, we recall a result of existence of strong solutions to the ideal nonhomogeneous incompressible MHD equations. Theorem 3.1 (see [13]) Let Ω be a bounded domain with boundary ∂Ω ∈ C ∞ and 0 < ρ∗ ≤ ρ00 ≤ ρ∗ < ∞, for two constants ρ∗ and ρ∗ . Moreover, assume that ρ00 , u00 , H00 ∈ H s (Ω) with integer s ≥ 3, div u00 = div H00 = 0 in Ω, and u00 · n = H00 · n = 0 on ∂Ω. Then there exists a positive time T ∗ such that the problem (1.6)–(1.7) has a unique solution (ρ0 , u0 , H 0 ) satisfying 0 < ρ∗ ≤ ρ0 ≤ ρ∗ < ∞,

(ρ0 , u0 , H 0 ) ∈ L∞ (0, T ∗ ; H s (Ω)).

(3.1)

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Furthermore, if u0 and H 0 satisfy ∇u0 , ∇H 0 ∈ L∞ (0, T ; L∞(Ω))

(3.2)

with 0 < T < ∞, then the solution (ρ0 , u0 , H 0 , π 0 ) can be extended beyond T > 0. Remark 3.2 By Sobolev embedding inequality and the regularity of the strong solution, we have that u0 ∈ L∞ ((0, T ∗ ) × Ω), (∇ρ0 , ∇u0 , ∇H 0 ) ∈ L∞ ((0, T ∗ ) × Ω), u0t ∈ L∞ ((0, T ∗ ) × Ω). Let (ρν,µ , uν,µ , H ν,µ ) be a weak solution, given by Theorem 2.4, of the system   ρν,µ + uν,µ · ∇ρν,µ = 0,  t     ρν,µ uν,µ + ρν,µ uν,µ · ∇uν,µ − H ν,µ · ∇H ν,µ + ∇π ν,µ − ν∆uν,µ = 0, t

 Htν,µ + uν,µ · ∇H ν,µ − H ν,µ · ∇uν,µ − µ∆H ν,µ = 0,      div uν,µ = div H ν,µ = 0

in Q, with the following initial-boundary conditions   ρν,µ (0) = ρν,µ  0 ,  ν,µ ν,µ (ρ u )(0) = v0ν,µ ,    ν,µ H (0) = H0ν,µ in Ω, and

 ν,µ   u · n = 0,     (Suν,µ · n)τ = −αuν,µ , τ

 H ν,µ · n = 0,      (SH ν,µ · n)τ = −βHτν,µ

(3.3)

(3.4)

(3.5)

on Σ. We now give the main result of this section.

Theorem 3.3 Suppose that the hypotheses of Theorem 2.4 and Theorem 3.1 hold. Let (ρν,µ , uν,µ , H ν,µ ) be the solution of the problem (1.1)–(1.3) given in Theorem 2.4, and (ρ0 , u0 , H 0 ) be the solution of the problem (1.6)–(1.7) given in Theorem 3.1. Assume further that ρ∗ ≤ ρν,µ ≤ ρ∗ for all ν, µ > 0, where ρ∗ and ρ∗ are the same constants of Theorem 3.1. 0 Then there exists C > 0 independent of ν > 0 and µ > 0 such that the following inequality holds kρ0 (t) − ρν,µ (t)k22 + ku0 (t) − uν,µ (t)k22 + kH 0 (t) − H ν,µ (t)k22  q v0ν,µ 2 ν,µ 0 2 p ρ u − k2 + kH00 − H0ν,µ k22 ≤ C kρ00 − ρν,µ k + k 0 2 0 0 ρν,µ 0  Z T∗ Z T∗ +ν ku0 k2H 1 (Ω) dτ + µ kH 0 k2H 1 (Ω) dτ . 0

(3.6)

0

In particular, if 2 kρ00 − ρν,µ 0 k2 + k

as ν, µ → 0, then sup 0<τ
q v0ν,µ 2 ν,µ 2 0 0 ρν,µ 0 u0 − p ν,µ k2 + kH0 − H0 k2 → 0 ρ0

kρ0 (τ ) − ρν,µ (τ )k22 + ku0 (τ ) − uν,µ (τ )k22

(3.7)

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+ kH 0 (τ ) − H ν,µ (τ )k22 → 0 as ν, µ → 0. (3.8) p p √ √ √ Remark 3.4 We point out that since ρ0 u0 − ρν,µ uν,µ = ( ρ0 − ρν,µ )u0 + ρν,µ (u0 − uν,µ ), we have p
√ sup k( ρ0 u0 )(τ ) − ( ρν,µ uν,µ )(τ )k22 → 0 as ν, µ → 0. 0<τ
Proof satisfy

The differences ρ = ρ0 − ρν,µ , u = u0 − uν,µ , H = H 0 − H ν,µ , and π = π 0 − π ν,µ  ν,µ 0   ρt + u · ∇ρ = −u · ∇ρ ,      ρν,µ [ut + uν,µ · ∇u] + ∇π = −ρ(u0t + u0 · ∇u0 ) − ρν,µ u · ∇u0 − ν∆uν,µ   + H · ∇H 0 + H ν,µ · ∇H,     H t + uν,µ · ∇H + u · ∇H 0 − H 0 · ∇u − H · ∇uν,µ + µ∆H ν,µ = 0,      div u = div H = 0

in Q, with the following initial-boundary conditions   u·n= H ·n= 0      ρ(0) = ρ0 − ρν,µ 0 0 0  u(0) = u0 − uν,µ  0     H(0) = H 0 − H ν,µ 0 0

(3.9)

on Σ, in Ω, in Ω,

(3.10)

in Ω.

Multiplying the first equation in (3.9) by ρ and integrating with respect to time and space variables, we have Z tZ 1 1 2 2 kρ(t)k2 ≤ kρ(0)k2 − (3.11) (u · ∇ρ0 )ρ dxdτ, 2 2 0 Ω where we have used the facts that div uν,µ = 0 in Q and uν,µ · n = 0 on Σ to deduce that the RtR integral 0 Ω (uν,µ · ∇ρ)ρ dxdτ vanishes. Based on H¨older’s and Young’s inequalities, we arrive at Z t
kρ(t)k22 ≤ kρ(0)k22 + k∇ρ0 k∞ kuk22 + kρk22 dτ. (3.12) 0

Additional, multiplying the second equation in (3.9) by u, integrating in space and time, and integrating by parts in the Laplacian term, we obtain that Z tZ Z tZ 1 √ ν,µ k( ρ u)(t)k22 − 2να uν,µ · u dσdτ − 2ν S(uν,µ ) : S(u) dxdτ 2 0 ∂Ω 0 Ω Z tZ q 1 v0ν,µ 2 0 p ≤ k ρν,µ u − k − (ρν,µ u · ∇u0 ) · u dxdτ 0 2 0 2 ρν,µ 0 Ω 0 Z tZ Z tZ 0 0 0 − ρ(ut + u · ∇u ) · u dxdτ + (H · ∇H 0 ) · u dxdτ 0 Ω 0 Ω Z tZ + (H ν,µ · ∇H) · u dxdτ. (3.13) 0

Ω

We observe that − 2να

Z tZ 0

∂Ω

uν,µ · u dσdτ − 2ν

Z tZ 0

Ω

S(uν,µ ) : S(u) dxdτ

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Z.P. Zhang: NONHOMOGENEOUS INCOMPRESSIBLE MHD EQUATIONS

Z tZ u0 2 u0 = 2να |u − | dσdτ + 2ν |S(u − )|2 dxdτ 2 2 0 ∂Ω 0 Ω Z tZ Z tZ να ν − |u0 |2 dσdτ − |S(u0 )|2 dxdτ. 2 0 ∂Ω 2 0 Ω

1675

Z tZ

Thus, by using H¨older’s inequality, we have Z t q √ 1 √ ν,µ 1 v0ν,µ 2 ν,µ 0 2 p k( ρ u)(t)k2 ≤ k ρ0 u0 − k + k∇u0 k∞ k ρν,µ uk22 dτ ν,µ 2 2 2 ρ0 0 Z t Z Z να t + ku0t k∞ kρk2 kuk2 dτ + |u0 |2 dσdτ 2 0 ∂Ω 0 Z t Z Z ν t + ku0 · ∇u0 k∞ kρk2 kuk2 dτ + |S(u0 )|2 dxdτ 2 0 0 Ω Z tZ Z t ν,µ 0 (H · ∇H) · u dxdτ. + k∇H k∞ kHk2 kuk2 dτ + 0

0

0

(3.15)

Ω

Similar to u, we can easily get the following inequality for H Z t 1 1 ν,µ 2 2 0 kH(t)k2 ≤ kH0 − H0 k2 + 2 k∇u0 k∞ kHk22 dτ 2 2 0 Z Z Z Z µ t µβ t 0 2 |H | dσdτ + |S(H 0 )|2 dxdτ + 2 0 ∂Ω 2 0 Ω Z t Z tZ 0 + k∇H k∞ kHk2 kuk2 dτ − (H ν,µ · ∇H) · u dxdτ. 0

(3.14)

(3.16)

Ω

Combining (3.12) and (3.15) with (3.16), and using trace inequality, we obtain that
1 √ kρ(t)k22 + k( ρν,µ u)(t)k22 + kH(t)k22 2  q  v0ν,µ 2 1 ν,µ 0 ν,µ 2 ν,µ 0 0 2 k ρ0 u0 − p ν,µ k2 + kH0 − H0 k2 ≤ kρ0 − ρ0 k2 + 2 ρ0 Z t Z t √ k∇ρ0 k∞ + k∇u0 k∞ + ku0t k∞ + k∇u0 k∞ k ρν,µ uk22 dτ + C 0

0



+ ku0 · ∇u0 k∞ + k∇H 0 k∞ kρk22 + kuk22 + kHk22 dτ  Z t Z t +ν ku0 k2H 1 (Ω) dτ + µ kH 0 k2H 1 (Ω) dτ , 0

(3.17)

0

where the positive constant C = C(α, β, Ω) is independent of ν and µ. Notice also that ρ∗ ≤ ρν,µ ≤ ρ∗ .

(3.18)

Furthermore, in view of (3.18), there exists a positive constant C, depending only on α, β, Ω, ρ∗ , ρ∗ , such that kρ0 (t) − ρν,µ (t)k22 + ku0 (t) − uν,µ (t)k22 + kH 0 (t) − H ν,µ (t)k22  q v0ν,µ 2 ν,µ 2 0 0 2 ρν,µ ≤ C kρ00 − ρν,µ k + k 2 0 0 u0 − p ν,µ k2 + kH0 − H0 k2 ρ0 Z t
+ k∇ρ0 k∞ + k∇u0 k∞ + ku0t k∞ + ku0 · ∇u0 k∞ + k∇H 0 k∞ kρk22 0  Z t Z t
kH 0 k2H 1 (Ω) dτ . + kuk22 + kHk22 dτ + ν ku0 k2H 1 (Ω) dτ + µ 0

0

(3.19)

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Now we apply Gronwall’s inequality to obtain that kρ0 (t) − ρν,µ (t)k22 + ku0 (t) − uν,µ (t)k22 + kH 0 (t) − H ν,µ (t)k22  q v0ν,µ 2 ν,µ 2 2 0 0 ≤ C kρ00 − ρν,µ ρν,µ k + k 2 0 0 u0 − p ν,µ k2 + kH0 − H0 k2 ρ0  Z T∗ Z T∗ +ν ku0 k2H 1 (Ω) dτ + µ kH 0 k2H 1 (Ω) dτ , 0

(3.20)

0

where the positive constant C depend only on α, β, Ω, ρ∗ , ρ∗ , ku0t kL∞ ((0,T ∗ )×Ω) , k∇(ρ0 , u0 , H 0 )kL∞ ((0,T ∗ )×Ω) , and ku0 · ∇u0 kL∞ ((0,T ∗ )×Ω) . Due to the regularity of u0 and H 0 in Theorem 3.1, all the above norms are finite. Thus, we complete the proof of Theorem 3.3.  Acknowledgements The author would like to show his appreciation to Professor Fucai Li for his valuable suggestions and encouragement during preparing this paper.

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