GLOBAL WEAK SOLUTIONS TO THE DENSITY-DEPENDENT HALL-MAGNETOHYDRODYNAMICS SYSTEM

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GLOBAL WEAK SOLUTIONS TO THE DENSITY-DEPENDENT

HALL-MAGNETOHYDRODYNAMICS SYSTEM

Jin Tan

To cite this version:

Jin Tan. GLOBAL WEAK SOLUTIONS TO THE DENSITY-DEPENDENT HALL-

MAGNETOHYDRODYNAMICS SYSTEM. 2021. �hal-03104648�

HALL-MAGNETOHYDRODYNAMICS SYSTEM

JIN TAN

Abstract. We are concerned with the global existence of finite energy weak solutions to 3D density-dependent magnetohydrodynamics (MHD) system with Hall-effect set in a general smooth bounded domain. The perfectly conducting wall boundary condition is imposed on the magnetic field. Due to the de- generacy of Hall-effect term (a tri-linear term) in vacuum, we assumed initial density lies in the bounded function space and having a positive lower bound.

Particularly, these bounds are preserved by the density transport equation which helps yield a satisfying compactness argument of the magnetic field.

1. Introduction and main results

1.1. Introduction. In this paper, we consider the following three dimensional density-dependent or inhomogeneous incompressible Magnetohydrodynamics sys- tem that includes the Hall-effect (Hall-MHD):

∂tρ+ div (ρu) = 0, (1.1)

∂t(ρu) + div (ρu⊗u)−div (2µd(u)) +∇P = curlB×B, (1.2)

divu= 0, (1.3)

∂tB+ curl (B×u+hcurlB×B

ρ ) =−curl (curlB

σ ). (1.4)

The unknowns are the density of the fluidρ, the fluid velocityu∈R³, the magnetic fieldB∈R³and the scalar pressureP.We denote byd(u) = ¹₂(∇u+∇u^T) the shear rate tensor, µ =µ(ρ) the fluid viscosity andσ =σ(ρ) its electrical conductivity (this dependence enables us to consider the motion of several immiscible fluids with various viscosities and conductivities), both conductivity and viscosity being positive continuous functions on the [0,∞) . The dimensionless numberhmeasures the magnitude of the so-called Hall-effect term: curl (hcurlB×B

ρ ), compared to the typical length scale of the fluid.

The above system is used to model the evolution of electrically conducting fluids such as plasmas or electrolytes (then, u represents the ion velocity) and takes into account the fact that in a moving conductive fluid, the magnetic field can induce currents which, in turn, polarize the fluid and change the magnetic field.

In the work of Acheritogaray, Degond, Frouvelle and Liu [1], they derived the following generalized Ohm’s law from the two-fluids Navier-Stokes-Maxwell model

2010Mathematics Subject Classification. 35D30; 35Q35; 76D05; 76W05.

Key words and phrases. Hall-MHD, Weak solutions, Weak-strong uniqueness.

1

under suitable scaling hypotheses:

j=σ(E+u×B+∇(lnρ)−hj×B ρ ),

wherej= curlBis the current density andEthe electric field. Jang and Masmoudi in [23] also gave a derivation of Hall-effect. The Maxwell-Farady equations:

∂tB+ curlE=0

with above generalized Ohm’s law then gives rise to (1.4). Compared with the clas- sical inhomogeneous incompressible MHD equations, the density-dependent Hall- MHD system have an additional Hall-effect term which is believed to be the key for understanding the problem of magnetic reconnection, as observed in space plas- mas, star formation, neutron stars and geo-dynamo (see for example [3, 17, 21, 22]).

Meanwhile, since Hall-effect term is a tri-linear term and degenerate in vacuum, it makes the mathematical analysis of the density-dependent Hall-MHD system much more complicated.

The primary aim of this paper is to establish the existence of global weak so- lutions that could be called “solutions a la Leray” to the density-dependent` Hall-MHD system by analogy with the classical global existence results for the incompressible homogeneous Navier-Stokes equations obtained by Leray [24] and the density-dependent Navier-Stokes equations by Lions [26]. Without taking into consideration of Hall-effect term (i.e. h= 0), many works have already been ad- dressed. For instance, with constant density, global existence for standard viscous resistive incompressible MHD system has been previously proved by Duvaut and Li- ons [13], see also Sermange and Temam [28], with inhomogeneous density Gerbeau and Le Bris [18] proved existence of a global weak solution. Consider incompressible homogeneous Hall-MHD system, the global existence of Leray-Hopf weak solutions to the periodic case is first studied in [1], later, Chae et al. [4] have treated R³ case as well as the local well-posedness of classical solutions with initial data in regular Sobolev spaces. Weak-strong uniqueness and energy identity have been in- vestigated by Dumas and Sueur in [12]. Partial regularity have been studied by Chae and Wolf in a series works [6, 7, 8] and later by Zeng and Zhang [32]. Very recently, Danchin and the author in [10, 11] have first established existence results for initial data with critical regularity in Besov spaces and Sobolev spaces, Dai [9]

shows a non-uniqueness result for weak solutions having Leray-Hopf type regularity.

And Zhang [31] proved existence of weak solutions that corresponds to Fujita-Kato solutions to the 3D inhomogeneous incompressible Navier-Stokes equations.

As far as we know, there are few existence results for the Hall-MHD system in a general bounded domain. Unlike MHD system in a bounded domain, the appear- ance of the Hall-effect term gives us additional difficulties in non-linear analysis and on boundary condition for magnetic field. Let Ω be a smooth, bounded, fixed connected open subset of R³.We shall denote by n the outward-pointing normal to Ω.For the boundary conditions, we consider the no-slip boundary condition for the fluid velocityu:

u= 0 on∂Ω, (1.5)

the perfectly conducting wall boundary condition for theBfield (see [1]) :







B·n= 0 on∂Ω,

(hcurlB×B

ρ +curlB

σ )×n= 0 on∂Ω, (1.6)

that is assuming zero normal component of the magnetic field and zero tangen- tial component of the electric field. This non-linear boundary condition has been emphasized by Chae et al. in [1, 5] for homogeneous case. Indeed, it seems hard to study Hall-MHD and related system in a general bounded domain since this non-linear boundary condition. To our best knowledge, the only well-posedness result of strong solutions with perfectly conducting wall boundary condition is due to Mulone and Solonnikov [25]. Recently, Han et al. [19] considered slip boundary condition foruand no-slip boundary condition forBand established global weak solutions to the Hall-MHD system with the ion-slip effect in a bounded domain.

Later, Han and Hwang [20] imposed a new boundary condition and proved local well-posedness of strong solutions with a regularity criteria. However, as we only consider solutions defined in the weak sense, the non-linear boundary condition is avoided.

Due to the degeneracy of Hall-effect term in vacuum, we shall suppose in this paper that initial density does not vanish. Thanks to the nature of (1.1) with (1.3) this property will be preserved along the time and it is crucial for our later analysis.

If there exists vacuum, we have no idea how to make sure the Hall-effect term is well-defined. Let us look at the compressible Hall-magnetohydrodynamics system in e.g. [15, 30] for a while,

∂_tρ+ div (ρu) = 0,

∂_t(ρu) + div (ρu⊗u)−µ∆u+∇P = curlB×B,

∂tB+ curl (B×u+hcurlB×B

ρ ) = ∆B,

divB= 0.

The celebrated results of Lions [27] and extended by Fereisl [16] on the weak solu- tions for compressible Navier-Stokes equations may not be successfully applied to the above model, since without the incompressibility condition (1.3) on the veloc- ity field we are not able to control vacuum regions even if we assume there is no vacuum at beginning. Still, global low-energy weak solutions of 3D compressible MHD equations with density positive and essentially bounded were established in [29]. This motivates us to study the existence of global weak solutions of the above Compressible Hall-MHD system with initial velocity small inL²(R³).

1.2. A priori estimate and functional spaces. Different from the work of Ger- beau and Le Bris, where they only assumed positive initial density, we need to assume initial density having a positive lower bound for technical reason (see Re- mark. 1). Given that infρ0>0, it is reasonable to make an initial hypothesis on velocity not momentum. We thus impose the following initial conditions:

ρ|t=0=ρ0≥ρ >0 on Ω, (1.7)

u|t=0=u0 on Ω, (1.8)

B|_t=0=B₀ on Ω. (1.9)

And we shall suppose in the sequel that forξ∈[ρ,∞),

0< µ≤µ(ξ)≤µ,¯ (1.10)

0< σ≤σ(ξ)≤¯σ. (1.11)

Now, we formally derive an a priori energy estimate. We first remark that (1.1) and the incompressibility condition (1.3) immediately imply that

ρ≤ρ(t, x)≤ kρ0kL^∞, a.e.t, x. Next, recall that for all velocity fieldsuand densityρ

div (ρu⊗u) =udiv (ρu) +ρu· ∇ u, (1.12) in the sense of distributions on Ω. Moreover, we shall make frequent use of the following formula of vector analysis : for all vector fieldsvandwin R³ we have

w

Ωcurlv·wdx=w

∂Ω(n×v)·wdS+w

Ωv·curlwdx, (1.13) whenever these integrals make sense. Here dS is the standard surface measure of

∂Ω.

Thanks to (1.1) and (1.12), we multiply (1.2) by u, integrate over Ω and use boundary condition (1.5) to get

1 2

d dt

w

Ωρ|u|²dx+1 2 w

Ωµ(ρ)|∇u+∇u^T|²dx=w

Ω(curlB×B)·udx. (1.14) Multiplying (1.4) byB, integrating over Ω,using (1.13), we have

1 2

d dt

w

Ω|B|²dx+w

Ω B×u+hcurlB×B

ρ +curlB σ(ρ)

·curlBdx

=−w

∂Ω n× B×u+hcurlB×B

ρ +curlB σ(ρ)

·BdS.

Thus, using the facts that

(curlB×B)·curlB= 0, (B×u)·curlB= (curlB×B)·u, and boundary conditions (1.5), (1.6) we get

1 2

d dt

w

Ω|B|²dx+w

Ω

|curlB|²

σ dx=−w

Ω(curlB×B)·udx. (1.15) Adding up (1.14) and (1.15), we formally have the following energy equality

1 2

d dt

w

Ωρ|u|²dx+w

Ω|B|²dx +1

2 w

Ωµ(ρ)|∇u+∇u^T|²dx+w

Ω

|curlB|²

σ(ρ) dx= 0.

(1.16) It is well-known and wide open in [24, 26], when dealing with global weak solutions of non-linear partial differential equations, the global weak solutions we obtained usually satisfy the energy inequality instead of energy equality

1 2

w

Ω(ρ|u|²+|B|²)dx+1 2

wT 0

w

Ωµ(ρ)|∇u+∇u^T|²dxdt+wT 0

w

Ω

|curlB|² σ(ρ) dxdt

≤ 1 2 w

Ω ρ0|u0|²+|B0|²

dx. (1.17)

Let us notice from (1.10) that w

Ωµ(ρ)|∇u+∇u^T|²dx≥2µw

Ω|∇u|²dx,

since divu= 0. Then thanks to (1.11) we have 1

2 w

Ω(ρ|u|²+|B|²)dx+µwT 0

w

Ω|∇u|²dxdt+ 1

¯ σ

wT 0

w

Ω|curlB|²dxdt

≤ 1 2

w

Ω ρ₀|u₀|²+|B₀|²

dx. (1.18)

In order to formulate our problem and the main results, let us recall the definition of some functional spaces that we shall use throughout this paper. The spaceD(Ω) is defined as the space of smooth functions compactly supported in the domain Ω, and D⁰(Ω) as the space of distributions on Ω. ForX a Banach space, p∈ [1,∞]

and T > 0, the notation L^p(0, T;X) or L^p_T(X) designates the set of measurable functions f : [0, T] → X with t 7→ kf(t)kX in L^p(0, T), endowed with the norm k·k_L^p

T(X):=kk·kXkL^p(0,T),and agree thatC([0, T];X) denotes the set of continuous functions from [0, T] toX. Slightly abusively, we will keep the same notations for multi-component functions. The spaceH¹(Ω) denotes the space ofL² functions of f on Ω such that∇f also belongs toL²(Ω).The Hilbertian norm is defined by

kfk²_H1(Ω):=kfk²_L2(Ω)+k∇fk²_L2(Ω).

The spaceH₀¹(Ω) is defined as the closure of D(Ω) for theH¹(Ω) norm, and the spaceH⁻¹(Ω) as the dual space ofH₀¹(Ω) for theD⁰×Dduality. Then we introduce the vector-valued spaces

V ={u∈ D(Ω), divu= 0}, H= the closure of V inL²(Ω)

={u∈L²(Ω), divu= 0, u·n|∂Ω= 0}, V1= the closure of V inH¹(Ω)

={u∈H₀¹(Ω), divu= 0},

W={B∈ C^∞( ¯Ω), divB= 0, B·n|∂Ω= 0}, W1= the closure of W in H¹(Ω)

={B∈H¹(Ω), divB= 0, B·n|∂Ω= 0}, and their norms

kukV1 =k∇ukL²(Ω), kBkW₁ =kcurlBk_L2(Ω). We equipH with the following inner product

(u,v) =w

Ωu·vdx, for allu,v∈H.

Just remark that, one can establish thatk · kV₁ andk · kW₁ defined two norms which are equivalent to that introduced byH¹(Ω) onV1andW1,respectively (cf. Duvaut and Lions [13], Sermange and Temam [28]).

In accordance with (1.3), we assume that divu₀= 0 and, for physical consistency, since a magnetic field has to be divergence free, we suppose that divB₀= 0, too, a property that is conserved through the evolution. With above notations and a priori estimate (1.18), we propose the following assumptions on the initial data:

ρ0∈L^∞, ρ0≥ρ >0, (1.19)

u₀∈ H, (1.20)

B0∈ H. (1.21) In a same fashion with [26, 18], we define our weak solutions as follows.

Definition 1.1. We say that (ρ,u,B) is a global weak solution of the problem (1.1)-(1.9) with the initial assumptions (1.19)-(1.21), if for any T > 0, (ρ,u,B) satisfies the following properties:

ρ≥ρ,¯ ρ∈L^∞((0, T)×Ω), ρ∈ C([0, T];L^p(Ω)), 1≤p <∞, u∈L²(0, T;V1), and ρ|u|²∈L^∞(0, T;L¹(Ω)),

B∈L²(0, T;W1)∩L^∞(0, T;L²(Ω)),

moreover, for any φ∈ C¹([0, T]×Ω)withφ(T,·) = 0,

−wT 0

w

Ω(ρ∂tφ+ρu· ∇φ)dxdt=w

Ωρ0φ(0, x)dx; (1.22) for any Φ∈ C¹([0, T]×Ω)with div Φ = 0andΦ(T,·) = 0,

wT 0

w

Ω(−ρu·∂tΦ−(ρu⊗u)· ∇Φ + 2µ(ρ)d(u)·d(Φ))dxdt

=wT 0

w

Ω(curlB×B)·Φdxdt+w

Ωm0·Φ(0, x)dx; (1.23) for any Ψ∈ C¹([0, T]×Ω)with Ψ(T,·) = 0,

wT 0

w

Ω −B·∂tΨ + (B×u+hcurlB×B

ρ +curlB

σ(ρ) )·curl Ψ dxdt

=w

ΩB0·Ψ(0, x)dx; (1.24)

and finally, the energy inequality (1.17) holds for allt∈[0, T].

1.3. Main results. In comparison with the models studied in [26, 18], the main difficulty of proving the existence of weak solutions lies in the Lorentz force term curlB×B in the Navier-Stokes equations (1.2) while B satisfies a quasi-linear parabolic equations with a tri-linear term involving density, we have to recover some compactness onBin order to pass to the limit in the non-linear terms.

We now state our main theorem.

Theorem 1.2. Under the regularity assumptions (1.19)-(1.21)on the initial data, the initial-boundary value problem to the density-dependent Hall-MHD system(1.1)- (1.9)has a weak solution in the sense of Definition 1.1. Furthermore, we have for all0≤α≤β ≤ ∞

meas{x∈R³, α≤ρ(t, x)≤β} is independent of t≥0, (1.25) and¹

u, B∈ Cw([0, T];L²(Ω)).

The next theorem states a weak-strong uniqueness property of the solution. We show that any global weak solution coincides with a more regular solution as long as the latter exists.

1The spaceCw([0, T];L²(Ω)) denotes continuity on the interval [0, T] with values in the weak topology ofL²(Ω)

Theorem 1.3. Assume µ, σ are locally Lipschitz continuous. Let (ρ,u,B) be a weak solution obtained from Theorem 1.2. If there exists a solution ( ˆρ,u,ˆ B)ˆ ∈ C([0, T]×Ω)of the problem (1.1)-(1.6), which satisfies

∇ρ, ∂ˆ _tu, ∂ˆ _tBˆ ∈L²(0, T;L³(Ω)) and ∇ˆu,∇Bˆ∈L²(0, T;L^∞(Ω)), (1.26) and at initial time

ρ|ˆt=0=ρ0, u|ˆt=0=u0, B|ˆ t=0=B0 in Ω, then we have (ρ,u,B)≡( ˆρ,u,ˆ B)ˆ a.e. in(0, T)×Ω.

In the rest of this paper, we first set up our approximation scheme and establish the existence of solutions to the approximation problem in Section 2. Then in Section 3, in order to recover the original system, we deduce some compactness results and finally finish the proof of our main theorem. In the end, we prove Theorem 1.3.

Throughout this paper, we use C to denote a general positive constant which may different from line to line.

2. Approximation Scheme

The essential idea to prove the existence of a weak solution to the density- dependent Hall-MHD system is to introduce an approximation problem, that allows one to define (1.1) as a classical transport equation. Now, presenting the approxi- mation problem and showing the existence of a regular solution to this problem is our purpose for the next two steps.

2.1. First step : a linearized problem. At this step, we prove a preliminary result for a linearized problem with prescribed density, which will be useful in subsection 2.2. For this purpose, we first define two finite dimensional spaces for n∈N^∗:

V_n = span{Θ_i}ⁿ_i=1, and W_n = span{Γ_i}ⁿ_i=1,

where{Θ_i}^∞_i=1 ⊂ V and {Γ_i}^∞_i=1⊂ W are orthonormal basis ofV₁ and W₁ respec- tively.

Forρ, v,Hand n, T >0 arbitrarily fixed such that

ρ, ∂_tρ∈ C([0, T],C¹( ¯Ω)), and 0< ρ₁≤ρ(t, x)≤ρ₂, (2.1) v∈ C([0, T];Vn) such that∂tρ+ div (ρv) = 0, (2.2)

H∈ C([0, T];Wn), (2.3)

the linearized problem is to find a couple of vector-valued functions (u,B) such thatw

Ωρ(∂tu+v· ∇u)·Φdx+w

Ω2µ(ρ)d(u)·d(Φ)dx=w

Ω(curlB×H)·Φdx, (2.4) for any Φ∈ Vn,

w

Ω∂tB·Ψdx+w

Ω

H×u+hcurlB×H

ρ +curlB σ(ρ)

·curl Ψdx= 0, (2.5) for any Ψ∈ Wn,and with initial condition (u|t=0,B|t=0) = (u0,B0).

To show a existence result for the above linearized problem is not difficult, it will based on the Galerkin’s method. We have

Proposition 2.1. Let (u0,B0) ∈ Vn × Wn. Under the assumptions (2.1)-(2.3), there exists a unique pair of solution(u,B)∈ C¹([0, T];Vn)× C¹([0, T];Wn) to the problem (2.4)-(2.5). Moreover, we have energy equality (1.16).

Proof. We look for a solution (u,B) under the form













 u=

n

X

i=1

α_i(t)Θ_i,

B=

n

X

i=1

βi(t)Γi.

(2.6)

Look at weak form (2.4) and (2.5), replace Φ by Θ_j and Ψ by Γ_j for j= 1, . . . , n, respectively, we find that scalar functionsαi and βi (i= 1, . . . , n) are solutions of the following linear ODEs :





















 w

ΩρΘ_i·Θ_jdxdα_i dt +w

Ω ρ(v· ∇Θ_i)·Θ_j+ 2µ(ρ)d(Θ_i)·d(Θ_j dx

α_i

−w

Ω(curl Γi×H)·Θjdx βi= 0, w

ΩΓi·Γjdxdβi

dt +w

Ω(hcurl Γi×H

ρ +curl Γi

σ(ρ) )·curl Γjdx βi

+w

Ω(H×Θi)·curl Γjdx αi = 0,

(2.7)

with initial data fori= 1, . . . , n,defined by







αi(0) =w

Ωu0·Θidx, βi(0) =w

ΩB0·Γidx.

(2.8)

Since the family (√

ρΘi)i=1,...n (resp. (√

ρΓi)i=1,...n) with ρ ≥ ρ is free, it fol- lows that the matrix r

ΩρΘ_i·Θ_jdx

n×n (resp. r

ΩρΓ_i ·Γ_jdx

n×n) is nonsin- gular for any t ∈ [0, T]. We note that the coefficients lie in the above ODEs are continuous on [0, T] from assumptions (2.1)-(2.3). Thanks to the Cauchy- Lipschitz theorem, (α(t), β(t)) exists uniquely and is continuous on [0, T], where α(t) = (α1(t), . . . , αn(t))^T ∈Rⁿ, β(t) = (β1(t), . . . , βn(t))^T ∈Rⁿ. Thus, we obtain with (2.7) and (2.6)

(u,B)∈ C¹([0, T],Vn)× C¹([0, T],Wn)

In view of this regularity, we multiply the first equations in (2.7) byαj and second equations byβj, then we sum them fori, j= 1, . . . , n. This yields

w

Ωρ(∂tu+v· ∇u)·udx+w

Ω2µ(ρ)|d(u)|²dx−w

Ω(curlB×H)·udx= 0, and

w

Ω∂tB·Bdx+w

Ω

|curlB|² σ(ρ) dx+w

Ω(H×u)·curlBdx= 0.

By integration by parts with (2.2), we get (1.16). This completes the proof of

Proposition 2.1.

2.2. Second step : a regularized approximation problem. In this step, we solve a non-linear regularized approximation problem by using the Schauder’s fixed- point theorem (see Theorem II.3.9 in [2]) and Proposition 2.1.

The following definition of mollifier can be found in e.g. [14] with several prop- erties. Define

Ω:={x∈Ω|dist(x, ∂Ω)> }, andη∈ C^∞(R³) the standard mollifier by

η(x) =







Cexp( 1

|x|²−1) if |x|<1

0 if |x| ≥1,

for some normalizing constantC such thatr

R³η dx= 1.Let∈(0,1),define η(x) := 1

³η(x ).

Now, we are ready to define approximate initial data. Since we assumed our initial density to be bounded below byρ,we set

ρe=

(ρ₀ in Ω ρ inR³\Ω, and

ρ₀=ρe0∗η.

Then the initial density for the approximate system is defined by

ρ_n|t=0=ρ₀. (2.9)

Thanks to assumption (1.19), it is clear for some universal constantC0independent of,we have

ρ≤ρ₀≤C0, (2.10)

andρ₀∈ C^∞(Ω)

→0limρ₀=ρ₀ in L^p(Ω) (1≤p <∞). (2.11) We would also like to regularizeµ(ξ), σ(ξ) like in [26]. Assumeµ(ξ) is aC([0,∞)) function bounded away from zero, which is constant forξ≥0 large and such that sup_[0,∞)|µ−µ|< .We set

µ](ρ) =

(µ(ρ) in Ω 1 in R³\Ω,

and defineµ=µ](ρ)∗η|Ω.We defineσfrom σlikeµ fromµ.

We set the initial condition for approximate velocity field and magnetic field as un|t=0=u_0,n=P_Vnu₀ (2.12) and

Bn|t=0=B_0,n=PW_nB₀, (2.13) where

u₀= ((u₀1_Ω)∗η), B₀= ((B₀1_Ω)∗η)

andP_Vn (resp. P_Wn) is the orthogonal projection inL²(Ω) ontoVn (resp. Wn).

Fixed,our approximation problem is stated as follows forn∈N^∗.

Definition 2.2. For any given T >0, we say(ρn,un,Bn),with ρn∈ C([0, T]×Ω), un ∈ C([0, T];Vn), Bn ∈ C([0, T];Wn), is a global weak solution of the following approximation problem

∂tρn+ div (ρnun) = 0, (2.14)

∂_t(ρ_nu_n) + div (ρu_n⊗u_n)−div (2µ^ε(ρ_n)d(u_n)) +∇P_n= curlB_n×B_n, (2.15)

divu_n= 0, (2.16)

∂tBn+ curl (Bn×un+hcurlBn×Bn

ρ_n ) =−curl (curlBn

σ(ρ_n)), (2.17)

divB_n = 0, (2.18)

with the initial conditions (2.9),(2.12),(2.13), and boundary conditions

un|∂Ω= 0, (2.19)







Bn·n= 0 on∂Ω,

(hcurlBn×Bn

ρn

+curlBn

σ(ρn))×n= 0 on ∂Ω, (2.20) if (2.14)-(2.20)are satisfied in the weak sense of Definition 1.1 with the test function spaces in (1.23)and (1.24)replaced by the restriction on Vn andWn, respectively.

With the above definition, we have the following existence result.

Theorem 2.3. There exists a global weak solution(ρn,un,Bn)to the above initial- boundary value problem.

Proof. In order to solve this non-linear approximation problem by the Schauder’s fixed-point theorem, we shall construct a operatorF_n:I_n→I_n,where the convex set

In:=n

(¯u,B)¯ ∈ C([0, T];Vn)× C([0, T];Wn); sup

t∈[0,T]

k(¯u,B)k¯ L²(Ω)≤R0

o , R0is a constant to be determined later. We denote the input ofFn to be (¯un,B¯n), and the corresponding outputFn(¯un,B¯n) to be (un,Bn).Then we define our op- erator as follows. At first, with the input (¯un,B¯n),we consider the following linear problem:

(∂t(ρn) + div (ρnu¯n) = 0,

ρn|t=0=ρ₀. (2.21)

This is a classical transport equation since, ¯u_n is regular, div ¯u_n= 0 and vanishes near∂Ω.Thusρ_n is uniquely given by

ρn(t, x) =ρ₀(X(0;x, t)), ∀(t, x)∈[0, T]×Ω,¯ whereX is the solution of the ODE





 dX

ds = ¯un(s, X), X(t;x, t) =x.

Obviously, we have

ρ≤ρn≤C0. (2.22)

Sinceρ_o is smooth, so ρ_n ∈ C¹([0, T]×Ω) and is bounded in this space uniformly¯ in (¯u_n,B¯_n).

Now, we setρ=ρn,v= ¯un,H= ¯Bn and replaceµbyµ(ρn), σbyσ(ρn) and we invoke Proposition 2.1 to define (un,Bn) as the solution of:

w

Ωρn(∂tun+ ¯un· ∇un)·Φdx+w

Ω2µ^ε(ρn)d(un)·d(Φ)dx=w

Ω(curlBn×B¯n)·Φdx, (2.23) for any Φ∈ Vn,

w

Ω∂tBn·Ψdx+w

Ω

B¯n×un+hcurlBn×B¯n

ρ_n +curlBn

σ(ρ_n)

·curl Ψdx= 0, (2.24) for any Ψ∈ W_n,whileu_n andB_n satisfy initial condition (2.12)-(2.13).

Next, let us choose R₀ such that (u_n,B_n) ∈ I_n. Thanks to Proposition 2.1, (u_n,B_n) belongs toC([0, T];V_n)× C([0, T];W_n) and satisfies

d dt

w

Ω(ρ_n|u_n|²+|B_n|²)dx+w

Ω

4µ(ρ_n)|d(u_n)|²+ 2|curlB_n|² σ(ρn)

dx= 0. (2.25) This with (2.22) and divun= 0 lead to

sup

t∈[0,T]

kunk_L2(Ω)+ sup

t∈[0,T]

kBnk_L2(Ω)+kunk_L2(0,T;V1)+kBnk_L2(0,T;W1)≤C1 (2.26) where C1 is a constant independent ofR0,u¯n,B¯n.Hence, by takingR0=C1, we have (un,Bn)∈In.

Then we prove the compactness of mapping Fn. In fact, with uniform bound (2.26) in hand, in view of the famous Aubin-Lions Lemma (see Theorem II. 5.16 in [2]), we only need to show some uniform bounds for∂tun and ∂tBn in suitable spaces. Thanks to Proposition 2.1 again, we know that actually (∂_tu_n, ∂_tB_n) be- longs to C([0, T];Vn)× C([0, T];Wn). Hence by taking Φ = ∂_tu_n, Ψ = ∂_tB_n in (2.23)-(2.24), we have

w

Ω(ρ_n|∂_tu_n|²+|∂_tB_n|²)dx

=−w

Ω (ρnu¯n· ∇un)·∂tun+ 2µ(ρn)d(un)·d(∂tun) dx

−w

Ω

B¯n×un+hcurlBn×B¯n

ρn

+curlBn

σ(ρn)

·curl (∂tBn)dx +w

Ω(curlBn×B¯n)·∂tundx.

Since all norms in a finite dimensional space are equivalent and thanks to (2.26), we obtain by H¨older’s inequality

ρk∂tunk²_L2(Ω)+k∂tBnk²_L2(Ω)

≤C0ku¯nk_L^∞_(Ω)k∇unk_L2(Ω)k∂tunk_L2(Ω)+ 2¯µk∇unk_L2(Ω)k∇(∂tun)k_L2(Ω)

+ kB¯nkL^∞(Ω)kunkL²(Ω)+h

ρkcurlBnkL²(Ω)kB¯nkL^∞(Ω)+1

σkcurlBnkL²(Ω)

· kcurl (∂tBn)k_L2(Ω)+kcurlBnk_L2(Ω)kB¯nk_L^∞_(Ω)k∂tunk_L2(Ω)

≤C(k∂_tu_nk_L2(Ω)+k∂_tB_nk_L2(Ω)) and thus

k∂tunk_L2(Ω)+k∂tBnk_L2(Ω)≤C, (2.27) where C is a constant independent on n, . We conclude that F_n is a compact operator onI_n to itself.

In order to apply the Schauder’s theorem, we still have to check the continu- ity of Fn. It suffices to prove that the mapping is sequentially continuous. Let {(¯u^m_n,B¯^m_n)}m≥1 ⊂ In be a sequence which strongly converges to (¯un,B¯n) in In. Recall our definition of mapping F_n, we denote ρ^m_n as the corresponding solu- tion to (2.21) and let (u^m_n,B^m_n) = F_n(¯u^m_n,B¯^m_n) as the corresponding solution to (2.23)-(2.24). It is clear when solving (2.21), we have that {ρ^m_n}_m≥1 is bounded in C¹([0, T]×Ω) uniformly in¯ {(¯u^m_n,B¯^m_n)}_m≥1. Thus Aubin-Lions Lemma implies that {ρ^m_n}_m≥1 is pre-compact in C([0, T]×Ω).¯ For {(u^m_n,B^m_n)}_m≥1, we know it is a subset of I_n and the control of {(∂_tu^m_n, ∂_tB^m_n)}_m≥1 in L^∞(0, T;L²(Ω)) can also be obtained by following a same procedure as to get (2.27). So one more ap- plication of the Aubin-Lions Lemma gives that {(u^m_n,B^m_n)}_m≥1 is pre-compact in C([0, T];L²(Ω)).Without loss of generality, as we can always replace our original se- quence by a weakly converging subsequence, we conclude that (u^m_n,B^m_n) converges to (un,Bn) a solution of (2.4)-(2.5) whenmgoes to ∞.Since problem (2.21) and (2.23)-(2.24) are all linear problems and the solutions are shown to be unique, one has (un,Bn) =Fn(¯un,B¯n),that is (u^m_n,B^m_n) converges toFn(¯un,B¯n) asm→ ∞.

From the Schauder’s fixed-point theorem, there exists a fixed point (un,Bn) of Fn inIn.It means that with input (un,Bn) one can well-defineρn as the solution of (2.14). Moreover, following the definition of output, by using (2.14) and the regularity ofu_n, ρ_n, one can rewrite (2.23) as

w

Ω∂t(ρnun)·Φdx−w

Ωρnun⊗un· ∇Φdx+w

Ω2µ^ε(ρn)d(un)·d(Φ)dx

=w

Ω(curlB_n×B_n)·Φdx, (2.28)

and get w

Ω∂_tB_n·Ψdx+w

Ω B_n×u_n+hcurlB_n×B_n ρn

+curlB_n σ(ρn)

·curl Ψdx= 0, (2.29) After time integration on (2.14), (2.28), (2.29) over [0, T],we infer that (un,Bn) satisfies the weak formulations in the Definition 2.2. Finally, we prove (1.25). Let γ_m be a function ofC¹(R;R). Multiplying (2.14) byγ_m⁰ (ρ_n) and using divu_n = 0 we have

∂tγm(ρn) +un· ∇γm(ρn) = 0.

We integrate this equation on [0, T]×Ω and use again that divun = 0, with un

vanishes near the boundary to obtain w

Ωγ_m(ρ_n(t, x))dx=w

Ωγ_m(ρ₀(x))dx. (2.30) For 0≤α≤β <∞we choose formlarge enough 0≤γ_m≤1 such that







γ_m(λ) = 0 if λ /∈[α, β], γm(λ) = 1 if λ∈[α+ 1

m, β− 1 m].

Lettingm→ ∞in (2.30) we deduce that (1.25) holds forρn, w

Ω1_[α,β](ρ_n(t, x))dx=w

Ω1_[α,β](ρ₀(x))dx

where 1_[α,β](λ) is the characteristic function on [α, β]. This completes the proof of

Theorem 2.3.

3. Convergence of the approximation problem

The aim of this last section is to prove our main Theorem 1.2 by passing to the limit in the regularized approximation problem stated in Definition 2.2 asn→ ∞ and →0. The fundamental tool is a compactness result due to P.-L. Lions [26]

that we recall here for its importance.

Theorem 3.1. We suppose that two sequences ρn and un are given satisfying ρn ∈ C([0, T];L¹(Ω)), 0 ≤ ρn ≤ C a.e. on (0, T)×Ω, un ∈ L²(0, T;H₀¹(Ω)), kunk_L2(0,T;H¹₀(Ω))≤Canddivun= 0(C denotes various constants independent of n). We assume :

(∂tρn+ div (ρnun) = 0 in D⁰((0, T)×Ω), ρn|t=0=ρ0n,

and

ρ0n→ρ0 in L¹(Ω),

un *u weakly in L²(0, T;H₀¹(Ω)).

Then :

(1) ρnconverges inC([0, T];L^p(Ω))for all1≤p <∞to the uniqueρbelonging toC([0, T];L¹(Ω)) bounded on(0, T)×Ωsolution of

(∂tρ+ div (ρu) = 0 in D⁰((0, T)×Ω), ρ|t=0=ρ0 a.e.in Ω.

(2) We assume in addition thatρ_n|u_n|²is bounded inL^∞(0, T;L¹(Ω))and that we have for some l≥1

|< ∂_t(ρ_nu_n),Φ>| ≤CkΦk_L2(0,T;H^l(Ω))

for allΦ∈ D((0, T)×Ω)such thatdiv Φ = 0 on(0, T)×Ω.Then:

√ρ_nu_n→√

ρu in L^q(0, T;L^r(Ω)) for 2< q <∞, 1≤r < 6q 3q−4

un→u in L^θ(0, T;L^3θ(Ω)) for 1≤θ <2 on the set{(t, x)|ρ(t, x)>0}.

3.1. Pass to the limit as n → ∞. For fixed , we denote by (ρ_n,u_n,B_n) the smooth approximate solution given by Theorem 2.3. It is clear that from (2.25), for any fixedT >0, one has

w

Ω(ρn|un|²dx+|Bn|²)dx+wT 0

w

Ω

4µ(ρn)|d(un)|²+ 2|curlBn|² σ(ρ_n)

dxdt

≤w

Ω(ρ₀|u₀|²+|B₀|²)dx. (3.1)

Note thatρnis bounded away from 0 uniformly, then we have the following bounds independent ofn:

kρn|un|²k_L∞(0,T;L¹(Ω))≤C, (3.2)

kunkL^∞(0,T;L²(Ω))≤C, (3.3)

kunkL²(0,T;V₁)≤C, (3.4)

kBnk_L^∞_(0,T;L2(Ω))≤C, (3.5)

kB_nk_L2(0,T;W₁)≤C, (3.6)

which implies that as n goes to infinity, up to extraction (we will extract subse- quence if necessary),

u_n *u weakly in L²(0, T;V₁), u_n *u weakly^∗ in L^∞(0, T;V) B_n*B weakly in L²(0, T;W₁), B_n*B weakly^∗ in L^∞(0, T;W).

In view of (2.11), (2.22) and (3.4), the first assertion of Theorem 3.1 implies that ρn→ρ in C([0, T];L^p(Ω)) with 1≤p <∞ (3.7) and

∂tρ+ div (ρu) = 0.

Our goal is now to pass to the limit as n goes to infinity in the following weak formulations for approximation problem that have stated in Definition 2.2 :

wT 0

w

Ω −ρnu_n·∂_tΦdxdt−ρ_nu_n⊗u_n· ∇Φ + 2µ^ε(ρ_n)d(u_n)·d(Φ) dx

=−wT 0

w

Ω(curlB_n×B_n)·Φdxdt+w

Ωρ₀u_0,n·Φ(0, x)dx, (3.8) for any Φ∈ C¹([0, T]× Vn) with div Φ = 0 and Φ(T,·) = 0,

wT 0

w

Ω −B_n·∂_tΨ + B_n×u_n+hcurlB_n×B_n ρn

+curlB_n σ(ρn)

·curl Ψ dxdt

=w

ΩB_0,n·Ψ(0, x)ds, (3.9)

for any Ψ∈ C¹([0, T]× Wn) with Ψ(T,·) = 0.

It turns out that we need to apply the second part of Theorem 3.1 to pass to the limit. Since we already get (3.2), let us estimate< ∂t(ρnun),Φ>from equality (2.28) with Φ∈ D((0, T)×Ω), div Φ = 0.

First, by Sobolev embeddings : H¹(Ω),→L⁶(Ω), H¹²(Ω),→L³(Ω), we have

|wT 0

w

Ωρnun⊗un· ∇Φdxdt|

≤kρnkL^∞((0,T)×Ω)kunkL^∞_T(L²)kunk_L²

T(L⁶)k∇Φk_L²

T(L³)

≤C0C²kΦk

L²_T(H³2), and by divun= 0, div Φ = 0,

|wT 0

w

Ω2µ(ρ_n)d(u_n)·d(Φ)dxdt|

≤Ckµk_L^∞_([0,T]×Ω)kd(un)k_L2

T(L²)kd(Φ)k_L2 T(L²)

≤C²kΦk_L²

T(H¹).

Using the inequalitykΦk_L^∞_(Ω)≤CkΦkH^s fors > ³₂, we have

|wT 0

w

Ω(curlBn×Bn)·Φdxdt|

≤kcurlBn)k_L²

T(L²)kBnkL^∞_T(L²)kΦk_L²

T(L^∞)

≤C²kΦk_L2 T(H^s).

Therefore for anyl > ³₂,the second assertion of Theorem 3.1 with strong conver- gence ofρn then imply that asn→ ∞

ρ_nu_n→ρu in L^q(0, T;L^r(Ω)) for 2< q <∞, 1≤r < 6q

3q−4 (3.10)

un →u in L^θ(0, T;L^3θ(Ω)) for 1≤θ <2. (3.11) Let us prove the strong convergence ofBn toBinL²(0, T;W) at this moment.

Indeed, from formula (2.29), for any Ψ∈L⁴(0, T;H²(Ω)), one has

|< ∂_tB_n·Ψ>|

≤|wT 0

w

Ω Bn×un+hcurlBn×Bn

ρ_n +curlBn

σ(ρ_n)

·curl Ψdxdt|

≤kBnk_L⁴

T(L³)kunk_L²

T(L⁶)k∇Φk_L⁴

T(L²)+h

ρkcurlB_nk_L²

T(L²)kBnk_L⁴

T(L³)k∇Φk_L⁴

T(L⁶)

+ 1

σkcurlB_nk_L2

T(L²)k∇Φk_L2 T(L²)

≤C²kΦk_L⁴

T(H²),

and thus{∂tB_n}n≥1 is bounded inL⁴³(0, T;H⁻²).

Remark 1. If there is no positive lower bound to initial density but onlyρ0 ≥0, for fixedwe can define the initial condition for approximate density as in [26, 18], whereρⁿ ≥is constructed. However, when passing to the limit as →0 we will lost above uniform bound with respect to due to the appearance of Hall-effect term. This is the reason why we need to assume infρ0>0.

Keeping (3.6) in mind, we infer from the Aubin-Lions Lemma that

Bn →B in L²(0, T;L²(Ω)). (3.12) The weak and strong convergences obtained forρ_n,u_n andB_nenable us to pass to the limit in the weak formulations (3.8)-(3.9) like in [18], except for the Hall-term.

To deal with it, we write curlBn×Bn

ρn

−curlB×B ρ

=ρ−ρn

ρ_nρ (curlBn×Bn) +curlBn×(Bn−B)

ρ +(curlBn−curlB)×B

ρ .

Thanks to (3.5), (3.6) and (3.7) we know that the first term strongly tends to 0 in L¹(0, T;L⁶⁵(Ω)),while the second one strongly tends to 0 inL¹(0, T;L¹(Ω)) due to (3.12). Finally, the third term weakly tends to 0 inL¹(0, T;L³²(Ω)) since curlB_n is weakly convergent to curlB inL²((0, T)×Ω) andB lies in L²(0, T;L⁶(Ω)), ρlies inL^∞((0, T)×Ω).

For the initial values, by definition ofu_0,n andB_0,n,we have w

Ωρ₀u_0,n·Φ(0, x)dx→w

Ωρ₀u₀·Φ(0, x)dx=w

Ωm₀·Φ(0, x)dx, w

ΩB_0,n·Ψ(0, x)dx→w

ΩB₀·Φ(0, x)dx.

In conclusion, passing to the limit in (3.8)-(3.9) and energy inequality (3.1), we have obtained the following result :

Proposition 3.2. For any T >0,there is a solution (ρ,u,B)which satisfies

∂tρ+ div (ρu) = 0,

∂_t(ρu) + div (ρu⊗u)−div (2µ^ε(ρ)d(u)) +∇P = curlB×B,

divu= 0,

∂_tB+ curl (B×u+hcurlB×B

ρ ) =−curl (curlB σ(ρ)), divB= 0,

in the sense of Definition 2.2 with the initial data

ρ|_t=0=ρ₀, u|_t=0=u₀ B|_t=0=B₀. Moreover, the solution satisfies the following energy inequality :

w

Ω(ρ|u|²dx+|B|²)dx+wT 0

w

Ωµ(ρ)|∇u|²dxdt+ 2wT 0

w

Ω

|curlB|² σ(ρ) dxdt

≤w

Ω(ρ₀|u₀|²+|B₀|²)dx, (3.13)

and w

Ω1[α,β](ρ(t, x))dx=w

Ω1[α,β](ρ₀(x))dx. (3.14) 3.2. Pass to the limit as →0. For this passage to the limit, there is no addi- tional difficulty compare to the previous step since ρ is still bounded away from zero byρuniformly. In particular, from energy inequality (3.13) we have as→0,

u*u weakly in L²(0, T;V1), u*u weakly^∗ in L^∞(0, T;V) B*B weakly in L²(0, T;W1), B*B weakly^∗ in L^∞(0, T;W), and by Theorem 3.1

ρ→ρ in C([0, T];L^p(Ω)) with 1≤p <∞, (3.15) ρu→ρu in L^q(0, T;L^r(Ω)) for 2< q <∞, 1≤r < 6q

3q−4, u→u in L^θ(0, T;L^3θ(Ω)) for 1≤θ <2.

Moreover, again by Aubin-Lions lemma,

B→B in L²(0, T;L²(Ω)).

Just remark thatρ bounded below byρis essential for getting uniform bound of

∂tB in L⁴³(0, T;H⁻²).

In view of the construction ofµ, σ,one has

µ→µ in C([0, T];L^p(Ω)), ∀1≤p≤ ∞, σ→σ in C([0, T];L^p(Ω)), ∀1≤p≤ ∞.

Hence, the above convergence properties with convergence for initial values ensure the existence part of Theorem 1.2. We now consider the initial value of energy inequality (3.13). We observe that since∇ ×q₀= 0, then

w

Ωρ₀|u₀|²dx=w

Ω

|m₀−q₀|² ρ₀ dx

=w

Ω

|m₀|²

ρ₀ −|q₀|²

ρ₀ −2u₀·q₀ dx

=w

Ω

|m₀|²

ρ₀ −|q₀|² ρ₀

dx.

Thanks to convergence property (??), letting→0 in (3.13) leads to the energy inequality (1.17). Look at (3.14), an /2 argument with convergence properties (2.11), (3.15) forρ₀ andρ₀then implies(1.25).

To prove that B ∈ Cw([0, T];L²(Ω)), one first need to notice that ∂_tB lies in L¹(0, T;H⁻²),in particularBis almost everywhere equal to a continuous function from [0, T] into H⁻²(Ω). Finally, B ∈L^∞(0, T;L²) and H²(Ω) is dense in L²(Ω) imply that B is weakly continuous from [0, T] into L²(Ω). u ∈ C_w([0, T];L²(Ω)) based on similar argument.

This concludes the proof of Theorem 1.2.

4. Weak-strong uniqueness

In this section, we prove a weak-strong uniqueness property for global weak solutions obtained from Theorem 1.2. We first remark that, in view of regularities ( ˆρ,u,ˆ B)ˆ ∈ C([0, T]×Ω) and (1.26) i.e.

∇ρ, ∂ˆ tˆu, ∂tBˆ ∈L²(0, T;L³(Ω)) and ∇ˆu,∇Bˆ ∈L²(0, T;L^∞(Ω)), we actually could takeuˆand Bˆ as test functions in the weak formulations (1.23), (1.24), and then get the following equalities for allt∈(0, T)

w

Ωρu·ˆudx+ 2wt 0

w

Ωµ(ρ)d(u)·d(ˆu)dxds−wt 0

w

Ω(curlB×B)·ˆudxds

=w

Ωm0·ˆu(0, x)dx+wt 0

w

Ωρu·(∂suˆ+u· ∇ˆu)dxds, (4.1) w

ΩB·Bˆdx+wt 0

w

Ω hcurlB×B

ρ +curlB

σ(ρ) )·curlBˆ dxds

=w

Ω|B₀|²dx+wt 0

w

ΩB·∂_sBˆdxds−wt 0

w

Ω(curlBˆ×B)·udxds. (4.2) Above, we have used the following two vector identities

(ρu⊗u)· ∇ˆu=ρu·(u· ∇ˆu), (B×u)·curlBˆ= (curlBˆ×B)·u.

Let us notice that there exists a gradient term∇Pˆ that belongs toL¹(0, T;L^∞(Ω)) +L²(0, T;H⁻¹(Ω)) coupled with ( ˆρ,ˆu,B).ˆ Next we write

ρ(∂tˆu+u· ∇ˆu)−div (2µ(ρ)d(ˆu)) +∇Pˆ−curlBˆ×Bˆ

= (ρ−ρ)(∂ˆ tuˆ+uˆ· ∇ˆu) +ρ(u−ˆu)· ∇ˆu−div 2(µ(ρ)−µ( ˆρ))d(ˆu)

, (4.3)

∂tBˆ+ curl (Bˆ×ˆu+hcurlBˆ×Bˆ ˆ

ρ ) + curl (curlBˆ

σ( ˆρ) ) = 0, (4.4)

and if we multiply (4.3), (4.4) byu, B,respectively, and integrate over (0, t)×Ω, then we have by integrations by parts

wt 0

w

Ωρu·(∂sˆu+u· ∇ˆu)dxds+ 2wt 0

w

Ωµ(ρ)d(ˆu)·d(u)dxds

=wt 0

w

Ω

curlBˆ×Bˆ+ (ρ−ρ)(∂ˆ _suˆ+ˆu· ∇ˆu) +ρ(u−ˆu)· ∇ˆu

·udxds +wt

0

w

Ω2(µ(ρ)−µ( ˆρ))d(ˆu)·d(u)dxds, (4.5)