# Singular Limit of the Rotational Compressible Magnetohydrodynamic Flows

Authors: Young-Sam Kwon
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We derive a rigorous quasi-geostrophic equation governed by magnetic field from the stratified flows of the rotational compressible magnetohydrodynamic flows with the well-prepared initial data and the tool of proof is based on the relative entropy. Furthermore, the convergence rates are obtained.


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## Body

## 1. Introduction

Magnetohydrodynamic flows arise in science and engineering in a variety of practical applications such as in plasma confinement, liquid-metal cooling of nuclear reactors, and electromagnetic casting. The fundamental concept behind MHD flows is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. The set of equations that describe MHD flows are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. These differential equations must be solved simultaneously, either analytically or numerically. Here we consider the viscous rotational compressible magnetohydrodynamic flows in the 2-dimensional whole spaceΩ≔R2:(1)∂tϱε+div⁡ϱεuε=0,(2)∂tϱϵuϵ+div⁡ϱϵuϵ⊗uϵ+1ε2γ∇ϱϵγ+1εϱϵuϵ⊥=μεΔuϵ+με+λε∇div⁡uϵ+Bε·∇Bε-12∇Bε2+1εϱϵ∇G,(3)∂tBε+div⁡uϵBε+uϵ·∇Bε-Bε·∇uϵ=νεΔBε,where uϵ is the vector field, ϱϵ is the density, Bε is the magnetic field, γ>3/2, and G∈Cc3(Ω), and we also assume that(4)με=εθ,νε=εσ,λε⟶0,with θ,σ>0, as ε tends to 0.We first notice that the global-in-time existence solutions for systems ((1)–(3)), supplemented with physically relevant constitutive relations, has been studied by Hu and Wang [1].It should be pointed out that the incompressible inviscid limit problems to the compressible Navier-Stokes equations and related models are very interesting. For the case without rotational force, Masmoudi [2] proved the convergence of the weak solution of isentropic Navier-Stokes equations to the strong solution of the incompressible Euler equations in the 2-dimensional whole space R2 and the space case by applying the related entropy method. Later, his result was extended to the isentropic compressible magnetohydrodynamic equations [3, 4]. Feireisl and Novotný [5] studied the inviscid incompressible limit to the full Navier-Stokes-Fourier system in the whole space.In this paper, we derive a rigorous quasi-geostrophic equation from the stratified flow of rotational compressible magnetohydrodynamic flows ((1)–(3)) on the 2-dimensional whole space with the well-prepared initial data. Our contribution of this paper is physically to derive a rigorous quasi-geostrophic equation from the stratified flows of the rotational compressible magnetohydrodynamic equations based on the relative entropy method. Recently, Feireisl and Novotný [6] have studied the asymptotic limit for the models with a rotational term originating from a Coriolis force with the mild stratification and with the well-prepared initial data. This result is based on their paper, but it is more a developed version than their result because we derive a quasi-geostrophic equation from a global weak solution of compressible MHD flows.Let the densityϱ~ be the solution of the static problem(5)∇ϱ~εγ=γεϱ~ε∇G,ϱ~εx⟶1asx⟶∞,where we have(6)ϱ~εx-1≤εGx,∇ϱ~εx≤ε∇Gx,ϱ~ε∈C3Ω,x∈Ω.Assume that the initial data have the following property at infinity:(7)ϱϵx⟶1,uϵx⟶0,Bεx⟶0asx⟶∞.Formally, we will investigate the limit(8)ϱϵ-ϱ~εε⟶q,ϱϵuϵ⟶v,Bε⟶B,as ϵ tends to 0 in the suitable sense such that the given limits (q,B) represent the unique local smooth strong solution of the following system on [0,T]: for q0∈Hk+1(Ω) and B0∈Hk(Ω),k≥3,(9)v=∇⊥q,∂tΔq-q+∇⊥q·∇Δq+∇⊥q·∇G=div⊥⁡B·∇B,q0,·=q0∂tB+∇⊥q·∇B-B·∇∇⊥q=0,B0,·=B0,where the notations are defined as follows:(10)v⊥=-v2,v1,∇⊥g=-∂g∂x2,∂g∂x1,div⊥⁡v=-∂v1∂x2+∂v2∂x1.Note that the existence of global strong solution of system (9) can be proven with the same method of [7].Theorem 1.
LetΩ be the 2-dimensional whole space R2. For the given initial q0∈Hk+1(Ω) and B0∈Hk(Ω),k≥3, there is T>0 such that system (9) has the unique local smooth solution (q,B) on [0,T], verifying the following regularity:(11)q∈C0,T;Hk+1Ω;R∩C10,T;HkΩ;R,(12)B∈C0,T;HkΩ;R2∩C10,T;Hk-1Ω;R2.The outline of this article is as follows. In Section2, we present the rigorous result for (8) and (9). In Section 3, we derive a rigorous proof of the rotational compressible magnetohydrodynamic flows ((1)–(3)).Definition 2.
We say that a quantity{ϱ,u,B} is a weak solution of the magnetohydrodynamic (MHD) flows ((1)–(3)) supplemented with the initial data {ϱ0,u0,B0} provided that the following hold:(i)
The densityϱ is a nonnegative function, where ϱ-1∈L∞0,T;Lγ+L2Ω, the velocity field u∈L20,T;W1,2Ω;R2, ϱ|u|2∈L∞0,T;L1Ω, and u represents a renormalized solution of equation (1) on (0,T)×Ω; that is, the integral identity(13)∫Ωϱ+bϱφT,·dx-∫Ωϱ0+bϱ0φ0,·dx=∫0T∫Ωϱ+bϱ∂tφ+ϱ+bϱu·∇φ+bϱ-b′ϱϱdiv⁡uφdxdtholds for any test functionφ∈D([0,T)×Ω) and any b such that(14)b∈C10,∞,b′r=0wheneverr≥rb.(ii)
The balance of momentum holds in distributional sense; namely,(15)∫Ωϱu·φ→T,·dx-∫Ωϱu0·φ→0,·dx=∫0T∫Ωϱu·∂tφ→+ϱu⊗u:∇φ→+1ϵ2γϱγdiv⁡φ→+B·∇B-12∇B2·φ→dxdt-∫0T∫Ω1εϱu⊥·φ→-με∇u:∇φ→-με+λεdiv⁡udiv⁡φ→+1εϱ∇G·φ→dxdt,for any test functionφ→∈D([0,T)×Ω;R2).(iii)
The total energy of the system holds:(16)∫Ω12ϱu2+12B2+1γγ-1ε2ϱγ-ϱ~εγ-γϱ~εγ-1ϱ-ϱ~εt,·dx+∫0t∫Ωμε∇u2+με+λεdiv⁡u2+νε∇B2dxdt≤E0,ϵholds for a.e.t∈(0,T), where(17)E0,ϵ=∫Ω12ϱ0u02+12B02+1γγ-1ε2ϱ0γ-ϱ~εγ-γϱ~εγ-1ϱ0-ϱ~εdx.(iv)
The Maxwell equation (3) with div⁡B=0 and the regularity B∈L2(0,T;W1,2(Ω))∩L∞(0,T;L2(Ω)) verifies(18)∫ΩB·φ→T,·dx-∫ΩB0·φ→0,·dx=∫0T∫ΩB·∂tφ→dxdt-∫0T∫Ωu×B·∇×φ→-ν∇×B·∇×φ→dxdt,for allφ→∈D([0,T)×Ω;R2).


## 2. Main Results

In this section, we introduce the main results.Theorem 3.
LetΩ=R2 be the 2-dimensional whole space and let (ϱϵ,uϵ,Bε) be a weak solution to ((1)–(3)) in the sense of Definition 2, verifying viscosity (4) with 0<θ+σ<2 and the initial data:(19)q0,ε-q0L2Ω2≤Cε,Hεq0,ε-q0L2Ω2≤Cε,(20)ϱ0,εu0,ε-∇⊥q0L2Ω2≤Cε,B0,ε-B0L2Ω2≤Cε,where(21)ϱ0,ε=ϱ~ε+εq0,ε,q0,ε∈L2∩L∞Ω,q0∈Hk+1Ω,v0,B0∈HkΩ;R3,for k≥3. Then, one has(22)ϱϵ-ϱ~εε-qL∞0,T;L1K≤CKεκϱϵuε-vL∞0,T;L2Ω;R22≤CεκBε-BL∞0,T;L2Ω2≤Cεκfor sufficiently small 0<ε<1, any T<T∗, and any compact K⊂⊂R2 such that (q,B) verifies (9). Furthermore, the numbers κ are defined by (23)κ=min⁡1,2γ,1-θ+σ2,53-θ2,θ,σ.


## 3. Proof of Theorem3

In this section, we are going to give the rigorous proof of Theorem3.Step 1.
In this part, we are going to derive some estimates on the sequence{ϱϵ,uϵ,Bε}ϵ>0.
From the energy inequality (16), we obtain(24)esssupt∈0,T⁡ϱϵuϵtL2Ω≤C,(25)esssupt∈0,T⁡ϱϵγ-ϱ~εγ-γϱ~εγ-1ϱϵ-ϱ~εL1Ω≤ϵ2C,(26)∇uϵL20,T×Ω≤Cε-θ/2,(27)esssupt∈0,T⁡BεtL2Ω≤C,∇BεL20,T×Ω≤Cε-σ/2.
We consider the properties of convex function:(28)ϱϵγ-ϱ~εγ-γϱϵ-ϱ~ε≥Cϱϵ-ϱ~ε2ifγ≥2,(29)ϱϵγ-ϱ~εγ-γϱϵ-ϱ~ε≥Cϱϵ-ϱ~ε2ifγ<2,x≤R,(30)ϱϵγ-ϱ~εγ-γϱϵ-ϱ~ε≥Cϱϵ-ϱ~εγifγ<2,x≥R,(31)ϱϵγ-ϱ~εγ-γϱϵ-ϱ~ε≥Cδϱϵ-ϱ~εγifϱϵ-ϱ~ε>δ>0,where we can see these properties in [8]. Following (28) and (29) together with (25), we get(32)esssupt∈0,T⁡ϱϵ-ϱ~εϵ1ϱϵ-ϱ~ε≤1/2L2Ω≤C.Using (31), we also derive(33)esssupt∈0,T⁡∫Ωϱϵ-ϱ~εγϵ21ϱϵ-ϱ~≥1/2dx≤C,which gives (34)esssupt∈0,T⁡∫Ω1+ϱϵγ1ϱϵ-ϱ~ε≥1/2dx≤ε2C.Note that(35)ϱϵ-ϱ~εL∞0,T;L2Ω≤εC,while using (28)–(31) implies that(36)∫Ωϱϵ-ϱ~ε2dx=∫ϱϵ-ϱ~≤1/2ϱϵ-ϱ~ε2dx+∫ϱϵ-ϱ~≥1/2ϱϵ-ϱ~ε2dx≤C∫ϱϵ-ϱ~≤1/2ϱϵ-ϱ~ε2dx+C∫ϱϵ-ϱ~≥1/2ϱϵ-ϱ~εγdx≤ε2C.The Sobolev embedding also gives(37)uϵL22≤C+Cε4/3∇uϵL22,and the proof is provided in [9].Step 2.
We introduce the relative entropy in the version of the magnetohydrodynamic flows. Let us set(38)hϱ=1γγ-1ϱγ-ϱ~εγ-γϱ~εγ-1ϱ-ϱ~ε,Hεqε=2εsign⁡qεhϱ~ε+εqε,ϱϵ=ϱ~ε+εqε,where(39)sign⁡qε=1,ifsign⁡qε>0,0,ifsign⁡qε=0,-1,ifsign⁡qε<0.
We define the relative entropy:(40)Eετ=12∫Ωϱϵuϵ-v2+Hεqε-q2+Bε-B2dx,where we put v≔∇⊥q. Adapting v as a test function to the moment equation (2) provides(41)-∫Ωϱϵuϵ·vτdx=-∫Ωϱ0,εu0,ε·v0dx+∫0τ∫Ω1εϱϵuϵ⊥·vdxdt-∫0τ∫Ωϱϵuϵ⊗uϵ:∇v+Bε·∇Bε·v-με∇uϵ:∇vdxdt-∫0τ∫Ωϱϵuϵ·∂tvdxdt-1ε∫0τ∫Ωϱϵ∇G·vdxdt.We also use q as a test function to the continuity equation (1) to deduce that(42)-∫Ωqεqdx=-∫Ωq0,εq0dx-∫0τ∫Ωqε∂tqdxdt-∫0τ∫Ω1εϱϵuϵ·∇qdxdt.To compute the relative entropy, we also use B as a test function to the magnetic field equation (3) and insert (9), which yields (43)-∫ΩBε·Bτdx=-∫ΩB0,ε·B0dx+∫0τ∫ΩBε·v·∇B-B·∇vdxdt+∫0τ∫Ωdiv⁡uϵBε+uϵ·∇Bε-Bε·∇uϵ·Bdxdt+∫0τ∫Ωνε∇Bε:∇Bdxdt.Adding (16), (41), (42), and (43) derives the following inequality:(44)Eετ+∫0τ∫Ωμε∇uϵ2+με+λεdiv⁡uϵ2+νε∇Bε2dxdt≤∑j=17Aj,where(45)A1=Eε0+∫Ωq0Hεq0,ε-q0,εdx-12∫Ωϱ0,ε-ϱ~ε∇⊥q02dxA2=12∫Ωϱϵ-ϱ~ε∇⊥q2dx-∫ΩqHεqε-qεdxA3=12∫Ωϱ~ε∇⊥q2-ϱ~ε∇⊥q02+q2-q02+B2-B02dxA4=∫0τ∫Ωμε∇uϵ:∇v+νε∇Bε:∇BdxdtA5=-∫0τ∫Ωϱϵuϵ·∂tvdxdt-∫0τ∫Ωϱϵuϵ⊗uϵ:∇vdxdt-∫0τ∫Ωqε∂tqdxdt-1ε∫0τ∫Ωϱϵ∇G·vdxdtA6=-∫0τ∫ΩBε·∇Bε·v+∫0τ∫ΩBε·v·∇B-B·∇vdxdtA7=∫0τ∫Ωdiv⁡uϵBε+uϵ·∇Bε-Bε·∇uϵ·Bdxdt.Step 3.
From (19), it is easily seen that it shows that(46)A1≤Cε.Indeed, using (6), (19), and (21), we get(47)ϱ0,εu0,ε-∇⊥q0L2Ω2≤Cϱ0,εu0,ε-∇⊥q0L2Ω2+Cϱ0,ε-ϱ~ε∇⊥q0L2Ω2+Cϱ~ε-1∇⊥q0L2Ω2≤Cε,where ∇⊥q0L∞Ω≤C and the constant depends on the support of G, and we also have(48)Hq0,ε-q0L2Ω2≤CHq0,ε-q0,εL2Ω2+Cq0,ε-q0L2Ω2≤Cε,where we have used (21). Similarly, we get(49)q0Hq0,ε-q0L2Ω2≤Cε,ϱ0,ε-ϱ~ε∇⊥q02L2Ω2≤Cε.By the regularity of q in (11) and (34), we can estimate the first term of A2 as follows:(50)∫Ωϱϵ-ϱ~ε∇⊥q2dx≤Cϱϵ-ϱ~ε1ϱϵ-ϱ~ε≤1/2L∞0,T;L2Ω+Cϱϵγ+11ϱϵ-ϱ~ε≥1/2L∞0,T;L2Ω≤Cε+ε2,where the Sobolev imbedding theorem H2⊂L∞ implies that ∇⊥q∈L∞(Ω). The estimate of the second term of A2 is also given in [10] such that(51)∫ΩqHεqε-qεdx≤Cε2/γ.
To handleA3, we multiply q to system (9), which yields(52)12∫Ω∇⊥q2+q2dx-12∫Ω∇⊥q02+q02dx=∫0τ∫ΩB·∇B·vdxdt,while(53)12∫0τ∫Ω∇⊥q·∇Gqdxdt=0.Multiplying B to the magnetic field equation, we obtain the following energy equation:(54)12∫ΩB2dx-12∫ΩB02dx=-∫0τ∫ΩB·∇B·vdxdt.Adding (52) and (54), it follows that(55)A3=12∫Ωϱ~ε-1∇⊥q2-∇⊥q02dx≤Cε,where we have used (6). Let us next show that the viscosity term A4 vanishes:(56)∫0τ∫Ωμε∇uϵ:∇v+νε∇Bε:∇Bdxdt≤με2∫0τ∫Ω∇uϵ2dxdt+νε2∫0t∫Ω∇Bε2dxdt+Cμε+νε,where we have here used (4) and (11).Step 4.
For this part, we need to estimate the convective term ofA5 and the term can be expressed in the following form:(57)-∫0τ∫Ωϱϵuϵ-v⊗uϵ-v:∇vdxdt-∫0τ∫Ωϱϵv⊗uϵ:∇vdxdt-∫0τ∫Ωϱϵuϵ⊗v:∇vdxdt+∫0τ∫Ωϱϵv⊗v:∇vdxdt≔∑j=14Ij.The first term I1 can be controlled by(58)I1≤C∫0τEεtdt,where, by (11), ∇v∈H2(Ω) together with the Sobolev embedding, which implies that(59)∇vL∞Ω≤C.I2 is reformulated by(60)I2=-∫0τ∫Ωϱϵuϵ·v·∇vdxdt.For I3, employing the estimates of (32) and (34) together with the continuity equation (1), we get(61)I3=-12∫0τ∫Ωϱϵuϵ·∇v2dxdt=12∫0τ∫Ωρϵ-ρ~ε∂tv2dxdt-12∫Ωϱϵ-ϱ~εvτ2dx+12∫Ωϱ0,ε-ϱ~εv02dx≤Cε+ε2.To handle I4, we use divv=0 to obtain(62)I4=-∫0τ∫Ωϱϵ-ϱ~εv·∇v·vdxdt-∫0τ∫Ωϱ~ε-1v·∇v·vdxdt≤Cε+ε2,where we have used (32) and (33). Thus, A5 is written by(63)A5≤-∫0τ∫Ωϱϵuϵ·∂tv+v·∇v+qε∇G·v+qε∂tqdxdt+C∫0τEεtdt+Cε+ε2,where(64)-1ε∫0τ∫Ωϱϵ∇G·vdxdt=-∫0τ∫Ωϱ~εε∇G·vdxdt+∫0τ∫Ωqε∇G·vdxdt=∫0τ∫Ωqε∇G·vdxdttogether with using (5) and divv=0. To handle (63), we need the following lemma.Lemma 4.
Let(ϱϵ,uϵ,Bε) be global weak solutions of (1)–(3) in the sense of Definition 2. Then, one has the following inequality:(65)∫0τ∫Ωϱϵuϵ-∇⊥qε·ψdxdt≤Cεfor any test function ψ.Proof.
We use the test functionψ⊥ to the moment equation (2) to deduce(66)∫0τ∫Ωϱϵuϵ-∇⊥qε·ψdxdt=ε∫Ωϱϵuϵτ·ψ⊥τdx-ε∫Ωϱ0,εu0,ε·ψ0⊥dx+ε∫0τ∫Ωϱϵuϵ·∂tψ⊥dxdt+ε∫0t∫Ωϱϵuϵ⊗uϵ:∇ψ⊥dxdt+ε∫0τ∫Ω1ε2γϱϵγ-ϱ~εγ-γϱ~εγ-1ϱϵ-ϱ~εdiv⁡ψ⊥dxdt-ε∫0τ∫Ωϱϵ-ϱ~εε∇G·ψ⊥dxdt-εμε∫0τ∫Ω∇uϵ:∇ψ⊥dxdt-εμε+λε∫0t∫Ωdiv⁡uϵdiv⁡ψ⊥dxdt+ε∫0τ∫ΩBε·∇Bε·ψ⊥+ε2∫0t∫ΩBε2div⁡ψ⊥dxdt.In virtue of the estimates in (19)–(21) and (24)–(27), the terms of the right-hand side in (66) can be controlled and so it proved (65).Making use of Lemma4 together with v=∇⊥q and ψ=∂t∇⊥q+(∇⊥q·∇)∇⊥q and integrating by parts, we obtain that(67)A5=-∫0τ∫Ωϱϵuϵ·∂t∇⊥q+∇⊥q·∇∇⊥qdxdt-∫0τ∫Ωqε∂tq+∇G·∇⊥qdxdt+C∫0τEεtdt+Cε+ε2≤∫0τ∫Ωqε∂tΔq-q+∇⊥q·∇Δq+∇G·∇⊥qdxdt+C∫0τEεtdt+Cε+ε2=∫0τ∫Ωqεdiv⊥⁡B·∇B-12∇B2dxdt+C∫0τEεtdt+Cε+ε2≤-∫0τ∫Ωϱϵuϵ·B·∇B-12∇B2dxdt+C∫0τEεtdt+Cε+ε2,where we have used ψ=(B·∇)B-1/2∇|B|2 as a test function on the last line of (67). Thus, the relative entropy gives(68)Eετ+12∫0τ∫Ωμε∇uϵ2+με+λεdiv⁡uϵ2+νε∇Bε2dxdt≤-∫0τ∫ΩBε·∇Bε·v+∫0τ∫ΩBε·v·∇B-B·∇vdxdt+∫0τ∫Ωdiv⁡uϵBε+uϵ·∇Bε-Bε·∇uϵ·Bdxdt-∫0τ∫Ωϱϵuϵ·B·∇B-12∇B2dxdt+∫0τEεtdt+Cε+ε2/γ+ε2+εθ+εσ.Step 5.
Finally, we now handle the part of magnetic field. We apply the integration by parts and the factsdivBε=0, divB=0, and divv=0 to obtain(69)Eεt+12∫0τ∫Ωμε∇xuϵ2+με+λεdiv⁡uϵ2+νε∇Bε2dxdt≤∫0τ∫Ωϱ~ε-ϱϵuϵ·B·∇Bdxdt+12∫0τ∫Ωϱ~ε-ϱϵuϵ·∇B2dxdt+∫0τ∫Ω1-ϱ~εuϵ·B·∇Bdxdt+12∫0τ∫Ω1-ϱ~εuϵ·∇B2dxdt+∫0τ∫ΩBε-B·∇v·Bε-Bdxdt+∫0τ∫ΩBε-B·∇B·uϵ-vdxdt-∫0τ∫Ωuϵ-v·∇B·Bε-Bdxdt+C∫0τEεtdt+Cε+ε2/γ+ε2+εθ+εσ≔∑j=17Dj+C∫0τEεtdt+Cε+ε2/γ+ε2+εθ+εσ.For the term D6, making use of (32), (33), and (37), together with the Sobolev embedding and Holder’s inequality, it follows that(70)∫0τ∫ΩBε-B·∇B·uϵ-vdxdt=∫0τ∫ΩBε-B·∇B·ϱϵuϵ-vdxdt+∫0τ∫Ωϱ~ε-ϱϵBε-B·∇B·uϵ-vdxdt+∫0τ∫Ω1-ϱ~εBε-B·∇B·uϵ-vdxdt≤Cϱϵ-ϱ~εL∞0,T;L2ΩBεL20,T;L6ΩuϵL20,T;L3Ω+Cϱϵ-ϱ~εL∞0,T;L2ΩuϵL20,T;L2Ω+BεL20,T;L2Ω+1+C1-ϱ~εL∞0,T;L2ΩBεL20,T;L6ΩuϵL20,T;L3Ω+C1-ϱ~εL∞0,T;L2ΩuϵL20,T;L2Ω+BεL20,T;L2Ω+1+C∫0τEεtdt≤Cε1-θ+σ/2+ε+ε5/3-θ/2+C∫0τEεtdt,where we have used the assumption of viscosity 0<θ+σ<2, (6), and the Sobolev embedding theorem:(71)BεL20,T;L6Ω≤C∇BεL20,T;L2Ω,uϵL20,T;L3Ω≤C∇uϵL20,T;L2Ω.Similarly, the term D7 can also be controlled by(72)D7≤C∫0τEεtdt+Cε1-θ+σ/2+ε+ε5/3-θ/2.Finally, we estimate D2, D3, D4, and D5 with the same method. We now rewrite D1 as follows:(73)D1=∫0τ∫Ωϱ~ε-ϱϵuϵ·B·∇Bdxdt=∫0τ∫Ωϱ~εϱ~ε-ϱϵuϵ·B·∇Bdxdt+∫0τ∫Ωϱ~ε-ϱϵϱϵuϵ·B·∇Bdxdt.Thus, from the estimates of (32), (33), and (37), together with the estimate of (24), we get(74)D1≤Cε+ε5/3-θ/2,where we have again used the assumption of viscosity 0<θ+σ<2 and so(75)∑j=14Dj≤Cε+ε5/3-θ/2.Consequently, the relative entropy in (69) is given by(76)Eετ+12∫0τ∫Ωμε∇xuϵ2+με+λεdiv⁡uϵ2+νε∇Bε2dxdt≤C∫0τEεtdt+Cε1-θ+σ/2+ε+ε2/γ+ε2+ε5/3-θ/2+εθ+εσ.Step 6.
Let us apply Grönwall’s inequality to (76) in order to obtain(77)Eετ≤Cε1-θ+σ/2+ε+ε2/γ+ε2+ε5/3-θ/2+εθ+εσ≤Cεκfor any τ∈[0,T], where the number κ is defined in (23). Note that(78)∫Kqε-qdx≤∫Ωqε-Hεqεdx+C∫KHεqε-q2dx≤C∫0TEεtdt+Cεκ,for any compact subset K⊂R2 and(79)∫Ωϱϵuϵ-∇⊥q2dx≤∫Ωϱϵuϵ-ϱϵ∇⊥q+ϱϵ-ϱ~ε∇⊥q2dx+C∫Ωϱ~ε-1∇⊥q2dx≤∫0TEεtdt+Cεκ,where T is the given time in Theorem 1. For the magnetic field, we obtain(80)∫ΩBε-B2dx≤∫0TEεtdt.Using (77), (78), (79), and (80) and passing to the limit for ε→0, we prove (22). In conclusion, we get the target equation (9) by passing to the limits as ε→0, but it is sufficient to show that, for any test function ϕ,(81)limε→0⁡∫0T∫Ω1εϱϵ∇G·∇⊥ϕdxdt=∫0T∫Ωq∇G·∇⊥ϕdxdt.Indeed,(82)limε→0⁡∫0T∫Ω1εϱϵ∇G·∇⊥ϕdxdt=limε→0⁡∫0T∫Ω1εϱ~ε+εqε∇G·∇⊥ϕdxdt=limε→0⁡∫0T∫Ω1γε2∇ϱ~εγ·∇⊥ϕdxdt+limε→0⁡∫0T∫Ωqε∇G·∇⊥ϕdxdt=∫0T∫Ωq∇G·∇⊥ϕdxdt,where we have used div⁡∇⊥=0 and (5).



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*Source: 9493186-2017-07-25.xml*