# Weak-Strong Uniqueness for Compressible Magnetohydrodynamic Equations with Coulomb Force

Authors: Lianhua He; Yonghui Zhou
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we consider the two-dimensional compressible magnetohydrodynamic system with Coulomb force. We apply the method of relative entropy to establish the weak-strong uniqueness property of this system.


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## Body

## 1. Introduction

In this paper, we consider the compressible magnetohydrodynamic equations with Coulomb force as follows:(1)∂tρ+divρu=0,(2)∂tρu+divρu⊗u+∇p+12M2−μΔu−λ+μ∇divu=divM⊗M+ρ∇Φ,(3)∂tM+divu⊗M−divM⊗u−νΔM=0,divM=0,(4)ΔΦ=ρ−ρ¯.Hereρ,u,M and Φ stand for the electron density, electron velocity, magnetic field and electrostatic potential, respectively. ρ¯>0 denotes the constant background ion density. p=pρ=aργ is the pressure function with the constant a>0 and the adiabatic exponent γ>1. The constants μ,λ are the viscosity coefficients satisfying μ>0 and λ+2μ/N≥0 where N is dimension. The constant ν>0 is the magnetic diffusivity of the fluid. Let Ω⊂R2 be a bounded smooth domain. This system can be applied to describe the dynamics of a charge transport where the compressible electron fluid interacts with its own electric field against a charged ion background together under the influence of the magnetic field. Usually, we denominate the equation (1) as the continuity equation, the equation (2) as the momentum balance equation and the equation (3) as the induction equation. The motion of such fluids is driven by two dominating body forces, namely, the Coulomb force (i.e. the electric field force) and the Lorentz force imposed on the fluid by the magnetic field. This is the reason why the physical phenomena of such fluids is more complicated.In this paper, we consider the initial-boundary conditions of MHD system (1)–(4) as following:
(5)ρ0,x=ρ0x∈LγΩ,ρ0x≥0;∫Ωρ0−ρ¯dx=0;ρ0,xu0,x=ρ0u0x,ρ0u02∈L1Ω;M0,x=M0x∈L2Ω,divM0=0inD′Ω;u∂Ω=0,M∂Ω=0,∇Φ·n∂Ω=0.where n denotes the outer normal vector of ∂Ω.When there are no electromagnetic effect and no Coulomb force, the system (1)–(5) reduces to the compressible Navier-Stokes system. Many papers are devoted to the study of Navier-Stokes equations (see [1–5] and the references cited therein). These papers are focused on the existence of classical solution or weak solution and large time asymptotic behavior of solution et al. The weak-strong uniqueness is an important aspect in mathematical theory research of Navier-Stokes equations. The relative entropy method is an important method to research weak-strong uniqueness property of compressible Navier-Stokes equations. With the assistance of the concept of relative entropy, Germain [6] introduced a class of weak solution to the compressible Naiver–Stokes equations satisfying the relative entropy inequality with respect to a strong solution emanating from the same initial condition, and established the weak-strong uniqueness of this weak solution. Unfortunately, the existence of this class of weak solution is not known. Feireisl et al. [7] established the global existence of suitable weak solutions for any finite-energy initial data to compressible Navier-Stokes equations and proved the weak-strong uniqueness property within this class of suitable weak solutions by relative entropy inequality. By means of the relative entropy method, Feireisl et al. not only proved the weak-strong uniqueness of finite-energy weak solution to compressible Navier-Stokes equations with monotone pressure in [8], but also proved the weak-strong uniqueness property for compressible Naiver-Stokes equations with non-monotone pressure in [9]. When the pressure function satisfies a hard sphere law, the weak-strong uniqueness of Navier-Stokes equations is established by Feireisl et al. in [10] and Chaudhur in [11] applying the method of relative entropy.When there is no electromagnetic effect, the MHD system (1)–(5) reduces to the compressible Navier-Stokes-Poisson system. Many researchers are interested to study the compressible NSP system (see [12–17] and the references cited therein). These papers are focused on the existence and asymptotic behavior of solution et al. For the weeak-strong uniqueness of the compressible Navier-Stokes-Poisson system, we have discussed in [18].Many researchers are interested in studying MHD system because of its physical importance, mathematical complexity and extensive applications(see [19–23] and references cited therein). There are also many references about the system (1)–(4)(see [19, 24–26]). In [24], Tan and Wang consider global existence and large time behavior of weak solution to the system (1)–(4). The existence of the finite-energy weak solution to the problem (1)–(5) is established by Feireisl et al. in [19] with γ≥4/3 and Jiang et al. in [25] with 9+33/12<γ≤4/3. Although the existence of weak solution has been established, the uniqueness problem is still an open problem. So far, there are very few results concerning the uniqueness of weak solutions to the initial-boundary value problem (1)–(5). Only when the dimension is three and the Coulomb force ρ∇Φ is not taken into account in the system (1)–(5), the weak-strong uniqueness of the finite-energy weak solution to the system (1)–(5) is established by Yang et al. in [26]. Then, our goal of this paper is to consider the uniqueness property of weak solution to two-dimensional MHD system with Coulomb force. In spirit of Feireisl [8], we will establish the uniqueness property of weak solution to the system (1)–(5) by using of the method of relative entropy. Compared with [8], the main difficulty is to deal with the items about magnetic field and electrostatic potential.The paper is organized as follows. In Section2, we recall the definition of finite-energy weak solution for the compressible MHD equations (1)–(5) and state the main results. In Section 3, we derive the relative entropy inequality to the system(1)–(5). In last section, we prove the weak-strong uniqueness property of the compressible MHD equations with Coulomb force (1)–(5).


## 2. Main Results

In this section, we define the finite-energy weak solution to the initial-boundary value problem (1)–(5) and state the main results.Definition 1.
ρ,u,Φ,Mis called as a finite-energy weak solution of (1)–(5), if
(i)
ρ,u,Φ,Mbelongs to the following class
(6)ρ≥0,ρ∈L∞0,T;LγΩ,u∈L20,T;H01Ω,ϕ∈L∞0,T;H1Ω,M∈L∞0,T;L2Ω∩L20,T;H01Ω,divMt=0fora.aont∈0,T.(ii)
Et∈Lloc10,T satisfies the energy inequality
(7)ddtEt+∫Ωμ∇u2+λ+μdivu2+ν∇M2t,·dx≤0in D′0,T, where the total energy Et is defined as
(8)Et=∫Ω12ρu2+aγ−1ργ+12M2+12∇Φ2t,·dx.(iii)
For anyt∈0,T, the pair ρ,Φ satisfies the following Poisson equation
(9)Δϕ=ρ−ρ¯inΩ,∇ϕ⋅n∂Ω=0,∫Ωϕdx=0.(iv)
Equation (1) holds in D′0,T×R2, i,e
(10)∫Ωρτ,·ϕτ,·dx−∫Ωρ0ϕ0,·dx=∫0τ∫Ωρ∂tϕ+ρu·∇ϕdxdt.for any ϕ∈C0∞0,T×R2 and τ∈0,T. Moreover, ρ,u satisfies the renormalized equation
(11)∫Ωbρϕτ,·dx−∫Ωbρ0ϕ0,·dx=∫0τ∫Ωbρ∂tϕ+bρu·∇ϕ+ϕbρ−b′ρρdivudxdt.where b∈C1R such that
(12)b′z=0forzlargeenough.(v)
Equation Equation (2) (3) holds in D′0,T×Ω, i,e
(13)∫Ωρuτ,··φτ,·dx−∫Ωρ0u0·φ0,·dx+∫0τ∫Ωμ∇u:∇φ+λ+μdivudivφdxdt=∫0τ∫Ωρu·∂tφ+ρu⊗u:∇φ+pρdivφ+12M2divφ−M⊗M:∇φ+ρ∇Φ·φdxdt.(14)∫ΩM·φτ,·dx−∫ΩM0·φ0,·dx+ν∫0τ∫Ω∇M:∇φdxdt=∫0τ∫ΩM·∂tφ+u⊗M:∇φ−M⊗u:∇φdxdtfor any φ∈C0∞0,T×Ω.Theorem 2.
Assume thatΩ∈R2 is a smooth domain. Let γ>9+33/12. Then for any given T>0, there exist a finite-energy weak solution ρ,u,Φ,M of the system (1)-(5).Remark 3.
(1)
Whenγ>4/3, we can apply the method in [19] without any modifications to prove the existence of finite-energy weak solution(2)
When9+33/12<γ≤4/3, Jiang proved the existence of finite-energy weak solution ρ,u,Φ,M of the system (1)–(5) in [25].
LetPρ=a/γ−1ργ. Then the following equalities hold
(15)P′ρρ−Pρ=pρ,P″ρρ=p′ρ.
Now we introduce main results of this paper.Theorem 4.
Letρ,u,Φ,M be a finite-energy weak solution to the MHD system (1)-(5) in the sense of Definition 1. Let r,Ψ∈C0∞0,T×Ω and U,H∈C0∞0,T×Ω. Assume that r,U,Ψ,H satisfy
(16)∂tH+divU⊗H−divH⊗U−νΔH=0,divH=0;(17)Δψ=r−ρ¯,∂tr+divrU=0,∇ψ⋅n∂Ω=0.
Then the following relative entropy inequality holds for a.eτ∈0,T(18)ερ,u,Φ,M∣r,U,Ψ,Hτ+∫0τ∫Ωμ∇u−U2+λ+μdivu−U2+ν∇M−H2dxdt≤ερ0,u0,Φ0,M0∣r0,·,U0,·,Ψ0,·,H0,·+∫0τRtdtwhere
(19)ερ,u,Φ,M∣r,U,Ψ,Hτ=∫Ω12ρu−U2+Pρ−P′rρ−r−Pr+12M−H2+12∇Φ−Ψ2τ,·dxand the remainder Rt is defined as
(20)Rt=∫Ωμ∇U:∇U−u+λ+μdivUdivU−udx+∫Ωρ∂tU+u·∇U·U−udx+∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdx+∫Ωr−ρ∂tP′r+rU−ρu·∇P′r+pr−pρdivUdx.Theorem 5.
Letγ>2. Assume that ρ,u,Φ,M is the finite-energy weak solution of the magnetohydrodynamic system (1)–(5) in the sense of Definition 1. And assume that r,U,Ψ,H is the strong solution emanating from the same initial data and satisfy
(21)0<r¯≤inf0,T×Ωrt,x≤rt,x≤sup0,T×Ωrt,x<∞,U∈L20,T;W2,qΩ,∇2U∈L10,T;L∞Ω,H∈L40,T;W2,qΩwith q>2. Then
(22)ρ=r,u=U,ϕ=ψ,M=Hin0,T×Ω.


## 3. Relative Entropy Inequality

In this section, we will establish the relative entropy inequality(18) to the system (1)–(5) and extend the admissible class of test function.


### 3.1. Proof of Theorem4

Proof.
For simplicity, we define symbolsr0, U0, H0 as
(23)r0=r0,⋅,U0=U0,⋅,H0=H0,⋅.
Taking1/2U2 as test function in (10), we have
(24)∫Ω12ρU2τ,·dx−∫Ω12ρ0U02dx=∫0τ∫ΩρU·∂tU+ρu·∇U·Udxdt.
Similarly, substitutingφ for U as a test function in (13), we can get
(25)∫Ωρu·Uτ,·dx−∫Ωρ0u0·U0dx+∫0τ∫Ωμ∇u:∇U+λ+μdivudivUdxdt=∫0τ∫Ωρu·∂tU+ρu⊗u:∇U+pρdivU+12M2divU−M⊗M:∇U+ρ∇Φ·Udxdt.
Takingφ=H in (14), we get
(26)∫ΩM·Hτ,·dx−∫ΩM0·H0dx+ν∫0τ∫Ω∇M:∇Hdxdt=∫0τ∫ΩM·∂tH+u⊗M:∇H−M⊗u:∇Hdxdt.
Multiplying (16) by H and integrating over 0,T×Ω, we can obtain
(27)∫Ω12H2τ,·dx−∫Ω12H02dx+ν∫0τ∫Ω∇H2dxdt=∫0τ∫ΩU⊗H:∇H−H⊗U:∇Hdxdt.
By virtue ofΦ,Ψ satisfying (4) and (17), one has
(28)ΔΦ−Ψ=ρ−r.
Differentiating (28) with respect to t, we get
(29)∂tΔΦ−Ψ=∂tρ−∂tr=divrU−ρu.
Multiplying (29) by Φ−Ψ and integrating the resulting equation over Ω, we can obtain
(30)∂t∫Ω12∇Φ−Ψ2dx=∫ΩrU−ρu·∇Φ−Ψdx.
From (7), we can deduce
(31)∫Ω12ρu2+aγ−1ργ+12M2τ,·dx+∫0τ∫Ωμ∇u2+λ+μdivu2+ν∇M2dxdt≤∫Ω12ρ0u02+aγ−1ρ0γ+12M02dx+∫0τ∫Ωρu·∇Φdxdt.
Summing up relations (24)–(31), we can infer
(32)∫Ω12ρu−U2+Pρ+12M−H2+12∇Φ−Ψ2τ,·dx+∫0τ∫Ωμ∇u:∇u−U+λ+μdivudivu−Udxdt+ν∫0τ∫Ω∇M2−∇M:∇H+∇H2dxdt≤∫Ω12ρ0u0−U02+Pρ0+12M0−H02+12∇Φρ0−Ψρ02dx+∫0τ∫ΩρU−u·∂tU+ρu·∇U·U−u−pρdivU−12M2divU+M⊗M:∇Udxdt−∫0τ∫ΩM·∂tH+u⊗M:∇H−M⊗u:∇H+H⊗U:∇H−U⊗H:∇Hdxdt+∫0τ∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdxdt.
Note that(33)∫Ω−P′rρ−r−Prτ,·dx−∫Ω−P′r0ρ0−r0−Pr0dx=∫0τ∫Ω∂t−P′rρ−r−Prdxdt=∫0τ∫Ωr−ρ∂tP′r−∂tρP′rdxdt=∫0τ∫Ωr−ρ∂tP′r−ρu·∇P′rdxdt,∫0τ∫ΩrU·∇P′r+prdivUdxdt=0.
Then we can rewrite (32) as
(34)∫Ω12ρu−U2+Pρ−P′rρ−r−Pr+12M−H2+12∇Φ−Ψ2τ,·dx+∫0τ∫Ωμ∇u:∇u−U+λ+μdivudivu−Udxdt+ν∫0τ∫Ω∇M2−∇M:∇H+∇H2dxdt≤∫Ω12ρ0u0−U02+Pρ0−P′r0ρ0−r0−Pr0+12M0−H02+12∇Φρ0−Ψρ02dx+∫0τ∫Ωρ∂tU+u·∇U·U−udxdt+∫0τ∫Ω−12M2divU+M⊗M:∇Udxdt−∫0τ∫ΩM·∂tH+u⊗M:∇H−M⊗u:∇H+H⊗U:∇H−U⊗H:∇Hdxdt+∫0τ∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdxdt+∫0τ∫Ωr−ρ∂tP′r+rU−ρu·∇P′r−pρ−prdivUdxdt.
For simplicity, we abbreviateερ,u,Φ,M∣r,U,Ψ,Hτ to ετ.
Replacing∂tH with (16), we can infer
(35)ετ+∫0τ∫Ωμ∇u:∇u−U+λ+μdivudivu−U+ν∇M−H2dxdt≤ε0+∫0τ∫Ωρ∂tU+u·∇U·U−udxdt+∫0τ∫ΩH⊗U−U⊗H:∇M+M⊗u−u⊗M:∇H+U⊗H−H⊗U:∇Hdxdt+∫0τ∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdxdt+∫0τ∫Ωr−ρ∂tP′r+rU−ρu·∇P′r−pρ−prdivUdxdt+∫0τ∫ΩU⊗M:∇M−M⊗U:∇Mdxdt.
Finally, we can get(36)ετ+∫0τ∫Ωμ∇u−U2+λ+μdivu−U2+ν∇M−H2dxdt≤ε0+∫0τRtdt.
The proof of Theorem4 is completed.☐


### 3.2. Extending the Admissible Class of Test Function

Making use of density argument, we can extend the class of test functionr,U,Ψ,H appeared in the relative entropy inequality (18), (20).For the left hand side of (18) to be well defined, the function r,U,Ψ,H must belong at least to the class:
(37)r∈C0,T;LωγΩ,U∈C0,T;Lω2γ/γ−1Ω;(38)∇UL20,T×Ω,U∂Ω=0;(39)H∈L20,T;H01Ω∩C0,T;Lω2Ω;(40)Ψ∈C0,T;Wω2,γΩ,∇Ψ·n∂Ω=0.Similarly, a short inspection of the integrals in (20) yields
(41)∂tU∈L10,T;L2γ/γ−1Ω;∇U∈L10,T;L∞Ω;∇H∈L20,T;L∞Ω.Moreover, the functionr must be bounded away from zero, and
(42)∂tP′r∈L10,T;Lγ/γ−1Ω;∇P′r∈L10,T;L2γ/γ−1Ω.It is easy to prove that the relative entropy inequality (18), (20) can be extended to r,U,Ψ,H satisfying (37)–(42) by density argument.


## 3.1. Proof of Theorem4

Proof.
For simplicity, we define symbolsr0, U0, H0 as
(23)r0=r0,⋅,U0=U0,⋅,H0=H0,⋅.
Taking1/2U2 as test function in (10), we have
(24)∫Ω12ρU2τ,·dx−∫Ω12ρ0U02dx=∫0τ∫ΩρU·∂tU+ρu·∇U·Udxdt.
Similarly, substitutingφ for U as a test function in (13), we can get
(25)∫Ωρu·Uτ,·dx−∫Ωρ0u0·U0dx+∫0τ∫Ωμ∇u:∇U+λ+μdivudivUdxdt=∫0τ∫Ωρu·∂tU+ρu⊗u:∇U+pρdivU+12M2divU−M⊗M:∇U+ρ∇Φ·Udxdt.
Takingφ=H in (14), we get
(26)∫ΩM·Hτ,·dx−∫ΩM0·H0dx+ν∫0τ∫Ω∇M:∇Hdxdt=∫0τ∫ΩM·∂tH+u⊗M:∇H−M⊗u:∇Hdxdt.
Multiplying (16) by H and integrating over 0,T×Ω, we can obtain
(27)∫Ω12H2τ,·dx−∫Ω12H02dx+ν∫0τ∫Ω∇H2dxdt=∫0τ∫ΩU⊗H:∇H−H⊗U:∇Hdxdt.
By virtue ofΦ,Ψ satisfying (4) and (17), one has
(28)ΔΦ−Ψ=ρ−r.
Differentiating (28) with respect to t, we get
(29)∂tΔΦ−Ψ=∂tρ−∂tr=divrU−ρu.
Multiplying (29) by Φ−Ψ and integrating the resulting equation over Ω, we can obtain
(30)∂t∫Ω12∇Φ−Ψ2dx=∫ΩrU−ρu·∇Φ−Ψdx.
From (7), we can deduce
(31)∫Ω12ρu2+aγ−1ργ+12M2τ,·dx+∫0τ∫Ωμ∇u2+λ+μdivu2+ν∇M2dxdt≤∫Ω12ρ0u02+aγ−1ρ0γ+12M02dx+∫0τ∫Ωρu·∇Φdxdt.
Summing up relations (24)–(31), we can infer
(32)∫Ω12ρu−U2+Pρ+12M−H2+12∇Φ−Ψ2τ,·dx+∫0τ∫Ωμ∇u:∇u−U+λ+μdivudivu−Udxdt+ν∫0τ∫Ω∇M2−∇M:∇H+∇H2dxdt≤∫Ω12ρ0u0−U02+Pρ0+12M0−H02+12∇Φρ0−Ψρ02dx+∫0τ∫ΩρU−u·∂tU+ρu·∇U·U−u−pρdivU−12M2divU+M⊗M:∇Udxdt−∫0τ∫ΩM·∂tH+u⊗M:∇H−M⊗u:∇H+H⊗U:∇H−U⊗H:∇Hdxdt+∫0τ∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdxdt.
Note that(33)∫Ω−P′rρ−r−Prτ,·dx−∫Ω−P′r0ρ0−r0−Pr0dx=∫0τ∫Ω∂t−P′rρ−r−Prdxdt=∫0τ∫Ωr−ρ∂tP′r−∂tρP′rdxdt=∫0τ∫Ωr−ρ∂tP′r−ρu·∇P′rdxdt,∫0τ∫ΩrU·∇P′r+prdivUdxdt=0.
Then we can rewrite (32) as
(34)∫Ω12ρu−U2+Pρ−P′rρ−r−Pr+12M−H2+12∇Φ−Ψ2τ,·dx+∫0τ∫Ωμ∇u:∇u−U+λ+μdivudivu−Udxdt+ν∫0τ∫Ω∇M2−∇M:∇H+∇H2dxdt≤∫Ω12ρ0u0−U02+Pρ0−P′r0ρ0−r0−Pr0+12M0−H02+12∇Φρ0−Ψρ02dx+∫0τ∫Ωρ∂tU+u·∇U·U−udxdt+∫0τ∫Ω−12M2divU+M⊗M:∇Udxdt−∫0τ∫ΩM·∂tH+u⊗M:∇H−M⊗u:∇H+H⊗U:∇H−U⊗H:∇Hdxdt+∫0τ∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdxdt+∫0τ∫Ωr−ρ∂tP′r+rU−ρu·∇P′r−pρ−prdivUdxdt.
For simplicity, we abbreviateερ,u,Φ,M∣r,U,Ψ,Hτ to ετ.
Replacing∂tH with (16), we can infer
(35)ετ+∫0τ∫Ωμ∇u:∇u−U+λ+μdivudivu−U+ν∇M−H2dxdt≤ε0+∫0τ∫Ωρ∂tU+u·∇U·U−udxdt+∫0τ∫ΩH⊗U−U⊗H:∇M+M⊗u−u⊗M:∇H+U⊗H−H⊗U:∇Hdxdt+∫0τ∫Ωρu−U·∇Φ+rU−ρu·∇Φ−Ψdxdt+∫0τ∫Ωr−ρ∂tP′r+rU−ρu·∇P′r−pρ−prdivUdxdt+∫0τ∫ΩU⊗M:∇M−M⊗U:∇Mdxdt.
Finally, we can get(36)ετ+∫0τ∫Ωμ∇u−U2+λ+μdivu−U2+ν∇M−H2dxdt≤ε0+∫0τRtdt.
The proof of Theorem4 is completed.☐


## 3.2. Extending the Admissible Class of Test Function

Making use of density argument, we can extend the class of test functionr,U,Ψ,H appeared in the relative entropy inequality (18), (20).For the left hand side of (18) to be well defined, the function r,U,Ψ,H must belong at least to the class:
(37)r∈C0,T;LωγΩ,U∈C0,T;Lω2γ/γ−1Ω;(38)∇UL20,T×Ω,U∂Ω=0;(39)H∈L20,T;H01Ω∩C0,T;Lω2Ω;(40)Ψ∈C0,T;Wω2,γΩ,∇Ψ·n∂Ω=0.Similarly, a short inspection of the integrals in (20) yields
(41)∂tU∈L10,T;L2γ/γ−1Ω;∇U∈L10,T;L∞Ω;∇H∈L20,T;L∞Ω.Moreover, the functionr must be bounded away from zero, and
(42)∂tP′r∈L10,T;Lγ/γ−1Ω;∇P′r∈L10,T;L2γ/γ−1Ω.It is easy to prove that the relative entropy inequality (18), (20) can be extended to r,U,Ψ,H satisfying (37)–(42) by density argument.


## 4. Weak-Strong Uniqueness

In this section, we will prove weak-strong uniqueness property of the MHD system (1)–(5). In order to prove the weak-strong uniqueness, we firstly rewrite the remainder Rt as another expression. Then we apply Gronwall’s inequality to prove the weak-strong uniqueness property.


### 4.1. Another Expression ofRt

Lemma 6.
Letr,U,Φ,H be a strong solution to MHD system (1)-(5). For r≥r¯>0, the remainder Rt can be rewritten as
(43)Rt=∫Ωρu−U·∇U·U−udx−∫Ωpρ−p′rρ−r−prdivUdx+∫Ωu−U·∇H·H−M+M−H·∇H·u−Udx+∫ΩM−H·∇H−M·U+U·∇M−H·M−Hdx+∫Ωρ−rrU−uμΔU+λ+μ∇divUdx+∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx+∫Ωr−ρU·∇Φ−Ψdx.Proof.
Becauser,U,Φ,H is a strong solution to MHD system (1)-(5), we can get
(44)∂tU+U·∇U=r−1μΔU+λ+μ∇divU−r−1∇pr−r−1∇H22+r−1divH⊗H+∇Ψ.
From (44), the remainder Rt is rewritten as
(45)Rt=∫Ωρ−rrU−u·μΔU+λ+μ∇divUdx+∫ΩρrU−u·−∇H22+divH⊗Hdx+∫Ωρu−U·∇U·U−udx+∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫Ωρu−U·∇Φ−Ψ+rU−ρu·∇Φ−Ψdx+∫Ωr−ρ∂tP′r+r−ρU·∇P′rdx+∫Ωpr−pρdivUdx.
Note that(46)r−ρ∂tP′r+r−ρU·∇P′r=p′rρ−rdivU.
Then we can rewriteRt as
(47)Rt=∫Ωρ−rrU−u·μΔU+λ+μ∇divUdx+∫ΩρrU−u·−∇H22+divH⊗Hdx+∫Ωρu−U·∇U·U−udx+∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫Ωr−ρU·∇Φ−Ψdx−∫Ωpρ−p′rρ−r−prdivUdx.
Besides, it is easy to infer that(48)∫Ω−ρrU−u·∇H22+ρrU−u·divH⊗Hdx=∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx+∫ΩH·∇H·U−u−U−u·∇H·Hdx(49)∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫ΩH·∇H·U−u−U−u·∇H·Hdx=∫Ωu·∇H·H−M+M−H·∇H·u+U·∇M·M−H+H−M·∇M·Udx=∫Ωu−U·∇H·H−M+U·∇H·H−M+U·∇M·M−Hdx+∫ΩM−H·∇H·u−U+M−H·∇H·U+H−M·∇M·Udx=∫Ωu−U·∇H·H−M+U·∇M−H·M−Hdx+∫ΩM−H·∇H−M·U+M−H·∇H·u−Udx.
On the basis of (48) and (49), we can infer
(50)Rt=∫Ωρu−U·∇U·U−udx−∫Ωpρ−p′rρ−r−prdivUdx+∫Ωu−U·∇H·H−M+M−H·∇H·u−Udx+∫ΩU·∇M−H·M−H+M−H·∇H−M·Udx+∫Ωρ−rrU−uμΔU+λ+μ∇divUdx+∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx+∫Ωr−ρU·∇Φ−Ψdx.
The proof of Lemma6 is completed.☐


### 4.2. Proof of Theorem5

To begin, it is easy to check that(51)Pρ−P′rρ−r−Pr≥crρ−r2,forr2≤ρ≤2r,1+ργ,otherwise.where cr is uniformly bounded for r belonging to compact sets in 0,∞.Proof.
There are two steps to prove Theorem5.
Step 1: We estimate the remainderRt.
For∇U∈L10,T;L∞Ω and (51), it is easy to infer that
(52)∣∫Ωρu−U·∇U·U−udx−∫Ωpρ−p′rρ−r−prdivUdx∣≤C∥∇U∥L∞Ωεt≤ηtεt.
By using of Ho¨lder’s inequality and Sobolev’s inequality, we can deduce
(53)∫Ωu−U⋅∇H⋅H−Mdx+∫ΩM−H⋅∇H⋅u−Udx≤C∫Ωu−U∇HM−Hdx≤CM−HL2Ω∇HLqΩu−UL2q/q−2Ω≤CM−HL2Ω∇HLqΩu−Uw01,2Ω≤C∇HLqΩ2∫ΩM−H2dx+δ∫Ω∇u−U2dx≤ηtεt+δ∫Ω∇u−U2dx,∣∫ΩU·∇M−H·M−Hdx∣=∣−∫ΩM−H22divUdx∣≤C∥divU∥L∞Ωεt≤ηtεthere δ>0 is sufficient small and q>2.
FordivM=divH=0, it is easy to get that
(54)∣∫ΩM−H·∇H−M·Udx∣=∣−∫ΩM−H·∇U·H−Mdx∣≤C∥∇U∥L∞Ω∫ΩM−H2dx≤ηtεt.
Now, we estimate the last three items ofRt.
(55)I=∫Ωρ−rrμΔU+λ+μ∇divUU−udx=∫r/2≤ρ≤2r+∫0≤ρ<r/2+∫ρ>2rρ−rrμΔU+λ+μ∇divUU−udx≔I1+I2+I3.
We can make use of Ho¨lder’s inequality, Sobolev’s inequality and (51) to obtain
(56)∣I1∣=∣∫r/2≤ρ≤2rr−1ρ−rμΔU+λ+μ∇divUU−udx∣≤C∥∇2U∥LqΩ∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥L2q/q−2Ω≤C∥∇2U∥LqΩ∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥W01,2Ω≤C∥∇2U∥LqΩ2∫r/2≤ρ≤2rρ−r2dx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx,∣I2∣=∣∫0≤ρ<r/2r−1ρ−rU−uμΔU+λ+μ∇divUdx∣≤C∥∇2U∥LqΩ∥1∥L20≤ρ<r/2∥U−u∥L2q/q−2Ω≤C∥∇2U∥LqΩ∥1∥L20≤ρ<r/2∥U−u∥W01,2Ω≤C∥∇2U∥LqΩ2∫0≤ρ<r/2Pρ−P′rρ−r−Prdx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx.where q>2.
Note that(57)∥ργ/2∥L2ρ>2r≤εt1/2and∣ρ−rρr∣ρ1/2−γ/2≤Casρ>2r≥2r¯>0,γ>1.
By Ho¨lder’s inequality and (57), we can obtain
(58)∣I3∣=∣∫ρ>2rρ−rrU−uμΔU+λ+μ∇divUdx∣≤C∫ρ>2r∣ρ−rρrρ1/2−γ/2∣·ργ/2·∣μΔU+λ+μ∇divU∣·ρ1/2∣U−u∣dx≤C∥∇2U∥L∞Ω∫ρ>2rργdx1/2∫ΩρU−u2dx1/2≤ηtεt.
Therefor, we can get the estimate ofI(59)∣I∣≤ηtεt+δ∫Ω∇U−u2dxwith δ>0 sufficiently small.
Similarly to the estimate ofI, we write
(60)J=∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx=∫r/2≤ρ≤2r+∫0≤ρ<r/2+∫ρ>2rρ−rrH·∇H·U−u−U−u·∇H·Hdx≔J1+J2+J3.
Applying Ho¨lder’s inequality, Sobolev’s inequality, (51) and (57), we can infer that
(61)∣J1∣=∣∫r/2≤ρ≤2rρ−rrH·∇H·U−u−U−u·∇H·Hdx∣≤∫r2≤ρ≤2r∣ρ−r∣·∣H∣·∣∇H∣·∣U−u∣dx≤C∥H∥L∞Ω∥∇H∥LqΩ∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥L2q/q−2Ω≤C∥H∥W1,qΩ2∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥W01,2Ω≤C∥H∥W1,qΩ4∫r/2≤ρ≤2rρ−r2dx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx;∣J2∣=∣∫0≤ρ<r/2ρ−rrH·∇H·U−u−U−u·∇H·Hdx∣≤∫0≤ρ<r/21·∣H∣·∣∇H∣·∣U−u∣dx≤C∥H∥L∞Ω∥∇H∥LqΩ∥1∥L20≤ρ<r/2∥U−u∥L2q/q−2Ω≤C∥H∥W1,qΩ2∥1∥L20≤ρ<r/2∥U−u∥W01,2Ω≤C∥H∥W1,qΩ4∫0≤ρ<r/21dx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx;∣J3∣=∣∫ρ>2rρ−rrH·∇H·U−u−U−u·∇H·Hdx∣≤C∫ρ>2r∣ρ−rρrρ1/2−γ/2∣·ργ/2·∣H∣·∣∇H∣·ρ1/2∣U−u∣dx≤C∥H∥L∞Ω∥∇H∥L∞Ω∥ργ/2∥Lρ>2r2∥ρ1/2U−u∥L2Ω≤C∥H∥W2,qΩ4εt≤ηtεt.with q>2.
Finally, we estimate the last item ofRt.
(62)K=∫Ωr−ρU·∇Φ−ψdx=∫r/2≤ρ≤2r+∫0≤ρ<r/2+∫ρ>2rr−ρU·∇Φ−ψdx≔K1+K2+K3.
From (51), we can get
(63)∣K1∣≤∫r/2≤ρ≤2rr−ρU·∣∇Φ−ψ∣dx≤C∥U∥L∞Ω∥r−ρ∥L2r/2≤ρ≤2r∇Φ−ψ≤C∥U∥L∞Ω∫r/2≤ρ≤2rPρ−P′rρ−r−Prdx1/2εt1/2≤ηtεt;∣K2∣≤∫0≤ρ<r1·∣U∣·∣∇Φ−ψ∣dx≤C∥U∥L∞Ω∥1∥L20≤ρ<r∇Φ−ψ≤C∥U∥L∞Ω∫0≤ρ<rPρ−P′rρ−r−Prdx1/2εt1/2≤ηtεt.
Using of Ho¨lder’s inequality and (57), we can infer that
(64)∣K3∣=∣∫ρ>2rr−ρU·∇Φ−ψdx∣≤C∫ρ>2r∣ργ/2∣·∣U∣·∣∇Φ−ψ∣dx≤C∥U∥L∞Ω∥ργ/2∥L2ρ>2r∇Φ−ψ≤ηtεtwith γ>2.
Step 2: Making use of Gronwall’s inequality, we can prove the weak-strong uniqueness of MHD system (1)–(5).
Substituting inequalities (52)–(64) into (18), we can obtain
(65)ετ+∫0τ∫Ωμ∇u−U2+λ+μdivu−U2+ν∇M−H2dxdt≤∫0τηtεtdtwith ηt∈L10,T. By Gronwall’s inequality, we can deduce εt=0 in 0,T. Then this implies ρ=r, u=U, Φ=Ψ, M=H. The proof of Theorem 5 is completed.☐Remark 7.
The strong solution we work with in the weak-strong uniqueness theorem is only a hypothetical object which we do not know if it even exist. In the future, we will consider the existence of this strong solution.


## 4.1. Another Expression ofRt

Lemma 6.
Letr,U,Φ,H be a strong solution to MHD system (1)-(5). For r≥r¯>0, the remainder Rt can be rewritten as
(43)Rt=∫Ωρu−U·∇U·U−udx−∫Ωpρ−p′rρ−r−prdivUdx+∫Ωu−U·∇H·H−M+M−H·∇H·u−Udx+∫ΩM−H·∇H−M·U+U·∇M−H·M−Hdx+∫Ωρ−rrU−uμΔU+λ+μ∇divUdx+∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx+∫Ωr−ρU·∇Φ−Ψdx.Proof.
Becauser,U,Φ,H is a strong solution to MHD system (1)-(5), we can get
(44)∂tU+U·∇U=r−1μΔU+λ+μ∇divU−r−1∇pr−r−1∇H22+r−1divH⊗H+∇Ψ.
From (44), the remainder Rt is rewritten as
(45)Rt=∫Ωρ−rrU−u·μΔU+λ+μ∇divUdx+∫ΩρrU−u·−∇H22+divH⊗Hdx+∫Ωρu−U·∇U·U−udx+∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫Ωρu−U·∇Φ−Ψ+rU−ρu·∇Φ−Ψdx+∫Ωr−ρ∂tP′r+r−ρU·∇P′rdx+∫Ωpr−pρdivUdx.
Note that(46)r−ρ∂tP′r+r−ρU·∇P′r=p′rρ−rdivU.
Then we can rewriteRt as
(47)Rt=∫Ωρ−rrU−u·μΔU+λ+μ∇divUdx+∫ΩρrU−u·−∇H22+divH⊗Hdx+∫Ωρu−U·∇U·U−udx+∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫Ωr−ρU·∇Φ−Ψdx−∫Ωpρ−p′rρ−r−prdivUdx.
Besides, it is easy to infer that(48)∫Ω−ρrU−u·∇H22+ρrU−u·divH⊗Hdx=∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx+∫ΩH·∇H·U−u−U−u·∇H·Hdx(49)∫ΩH⊗U−U⊗H:∇M−H+M⊗u−u⊗M:∇H+U⊗M−M⊗U:∇Mdx+∫ΩH·∇H·U−u−U−u·∇H·Hdx=∫Ωu·∇H·H−M+M−H·∇H·u+U·∇M·M−H+H−M·∇M·Udx=∫Ωu−U·∇H·H−M+U·∇H·H−M+U·∇M·M−Hdx+∫ΩM−H·∇H·u−U+M−H·∇H·U+H−M·∇M·Udx=∫Ωu−U·∇H·H−M+U·∇M−H·M−Hdx+∫ΩM−H·∇H−M·U+M−H·∇H·u−Udx.
On the basis of (48) and (49), we can infer
(50)Rt=∫Ωρu−U·∇U·U−udx−∫Ωpρ−p′rρ−r−prdivUdx+∫Ωu−U·∇H·H−M+M−H·∇H·u−Udx+∫ΩU·∇M−H·M−H+M−H·∇H−M·Udx+∫Ωρ−rrU−uμΔU+λ+μ∇divUdx+∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx+∫Ωr−ρU·∇Φ−Ψdx.
The proof of Lemma6 is completed.☐


## 4.2. Proof of Theorem5

To begin, it is easy to check that(51)Pρ−P′rρ−r−Pr≥crρ−r2,forr2≤ρ≤2r,1+ργ,otherwise.where cr is uniformly bounded for r belonging to compact sets in 0,∞.Proof.
There are two steps to prove Theorem5.
Step 1: We estimate the remainderRt.
For∇U∈L10,T;L∞Ω and (51), it is easy to infer that
(52)∣∫Ωρu−U·∇U·U−udx−∫Ωpρ−p′rρ−r−prdivUdx∣≤C∥∇U∥L∞Ωεt≤ηtεt.
By using of Ho¨lder’s inequality and Sobolev’s inequality, we can deduce
(53)∫Ωu−U⋅∇H⋅H−Mdx+∫ΩM−H⋅∇H⋅u−Udx≤C∫Ωu−U∇HM−Hdx≤CM−HL2Ω∇HLqΩu−UL2q/q−2Ω≤CM−HL2Ω∇HLqΩu−Uw01,2Ω≤C∇HLqΩ2∫ΩM−H2dx+δ∫Ω∇u−U2dx≤ηtεt+δ∫Ω∇u−U2dx,∣∫ΩU·∇M−H·M−Hdx∣=∣−∫ΩM−H22divUdx∣≤C∥divU∥L∞Ωεt≤ηtεthere δ>0 is sufficient small and q>2.
FordivM=divH=0, it is easy to get that
(54)∣∫ΩM−H·∇H−M·Udx∣=∣−∫ΩM−H·∇U·H−Mdx∣≤C∥∇U∥L∞Ω∫ΩM−H2dx≤ηtεt.
Now, we estimate the last three items ofRt.
(55)I=∫Ωρ−rrμΔU+λ+μ∇divUU−udx=∫r/2≤ρ≤2r+∫0≤ρ<r/2+∫ρ>2rρ−rrμΔU+λ+μ∇divUU−udx≔I1+I2+I3.
We can make use of Ho¨lder’s inequality, Sobolev’s inequality and (51) to obtain
(56)∣I1∣=∣∫r/2≤ρ≤2rr−1ρ−rμΔU+λ+μ∇divUU−udx∣≤C∥∇2U∥LqΩ∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥L2q/q−2Ω≤C∥∇2U∥LqΩ∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥W01,2Ω≤C∥∇2U∥LqΩ2∫r/2≤ρ≤2rρ−r2dx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx,∣I2∣=∣∫0≤ρ<r/2r−1ρ−rU−uμΔU+λ+μ∇divUdx∣≤C∥∇2U∥LqΩ∥1∥L20≤ρ<r/2∥U−u∥L2q/q−2Ω≤C∥∇2U∥LqΩ∥1∥L20≤ρ<r/2∥U−u∥W01,2Ω≤C∥∇2U∥LqΩ2∫0≤ρ<r/2Pρ−P′rρ−r−Prdx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx.where q>2.
Note that(57)∥ργ/2∥L2ρ>2r≤εt1/2and∣ρ−rρr∣ρ1/2−γ/2≤Casρ>2r≥2r¯>0,γ>1.
By Ho¨lder’s inequality and (57), we can obtain
(58)∣I3∣=∣∫ρ>2rρ−rrU−uμΔU+λ+μ∇divUdx∣≤C∫ρ>2r∣ρ−rρrρ1/2−γ/2∣·ργ/2·∣μΔU+λ+μ∇divU∣·ρ1/2∣U−u∣dx≤C∥∇2U∥L∞Ω∫ρ>2rργdx1/2∫ΩρU−u2dx1/2≤ηtεt.
Therefor, we can get the estimate ofI(59)∣I∣≤ηtεt+δ∫Ω∇U−u2dxwith δ>0 sufficiently small.
Similarly to the estimate ofI, we write
(60)J=∫Ωρ−rrH·∇H·U−u−U−u·∇H·Hdx=∫r/2≤ρ≤2r+∫0≤ρ<r/2+∫ρ>2rρ−rrH·∇H·U−u−U−u·∇H·Hdx≔J1+J2+J3.
Applying Ho¨lder’s inequality, Sobolev’s inequality, (51) and (57), we can infer that
(61)∣J1∣=∣∫r/2≤ρ≤2rρ−rrH·∇H·U−u−U−u·∇H·Hdx∣≤∫r2≤ρ≤2r∣ρ−r∣·∣H∣·∣∇H∣·∣U−u∣dx≤C∥H∥L∞Ω∥∇H∥LqΩ∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥L2q/q−2Ω≤C∥H∥W1,qΩ2∥ρ−r∥L2r/2≤ρ≤2r∥U−u∥W01,2Ω≤C∥H∥W1,qΩ4∫r/2≤ρ≤2rρ−r2dx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx;∣J2∣=∣∫0≤ρ<r/2ρ−rrH·∇H·U−u−U−u·∇H·Hdx∣≤∫0≤ρ<r/21·∣H∣·∣∇H∣·∣U−u∣dx≤C∥H∥L∞Ω∥∇H∥LqΩ∥1∥L20≤ρ<r/2∥U−u∥L2q/q−2Ω≤C∥H∥W1,qΩ2∥1∥L20≤ρ<r/2∥U−u∥W01,2Ω≤C∥H∥W1,qΩ4∫0≤ρ<r/21dx+δ∫Ω∇U−u2dx≤ηtεt+δ∫Ω∇U−u2dx;∣J3∣=∣∫ρ>2rρ−rrH·∇H·U−u−U−u·∇H·Hdx∣≤C∫ρ>2r∣ρ−rρrρ1/2−γ/2∣·ργ/2·∣H∣·∣∇H∣·ρ1/2∣U−u∣dx≤C∥H∥L∞Ω∥∇H∥L∞Ω∥ργ/2∥Lρ>2r2∥ρ1/2U−u∥L2Ω≤C∥H∥W2,qΩ4εt≤ηtεt.with q>2.
Finally, we estimate the last item ofRt.
(62)K=∫Ωr−ρU·∇Φ−ψdx=∫r/2≤ρ≤2r+∫0≤ρ<r/2+∫ρ>2rr−ρU·∇Φ−ψdx≔K1+K2+K3.
From (51), we can get
(63)∣K1∣≤∫r/2≤ρ≤2rr−ρU·∣∇Φ−ψ∣dx≤C∥U∥L∞Ω∥r−ρ∥L2r/2≤ρ≤2r∇Φ−ψ≤C∥U∥L∞Ω∫r/2≤ρ≤2rPρ−P′rρ−r−Prdx1/2εt1/2≤ηtεt;∣K2∣≤∫0≤ρ<r1·∣U∣·∣∇Φ−ψ∣dx≤C∥U∥L∞Ω∥1∥L20≤ρ<r∇Φ−ψ≤C∥U∥L∞Ω∫0≤ρ<rPρ−P′rρ−r−Prdx1/2εt1/2≤ηtεt.
Using of Ho¨lder’s inequality and (57), we can infer that
(64)∣K3∣=∣∫ρ>2rr−ρU·∇Φ−ψdx∣≤C∫ρ>2r∣ργ/2∣·∣U∣·∣∇Φ−ψ∣dx≤C∥U∥L∞Ω∥ργ/2∥L2ρ>2r∇Φ−ψ≤ηtεtwith γ>2.
Step 2: Making use of Gronwall’s inequality, we can prove the weak-strong uniqueness of MHD system (1)–(5).
Substituting inequalities (52)–(64) into (18), we can obtain
(65)ετ+∫0τ∫Ωμ∇u−U2+λ+μdivu−U2+ν∇M−H2dxdt≤∫0τηtεtdtwith ηt∈L10,T. By Gronwall’s inequality, we can deduce εt=0 in 0,T. Then this implies ρ=r, u=U, Φ=Ψ, M=H. The proof of Theorem 5 is completed.☐Remark 7.
The strong solution we work with in the weak-strong uniqueness theorem is only a hypothetical object which we do not know if it even exist. In the future, we will consider the existence of this strong solution.



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