On weak-strong uniqueness property for the full compressible magnetohydrodynamics flows

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by  Weiping Yan

Noname manuscript No.
(will be inserted by the editor)
On weak-strong uniqueness property for the full
compressible magnetohydrodynamics flows
Weiping Yan
3
1 Received: date/Accepted: date
0
2
Abstract This paper is devoted to the study of the weak-strong uniqueness
n
a propertyforthefullcompressiblemagnetohydrodynamicsflows.Thegoverning
J equations for magnetohydrodynamic flows are expressed by the full Navier-

5 Stokes system for compressible fluids enhanced by forces due to the presence
1 of the magnetic field as well as the gravity and with an additional equation

whichdescribesthe evolutionofthe magnetic field.Using the relativeentropy
]
P inequality, we prove that a weak solution coincides with the strong solution,
A emanating from the same initial data, as long as the latter exists.
.
h Keywords magnetohydrodynamic flows · weak solution · strong solution ·
t entropy
a
m
Mathematics Subject Classification (2010) 76W05 · 35D30 · 35D35
[
and 54C70

3
v
5
0 1 Introduction and Main results
4
2 This paper studies the weak-strong uniqueness property of the viscous com-
.
3 pressible magnetohydrodynamic flows
0
2
∂ ρ+div (ρu)=0, (1)
t x
1
: ∂t(ρu)+divx(ρu⊗u)+∇xP(ρ,θ)=divxS+J×H, (2)
v
q
i ∂ (ρs(ρ,θ))+div (ρs(ρ,θ)u)+div ( )=σ, (3)
X t x x θ
r ∂tH−∇×(u×H)+∇×(ν∇×H)=0. (4)
a
CollegeofofMathematics,JilinUniversity,Changchun 130012,P.R.China.
BeijingInternationalCenterforMathematicalResearch,PekingUniversity,Beijing100871,
P.R.China.
E-mail:[email protected]
2 WeipingYan
where u is the vector field, ρ is the density, θ is the temperature, J is the
electroniccurrent,e(ρ,θ)isthe(specific)internalenergyandHisthemagnetic
field. The electronic current satisfies Amp`ere’s law
J=∇×H, (5)
whereas the Lorentz force is given by
1 1
J×H=div ( H⊗H− |H|2I), (6)
x
µ 2µ
withµbeingapermeabilityconstantoffreespace,whichhereisassumedtobe
µ=1 for simplicity of the presentation. The electronic current J, the electric
field E and the magnetic field H are related through Ohm’s law
J=σ(E+u×H). (7)
Theinteractiondescribedbythetheoryofmagnetohydrodynamics,“collective
effects,” is governedby the Faraday’s law,
∂ H+∇×E=0, div H=0. (8)
t x
Taking into consideration (7) we are able to write (8) in the following form
∂ H+∇×(H×u)+∇×(ν∇×H)=0, (9)
t
where ν = 1.
σ
Motivatedbyseveralrecentstudies devotedtothe scaleanalysisaswellas
numericalexperimentsrelatedtotheproposedmodel(seeKleinetal.[13]),we
suppose that the viscous stress S is a linear function of the velocity gradient,
therefore described by Newton’s law
2
⊥
S(θ,∇ u)=µ(θ)(∇ u+∇ u− div uI)+η(θ)div uI, (10)
x x x 3 x x
while q is the heat flux satisfying Fourier’s law
q=−κ(θ)∇ θ, (11)
x
and σ stands for the entropy production rate which is non-negative measure
given by
1 q(θ,∇ θ)·∇ θ
σ ≥ (S(θ,∇ u):∇ u− x x ). (12)
x x
θ θ
Wesupplementcompressiblemagnetohydrodynamicflows(1)-(4)withconser-
vation boundary condition:
u| =q·n| =0, (13)
∂Ω ∂Ω
and
H| =0. (14)
∂Ω
TitleSuppressedDuetoExcessiveLength 3
The concept of weak solution in fluid dynamics was introduced by Leray [16]
in the context of incompressible, linearly viscous fluids. The original ideas of
Lerayhavebeenputintotheelegantframeworkofgeneralizedderivatives(dis-
tributions) and the associated abstract function spaces of Sobolev type (For
example, see Ladyzhenskaya [15] and Temam [22]). Lions [17] extended the
theory to the class of barotropic flows (see also [6]). One of meaningful com-
pressibleflowmodelsisthecompressiblemagnetohydrodynamics(MHD).Itis
a combination of the compressible Navier-Stokes equations of fluid dynamics
andMaxwell’sequationsofelectromagnetism.DucometandFeireisl[4]proved
thattheexistenceofglobalintimeweaksolutionstoamulti-dimensionalnon-
isentropic MHD system for gaseous stars coupled with the Poisson equation
with all the viscosity coefficients and the pressure depending on temperature
anddensity asymptotically,respectively.Hu andWang[11]studied the global
variationalweak solutionto the three-dimensionalfull magnetohydrodynamic
equationswithlargedatabyanapproximationschemeandaweakconvergence
method. Jiang, et all. [12] obtained that the convergence towards the strong
solutionoftheidealincompressibleMHDsysteminthe periodicdomains.Re-
cently,Kwon,etall[14]establishedtheincompressiblelimitsofweaksolutions
tothecompressiblemagnetohydrodynamicsflows(1)-(4)onbothboundedand
unbounded domains.
The physical properties of the magnetohydrodynamics flows are reflected
through various constitutive relations which are expressed as typically non-
linear functions relating the pressure P =P(ρ,θ), the internal energy e(ρ,θ),
the specific entropy s = s(ρ,θ) to the macroscopic variables ρ, u, and θ. Ac-
cordingtothefundamentalprinciplesofthermodynamics,thespecificinternal
energyeisrelatedtothepressureP,andthespecificentropysthroughGibbs’
relation
1
θDs(ρ,θ)=De(ρ,θ)+P(ρ,θ)D( ), (15)
ρ
where D denotes the differential with respect to the state variables ρ and θ.
Since the lack of informationresulting from the inequality sign in (12), we
need supplement the resulting system with the energy inequality,
d 1 1
( ρ|u|2+ρe(ρ,θ)+ |H|2)dx+ (|∇u|2+ν|∇×H|2)≤0. (16)
dtZ 2 2 Z
Ω Ω
Thus the total energy E is given by
1 1
E = ρ|u|2+ρe(ρ,θ)+ |H|2. (17)
2 2
Underthesecircumstances,itcanbeshown(see[7],Chapter2)thatanyweak
solutionof(1)thatissufficientlysmoothsatisfies,insteadof(12),thestandard
relation
1 q(θ,∇ θ)·∇ θ
σ = S(θ,∇ u):∇ u− x x . (18)
x x
θ (cid:18) θ (cid:19)
4 WeipingYan
The pressure P =P(ρ,θ) is here expressed as
a
P =P +P , P = θ4, a>0, (19)
F R R
3
whereP denotestheradiationpressure.Moreover,weshallassumethatP =
R F
P +P , where P is the classical molecular pressure obeying Boyle’s law,
M E M
while P is the pressure of electron gas constituent behaving like a Fermi
E
gas in the degenerate regime of high densities and/or low temperatures (see
Chapters 1, 15 in Eliezer et al. [5]). Thus necessarily P takes the form
F
5 ρ
PF =θ2p( 3), (20)
θ2
where p∈C1[0,∞) satisfies
′
p(0)=0, p(Z)>0 for all z ≥0. (21)
In agreement with Gibbs’ relation (15), the internal energy can be taken as
θ4
e=e +e , with e =a , (22)
F R R ρ
where e =e (ρ,θ), P (ρ,θ) are interrelated through the following equation
F F F
of state
2
P (ρ,θ)= ρe (ρ,θ). (23)
F F
3
We need the thermodynamic stability hypothesis:
∂P(ρ,θ) ∂e(ρ,θ)
>0, >0 for all ρ,θ >0. (24)
∂ρ ∂θ
The second inequality in thermodynamic stability hypothesis (24) gives that
5p(Z)−p′(Z)Z
0< 3 <c for all Z >0, (25)
Z
which implies that the function Z 7→ p(Z) is decreasing and we suppose that
5
Z3
p(Z)
Z−li→m∞ Z35 =p∞ >0. (26)
In accordance with (15) and (22) we set the entropy as
ρ 4a
s=s +s , wtih s =S( ), s = θ3. (27)
F R F θ32 R 3ρ
Furthermore, by the Third law of thermodynamics,
3 5p(Z)−p′(Z)Z
S′(Z)=− 3 <0, lim S(Z)=0. (28)
2 Z2 Z−→∞
TitleSuppressedDuetoExcessiveLength 5
We choose the transport coefficients in the form
µ(θ)=µ +µ θ, µ ,µ >0, η ≡0, (29)
0 1 0 1
κ(θ)=κ +κ θ2+κ θ3, κ >0, i=0,2,3. (30)
0 2 3 i
A fundamental test of admissibility of a class of weak solutions to a given
evolutionary problem is the property of weak-strong uniqueness. More specif-
ically, the weak solution must coincide with a (hypothetical) strong solution
emanatingfromthesameinitialdataaslongasthelatterexists.Thisproblem
has been intensively studied for the incompressible Navier-Stokes system, for
example, see [2,18,21]. It is a bit more delicate in the case of compressible
cases. The weak- strong uniqueness of compressible barotropic Navier-Stokes
system and isentropic compressible Navier-Stokes system were established in
[8,9] and [19], respectively. P. Germain [19] provides only a partial and con-
ditional answer to the weak-strong uniqueness problem for the compressible
Navier-Stokesequations.Thisquestionisdefinitelysolvedin[9].Morerecently,
FeireislandNovotny´[10]extendedtheproblemtocompressibleNavier-Stokes-
Fourier system by the relative entropy inequality. The relative entropy in [10]
is reminiscent to C.M. Dafermos [3] (who introduced the relatives entropies
via the entropy flux pairs for the conservation laws), but is different from the
C.M. Dafermos concept (in contrast to [3], it is based on the thermodynamic
stability conditions).
Inspiredby the workofFeireislandNovotny´ [10],we provethatthe weak-
strong uniqueness of compressible three-dimensional magnetohydrodynamic
equations. Our contribution is to construct suitable relative entropy inequal-
ity to (1)-(3). Then we overcome the presence of the magnetic field and its
interaction with the hydrodynamic motion in the MHD flow of large oscilla-
tion.
We organize the rest of this paper as follows. In section 2, we recall the
definition of the weak solutions and strong solutions to the magnetohydrody-
namic flows on bounded domains. Meanwhile, the relative entropy inequality
of(1)-(4)isderived.Inthelastsection,wegivetherigorousproofoftheweak-
strong uniqueness property for the compressible magnetohydrodynamic flows
on bounded domains in the spirit of Feireisl and Novotny´ [10].
2 Relative entropy and Main result
Let Ω ⊂R3 be a bounded Lipschitz domain. We recall the definition of weak
solution for (1)-(4).
Definition 1 We say that a quantity (ρ,u,θ,H) is a weak solution of the
fullmagnetohydrodynamicflows(MHD)(1)-(4)supplementedwiththeinitial
data (ρ ,u ,s(ρ ,θ ),H ), and ρ ≥ 0,θ > 0 provided that the following
0 0 0 0 0 0 0
holds.
5
i) The density ρ is a non-negative function, ρ ∈ Cweak([0,T];L3(Ω)),
the velocity field u ∈ L2(0,T;W10,2(Ω;R3)), ρu ∈ Cweak([0,T];L45(Ω;R3)).
6 WeipingYan
Equation (1) is replaced by a family of integral identities
τ
ρ(τ,·)ϕ(τ,·)dx− ρ ϕ(0,·)dx= (ρ∂ ϕ+ρu·∇ ϕ)dxdt (31)
0 t x
Z Z Z Z
Ω Ω 0 Ω
for any ϕ∈C1([0,T]×Ω¯), and any τ ∈[0,T].
ii) The balance of momentum holds in distributional sense, namely
ρu(τ,·)·ϕ(τ,·)dx− ρ u ·ϕ(0,·)dx
0 0
Z Z
Ω Ω
τ
= (ρu·∂ ϕ+ρu⊗u:∇ ϕ+Pdiv ϕ−S:∇ ϕ
t x x x
Z Z
0 Ω
+[(∇×H)×H]·ϕ)dxdt (32)
for any ϕ∈C1([0,T]×Ω¯;R3), ϕ| =0 and any τ ∈[0,T].
∂Ω
iii) The entropy balance (3) and (12) are replaced by a family of integral
inequalities
ρs(ρ ,θ )ϕ(0,·)dx− ρs(ρ,θ)(τ,·)ϕ(τ,·)dx
0 0
Z Z
Ω Ω
τ ϕ q·∇ θ
x
+ (S:∇ u− dxdt
x
Z Z (cid:18)θ θ (cid:19)
0 Ω
τ q·∇ ϕ
≤− ρs(ρ,θ)∂ ϕ+ρs(ρ,θ)u·∇ ϕ+ x dxdt (33)
t x
Z Z (cid:18) θ (cid:19)
0 Ω
foranyϕ∈C1([0,T]×Ω¯),ϕ≥0andalmostallτ ∈[0,T].Herethequantities
Sandqaregiventhroughtheconstitutiveequations(10)and(11).Moreover,
similarlytothe above,allquantities mustbe atleastintegrableon(0,T)×Ω.
In particular, θ belongs to L∞(0,T;L4(Ω))∩L2(0,T;W1,2(Ω)). In addition,
we require θ to be positive for almost all (t,x)∈(0,T)×Ω.
iv) The total energy of the system satisfies the following inequality
1 1 τ
ρ|u|2+ρe(ρ,θ)+ |H|2 dx + (|∇u|2+ν|∇×H|2)dxdt
Z (cid:18)2 2 (cid:19) Z Z
Ω 0 Ω
1 1
≤ ρ |u |2+ρ e(ρ ,θ )+ |H |2 dx (34)
0 0 0 0 0 0
Z (cid:18)2 2 (cid:19)
Ω
for almost all τ ∈[0,T].
v) The magnetic field H ∈ L2(0,T;W1,2(Ω;R3)). The Maxwell equation
(4) verifies
H(τ,·)ϕ(τ,·)dx − H ϕ dx
0 0
Z Z
Ω Ω
τ
= (H·∂ ϕ−(H×u+ν∇×H)·(∇×ϕ))dxdt, (35)
t
Z Z
0 Ω
where ϕ∈C1([0,T]×Ω¯;R3), ϕ| =0 and any τ ∈[0,T].
∂Ω
TitleSuppressedDuetoExcessiveLength 7
The definition of strong solution is
Definition 2 We saythat(ρ′,u′,θ′,H′)is a classical(strong)solutionto the
full magnetohydrodynamic system (1)-(4) in (0,T)×Ω if
ρ′ ∈C1([0,T]×Ω¯), θ′,∂ θ′,∇2θ′ ∈C([0,T]×Ω),
t
u′,∂ u′,∇2u′ ∈C([0,T]×Ω;R3), H′,∂ H′,∇2H′ ∈C([0,T]×Ω;R3),
t t
′ ′ ′
ρ(t,x)≥ρ>0, θ (t,x)≥θ >0, for all (t,x), (36)
0
and ρ′,u′,θ′,H′ satisfy equations (1)-(4), (18), together with the boundary
conditions (13)-(14). Observe that hypothesis (36) implies the following regu-
larity properties of the initial data:
ρ(0)=ρ ∈C1(Ω¯), ρ ≥ρ′ >0,
0 0 0
u(0)=u ∈C2(Ω¯),
0
θ(0)=θ ∈C2(Ω¯), θ ≥θ′ >0,
0 0 0
H(0)=H ∈C2(Ω¯). (37)
0
Beforegivingthemainresult,wededucearelativeentropyinequalitywhich
issatisfiedbyanyweaksolutiontothefullmagnetohydrodynamicsystem(1)-
(4).
Let{A,B,C,D}beaquantityofsmoothfunction,AandC boundedbelow
away from zero in [0,T]×Ω, and B| = D| = 0. Moreover, we assume
∂Ω ∂Ω
that smooth functions B and D satisfy that
∂ D−∇×(B×D)+∇×(ν∇×D)=0. (38)
t
Taking ϕ= 1|B|2, ϕ=B and ϕ=C >0 as a test function in (31), (32) and
2
the entropy inequality (33), respectively, we get
1 1 τ
ρ|B|2(τ,·)dx− ρ |B|2(0,·)dx= (ρB·∂ B+ρu·∇ B·B)dxdt, (39)
0 t x
Z 2 Z 2 Z Z
Ω Ω 0 Ω
τ
ρu·B(τ,·)dx− ρ u ·B(0,·)dx = (ρu·∂ B+ρu⊗u:∇ B+P(ρ,θ)div B
0 0 t x x
Z Z Z Z
Ω Ω 0 Ω
−S(θ,∇ u):∇ B+((∇×H)×H)·B)dxdt(40)
x x
and
τ C q(θ,∇ θ)·∇ θ
ρ s(ρ ,θ )C(0,·)dx − ρs(ρ,θ)C(τ,·)dx+ (S(θ,∇ u):∇ u− x x )dxdt
Z 0 0 0 Z Z Z θ x x θ
Ω Ω 0 Ω
τ q(θ,∇ θ)
≤− (ρs(ρ,θ)∂ C+ρs(ρ,θ)u·∇ C+ x ·∇ C)dxdt. (41)
t x x
Z Z θ
0 Ω
8 WeipingYan
It follows from (39), (40) and the energy inequality (34) that
1 1 τ
( ρ|u−B|2 +ρe(ρ,θ)+ |H|2)(τ,·)dx+ (|∇u|2+ν|∇×H|2)dxdt
Z 2 2 Z Z
Ω 0 Ω
1 1
≤ ( ρ |u −B(0,·)|2+ρ e(ρ ,θ )+ |H |2)dx
0 0 0 0 0 0
Z 2 2
Ω
τ
+ ((ρ∂ B+ρu·∇ B)·(B−u)−P(ρ,θ)div B
t x x
Z Z
0 Ω
+S(θ,∇ u):∇ B−((∇×H)×H)·B)dxdt. (42)
x x
Then summing up (41) and (42), we deduce that
1 1
( ρ|u−B|2+ρe(ρ,θ)+ |H|2−Cρs(ρ,θ))(τ,·)dx
Z 2 2
Ω
τ τ C q(θ,∇ θ)·∇ θ
+ (|∇u|2+ν|∇×H|2)dxdt+ (S(θ,∇ u):∇ u− x x )dxdt
x x
Z Z Z Z θ θ
0 Ω 0 Ω
1 1
≤ ( ρ |u −B(0,·)|2+ρ e(ρ ,θ )+ |H |2+C(0,·)ρ s(ρ ,θ ))dx
0 0 0 0 0 0 0 0 0
Z 2 2
Ω
τ
+ ((ρ∂ B+ρu·∇ B)·(B−u)−P(ρ,θ)div B+S(θ,∇ u):∇ B−((∇×H)×H)·B)dxdt
t x x x x
Z Z
0 Ω
τ q(θ,∇ θ)
− (ρs(ρ,θ)∂ C+ρs(ρ,θ)u·∇ C+ x ·∇ C)dxdt. (43)
t x x
Z Z θ
0 Ω
Taking a test function ϕ=D in (35) and ϕ=∂ H (A,C) in (31), we have
ρ C
τ
H(τ,·)D(τ,·)dx− H D dx= (H·∂ D−(H×u+ν∇×H)·(∇×D))dxdt, (44)
0 0 t
Z Z Z Z
Ω Ω 0 Ω
ρ∂ρHC(A,C)(τ,·)dx − ρ0∂ρHC(0,·)(A(0,·),C(0,·))dx
Z Z
Ω Ω
τ
= (ρ∂ (∂ H (A,C)))+ρu·∇ (∂ H (A,C))dxdt, (45)
t ρ C x ρ C
Z Z
0 Ω
Multiplying (38) by D and integrate over (0,τ)×Ω, we find
1 1 τ
|D|2(τ,·)dx− |D |2dx=− ((D×B+ν∇×D)·(∇×D))dxdt.(46)
0
Z 2 Z 2 Z Z
Ω Ω 0 Ω
So by (43)-(46), we have
1 1
( ρ|u−B|2+ |H−D|2+H (ρ,θ)−∂ (H )(A,C)(ρ−A)−H (A,C))(τ,·)dx
C ρ C C
Z 2 2
Ω
τ
+ |∇u|2+ν(|∇×D|2−|∇×D||∇×H|+|∇×H|2) dxdt
Z Z
0 Ω(cid:0) (cid:1)
τ C q(θ,∇ θ)·∇ θ
+ (S(θ,∇ u):∇ u− x x )dxdt
x x
Z Z θ θ
0 Ω
TitleSuppressedDuetoExcessiveLength 9
1 1
≤Z (2ρ0|u0−B(0,·)|2+ 2|H0−D0|2+(HC(0,·)(ρ(0,·),θ(0,·))
Ω
−∂ρ(HC(0,·))(A(0,·),C(0,·))(ρ0−A(0,·))−HC(0,·)(A(0,·),C(0,·))))dx
τ
+ ((ρ∂ B+ρu·∇ B)·(B−u)−P(ρ,θ)div B+S(θ,∇ u):∇ B)dxdt
t x x x x
Z Z
0 Ω
τ
+ (−H·∂ D−((∇×H)×H)·B−(D×B)·(∇×D)+(H×u)·(∇×D))dxdt
t
Z Z
0 Ω
τ q(θ,∇ θ)
− (ρs(ρ,θ)∂ C+ρs(ρ,θ)u·∇ C+ x ·∇ C)dxdt
t x x
Z Z θ
0 Ω
τ
− (ρ∂ (∂ (H )(A,C)))+ρu·∇ (∂ H (A,C))dxdt
t ρ C x ρ C
Z Z
0 Ω
τ
+ ∂ (A∂ (H )(A,C)−H (A,C))dxdt. (47)
t ρ C C
Z Z
0 Ω
Replacing ∂ D by (38) in (47) to find
t
1 1
( ρ|u−B|2+ |H−D|2+H (ρ,θ)−∂ (H )(A,C)(ρ−A)−H (A,C))(τ,·)dx
C ρ C C
Z 2 2
Ω
τ τ C q(θ,∇ θ)·∇ θ
+ |∇u|2+ν|∇×D−∇×H|2 dxdt+ (S(θ,∇ u):∇ u− x x )dxdt
x x
Z Z Z Z θ θ
0 Ω(cid:0) (cid:1) 0 Ω
1 1 1
≤Z (2ρ0|u0−B(0,·)|2+ 2|H0−D0|2+ 2ρ0|B0|2+(HC(0,·)(ρ(0,·),θ(0,·))
Ω
−∂ρ(HC(0,·))(A(0,·),C(0,·))(ρ0−A(0,·))−HC(0,·)(A(0,·),C(0,·))))dx
τ
+ ((ρ∂ B+ρu·∇ B)·(B−u))−P(ρ,θ)div B+S(θ,∇ u):∇ B)dxdt
t x x x x
Z Z
0 Ω
τ
+ (−((∇×H)×H)·B−(D×B)·(∇×D)+(H×u)·(∇×D))dxdt
Z Z
0 Ω
τ τ q(θ,∇ θ)
− (∇×(B×D))·Hdxdt− (ρs(ρ,θ)∂ C+ρs(ρ,θ)u·∇ C+ x ·∇ C)dxdt
t x x
Z Z Z Z θ
0 Ω 0 Ω
τ
− (ρ∂ (∂ (H )(A,C)))+ρu·∇ (∂ H (A,C))dxdt
t ρ C x ρ C
Z Z
0 Ω
τ
+ ∂ (A∂ (H )(A,C)−H (A,C))dxdt. (48)
t ρ C C
Z Z
0 Ω
Note that
((∇×H)×H)·Bdx=− H⊤BH+ 1∇(|H|2)·B dx, (49)
Z Z (cid:18) 2 (cid:19)
Ω Ω
(∇×(B×H))·Bdx= H⊤BH+ 1∇(|H|2)·B dx.
Z Z (cid:18) 2 (cid:19)
Ω Ω
10 WeipingYan
So direct calculation shows that
τ
(−((∇×H)×H)·B−(D×B)·(∇×D)+(H×u)·(∇×D))dxdt
Z Z
0 Ω
τ
− ∇×(B×D)·Hdxdt
Z Z
0 Ω
τ
= (−((∇×H)×H)·B−∇×(B×D)·H)dxdt
Z Z
0 Ω
τ
+ (−(D×B)·(∇×D)+(H×u)·(∇×D))dxdt
Z Z
0 Ω
τ 1
= ((H−D)⊤∇B(H−D)+ ∇(|H−D|2)·B)dxdt
Z Z 2
0 Ω
τ 1
⊤
+ (D−H) ∇(u−B)D+ ∇(D(D−H))(u−B)dxdt
Z Z 2
0 Ω
τ 1
− (D⊤∇(u−B)D+ ∇(|D|2)(u−B))dxdt. (50)
Z Z 2
0 Ω
Note that
∂ (∂ H (A,C)) = −s(A,C)∂ C−A∂ s(A,C)∂ C+∂2 H (A,C)∂ ρ
y ρ C y ρ y ρ,ρ C y
+∂2 H (A,C)∂ C, for y =t,x.
ρ,θ C y
Thus it follows from (48) and (50) that
1 1
( ρ|u−B|2+ |H−D|2+H (ρ,θ)−∂ (H )(A,C)(ρ−A)−H (A,C))(τ,·)dx
C ρ C C
Z 2 2
Ω
τ τ C q(θ,∇ θ)·∇ θ
+ |∇u|2+ν|∇×D−∇×H|2 dxdt+ (S(θ,∇ u):∇ u− x x )dxdt
x x
Z Z Z Z θ θ
0 Ω(cid:0) (cid:1) 0 Ω
1 1
≤Z (2ρ0|u0−B(0,·)|2+ 2|H0−D0|2+(HC(0,·)(ρ(0,·),θ(0,·))
Ω
−∂ρ(HC(0,·))(A(0,·),C(0,·))(ρ0−A(0,·))−HC(0,·)(A(0,·),C(0,·))))dx
τ
+ ((ρ∂ B+ρu·∇ B)·(B−u)−P(ρ,θ)div B+S(θ,∇ u):∇ B)dxdt
t x x x x
Z Z
0 Ω
τ 1
+ ((H−D)⊤∇B(H−D)+ ∇(|H−D|2)·B)dxdt
Z Z 2
0 Ω
τ 1
⊤
+ (D−H) ∇(u−B)D+ ∇(D(D−H))(u−B)dxdt
Z Z 2
0 Ω
τ 1
− (D⊤∇(u−B)D+ ∇(|D|2)(u−B))dxdt
Z Z 2
0 Ω
τ q(θ,∇ θ)
x
− (ρ(s(ρ,θ)−s(A,C))∂ C+ρ(s(ρ,θ)−s(A,C))u·∇ C+ ·∇ C)dxdt
t x x
Z Z θ
0 Ω
τ
+ ρ(A∂ s(A,C)∂ C+r∂ s(A,C)u·∇ C)dxdt
ρ t ρ x
Z Z
0 Ω