三维有界域上不可压缩无磁耗散MHD方程弱解的能量守恒 Energy Conservation for the Weak Solutions to the Three-Dimensional Incompressible Magnetohydrodynamic Equations of Viscous Non-Resistive Fluids in a Bounded Domain

doi:10.12677/PM.2021.115089

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PureMathematicsnØêÆ,2021,11(5),739-751

PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.115089

n‘k.•þØŒØ Ã^ÑÑMHD•§f

)UþÅð

<<<

uHnóŒÆ,êÆÆ,2À2²

ÂvFÏµ2021c47F¶¹^FÏµ2021c511F¶uÙFÏµ2021c518F

Á‡

©ïÄn‘k.•þØŒØ Ã^ÑÑMHD•§f)UþÅð¯K.ké•§?1N

1,2ä¼ê,,'uδ,ε,τ4•,lUþª.•UþÅð,éf)

(u,b,P) \^‡:u∈L

,b∈L

…∇b∈L

,P∈L

'…c

ØŒØ MHD•§§Ã^ÑÑ§k.«•§f)§UþÅð

EnergyConservationfortheWeak

SolutionstotheThree-Dimensional

IncompressibleMagnetohydrodynamic

EquationsofViscousNon-Resistive

FluidsinaBoundedDomain

XiongWang

SchoolofMathematics,SouthChinaUniversityofTechnology,GuangzhouGuangdong

©ÙÚ^:<.n‘k.•þØŒØ Ã^ÑÑMHD•§f)UþÅð[J].nØêÆ,2021,11(5):739-751.

DOI:10.12677/pm.2021.115089

<

Received:Apr.7

,2021;accepted:May11

,2021;published:May18

,2021

Abstract

Inthispaper,wemainlystudytheenergyconservationfortheweaksolutionsto

thethree-dimensionalincompressiblemagnetohydrodynamic equationsofviscousnon-

resistiveﬂuidsinaboundeddomain.Togetenergyconservation,weﬁrstusetheglobal

molliﬁcationmethodtotheequation,nexttakecut-oﬀfunction,thengetthelimitof

δ,ε,τ.Weproposeaconditionfor (u,b,P):u ∈L

,b ∈L

,∇b ∈L

andP∈

Keywords

IncompressibleMagnetohydrodynamicEquations,ViscousNon-ResistiveFluids,

BoundedDomain,WeakSolutions,EnergyConservation

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.ïÄµ9Ì‡(Ø

n‘ØŒØ Ã^ÑÑMHD•§











divu= 0,

+div(u⊗u)+∇P= (∇×b)×b+µ∆u,

−∇×(u×b) = 0,

divb= 0,

(1.1)

Ù ¥,u =(u

)(x,t) L«6N„Ý,b =(b

)(x,t) L«^|,P=P(x,t) L«Øå.

©•Ä´þãn‘ØŒØ Ã^ÑÑMHD•§|(1.1)3k .m«•Ω⊂R

äkeã>Š^

‡

∂Ω

= 0,

(1.2)

DOI:10.12677/pm.2021.115089740nØêÆ

<

ÚÐŠ^‡

u(x,0) = u

(x),b(x,0) = b

(x),x∈Ω.(1.3)

f)UþÅð¯K.

½Â1.1.éu‰½T>0,eÐŠ÷vu

∈L

(Ω),XJk±eA:¤á

•¯K(1.1)-(1.3) 3˜mD

(Ω×[0,T))¥¤á…÷v











u∈L

∞

(0,T;L

(Ω)),

u∈L

(0,T;H

(Ω)),

b∈L

∞

(0,T;L

(Ω)),

P∈L

∞

(0,T;L

loc

(Ω));

(1.4)

•é?¿ϕ∈C

∞



Ω×[0,T)



Ω



u·ϕ

+u⊗u: ∇ϕ+Pdivϕ−b⊗b: ∇ϕ+

|b|

divϕ−µ∇u: ∇ϕ



dxdt

Ω

(x)ϕ(x,0)dx= 0;

•eãUþØªéA¤kt∈[0,T] ¤á

Ω

(

|u|

|b|

)dx+

Ω

µ|∇u|

dxds

≤

Ω

(

)dx;

(1.5)

K¡(u,b,P)´¯K(1.1)-(1.3) 3Ω×[0,T]þ˜‡f).

¯¢þ,XJ)¿©1w,'Xr)½ö²;),¬keãUþª¤á

Ω

(

|u|

|b|

)dx+

Ω

µ|∇u|

dxds

Ω

(

)dx.

(1.6)

3¯K(1.1)¥,b=0 ,KT¯K=z¤ØŒØ Navier-Stokes •§3k.•þUþ

Åð¯K,T¯K®3©Ù[1]N¹¥‰Ñ‰Y.É©Ù[1]éu,·‚ïÄØŒØ Ã^Ñ

ÑMHD•§3k.•þUþÅð¯K.

©Ì‡(JXe:

DOI:10.12677/pm.2021.115089741nØêÆ

<

½n1.1.-Ω ´k.m8,>.∂Ω÷vC

1w5^‡.b







u(x,0) = u

(x),divu

= 03˜mD

(Ω)¥¤á;

b(x,0) = b

(x),divb

= 03˜mD

(Ω)¥¤á.

(1.7)

P(u,b,P)´•§|÷v½Â1.1. f).XJ

P∈L



0,T;L

(Ω)



,(1.8)

u∈L

(0,T;L

(Ω)),p≥4,q≥6,(1.9)

…

b∈L



0,T;L

(Ω)



,∇b∈L



0,T;L

(Ω)



,(1.10)

Ké?¿t∈[0,T] , Uþª(1.6) ¤á.

2.PÒ`²9~^Ún

Ún2.1.(Hardy.i\Øª[2])XJf∈W

1,p

(Ω),p∈[1,∞),…•3˜‡~êC= C(p,Ω)

†pÚΩk',@o



f(x)

dist(x,∂Ω)



(Ω)

≤Ckfk

1,p

(Ω)

e¡Ú\†1k'PÒÚ(Ø,ë•Chen-Liang-Wang-Xu©Ù[1]ÚEvansÖ[3].

é?¿(x,t) ∈Ω

×(ε,T−ε) ,Ù¥Ω

⊂⊂Ω ,½ÂòÈ

(x,t) =

Ω

u(y,s)η

(x−y,t−s)dyds,η

(x,t) =

η(

(2.1)

ùpη(x,t) ´|83ü ¥þIO1Ø.

·K2.1.3˜mL

loc



0,T;W

1,p

(Ω

)



¥,ε→0žk

(x,t) →u(x,t),∀p,q∈[1,+∞).

(2.2)

du∂Ω∈C

,é½:x

∈∂Ω,•3¢êr

>09¼êh:R

→RPV

=Ω∩

B(x

)={x∈B(x

):x

>h(x

)}éu˜‡éêε,0<ε<

.½Â²£:

:= x−εn(x

),∀x∈V

, Ù¥n(x

)´∂Ω3:x

?ü {•þ, KB(x

,ε) ⊂B(x

)∩Ω.

½Â²£¼ê

(x,t) = u(x

,t),∀x∈V

DOI:10.12677/pm.2021.115089742nØêÆ

<

é?¿(x,t) ∈V

×(ε,T−ε) …V

:= B(x

+2ε)∩Ω,k

(x,t) =

u(y,s)η

(x−y,t−s)dyds

−ε~n(x

)

u(y,s)η

−y,t−s)dyds.

(2.3)

·K2.2.3˜mL

loc



0,T;W

1,p

)



¥,ε→0žk

(x,t) →u(x,t),∀p,q∈[1,+∞).

du∂Ω ;5,∂Ω mCXUék•fCX,=Uék•‡:x

∈∂Ω ,Œ»r

>0 9

éA8ÜV

= Ω∩B(x

),i∈{1,2,...,k},¦∂Ω ⊂

i=1

…

∈C

∞



).•3V

⊂⊂Ω ,

¦Ω ⊂

i=0

-{ψ

}

i=0

´láum8x{V

,B(x

),...,B(x

)}ü ©),=











0 ≤ψ

≤1,i∈{0,1,2,...,k}

∈C

∞





,suppψ

⊂V

∈C

∞



B(x

)



,suppψ

⊂B(x

),i∈{1,2,...,k},

i=0

= 1.

(2.4)

½Â

[u]

(x,t) := ψ

(x)u

(x,t)+

i=1

(x)

(x,t),∀x∈Ω.

(2.5)

K[u]

(x,t) ∈C

∞



0,T;C

∞

(Ω)



·K2.3.3˜mL

loc



0,T;W

1,p

(Ω)



¥,ε→0ž

[u]

→u∀p,q∈[1,+∞).

(2.6)

?˜Ú,ε→0žk

3˜mL

loc



0,T;W

1,p

)



¥,[u]

−u

→0;

3˜mL

loc



0,T;W

1,p

)



¥,[u]

−

→0.

(2.7)

Ún2.2.(©z[4]¥Ún2.3)XJf∈W

1,r

(Ω×[0,T]), g∈L

(Ω×[0,T]), Ù¥1 ≤r,r

≤

∞,

,Kéu,‡†ε,f,gÃ'~êC>0 ,keã(Ø¤á

k∂(fg)

−∂(fg

loc

(Ω×(0,T))

≤Ckgk

(Ω×[0,T])



k∂

(Ω×[0,T])

+k∇fk

(Ω×[0,T])



(2.8)

DOI:10.12677/pm.2021.115089743nØêÆ

<

d?∂= ∂

½∂= ∂

½Â´d(2.1)‰Ñ.•?˜Ú,ε→0ž,k

3˜m L

loc



Ω×(0,T)



¥,∂(fg)

−∂(fg

) →0.

d?,r

<∞ž,r= r¶r

= ∞ž,r<r.

Ún2.3.(©Ù[1]¥íØ2.1 ) XJf∈W

1,r

(Ω×[0,T]), g∈L

(Ω×[0,T])Ù¥1 ≤r,r

≤

∞,

k∂

((

f˜g)

−f˜g

loc

(0,T;L

))

≤Ckgk

(Ω×[0,T])



k∂

(Ω×[0,T])

+k∇fk

(Ω×[0,T])



…

k∂

((

f˜g)

−f˜g

loc

(0,T;L

))

≤Ckgk

(Ω×[0,T])



k∂

(Ω×[0,T])

+k∇fk

(Ω×[0,T])



d?˜g

½Â´d(2.3)‰Ñ.•?˜Ú,ε→0ž,k

3˜mL

loc

(0,T;L

))¥,∂((

f˜g)

−f˜g

) →0,

d?,r

<∞ž,r= r¶r

= ∞ž,r<r.

Ún2.4.(Aubin-LionsÚn,©Ù[4])XJX´g‡Banach˜m,Y´Banach˜m,

X→Y,Y

Œ©…3X

¥È—.b¼êS{f

}÷v







∈L

∞

(0,T;X),∂

∈L

(0,T;Y),1 <p≤∞,

∞

(0,T;X)

,k∂

(0,T;Y)

≤C,∀n≥1.

([0,T],X

)¥ƒé;.

3.½n1.1y²

•y²½n1.1.UþÅð(Ø,e¡·‚æ^©Ù[1]¥•{ò•§(1.1)

1,

∂



i=1





div(u⊗u)

i=1

div(

u⊗





∇P

i=1

∇



−



div(b⊗b)

i=1

div(

b⊗





∇(

|b|

)

i=1

∇(

)



−µ



∆u

i=1

∆



= 0.

(3.1)

DOI:10.12677/pm.2021.115089744nØêÆ

<

ä¼êξ

[u]

,Ù¥ξ

(t) ∈C



(τ,T−τ)



,η

(x) ∈C

(Ω)…











0 ≤η

(x) ≤1,η

(x) = 1x∈Ω…dist(x,∂Ω) ≥δž;

→1δ→0ž,…|∇η

|≤

dist(x,∂Ω)

ò•§(3.1)ü>¦þξ

[u]

¿3Ω×(0,T) þÈ©,

Ω

[u]

·∂



i=1



dxdt

Ω

[u]



div(u⊗u)

i=1

div(

u⊗



dxdt

Ω

[u]



∇P

i=1

∇



dxdt(3.2)

−

Ω

[u]



div(b⊗b)

i=1

div(

b⊗



dxdt

Ω

[u]



∇(

|b|

)

i=1

∇(

)



dxdt

−µ

Ω

[u]



∆u

i=1

∆



dxdt= 0.

e5é(3.2)¥ˆ‘'uδÚε4•.Ù¥1‘,1n‘,18‘©Oë•©Ù[1]¥

ÚnA.1,ÚnA.2,Ún3.3ÚÙ!3.2,

lim

δ→0

lim

ε→0

Ω

[u]



div(u⊗u)

i=1

div(

u⊗



dxdt= 0;

(3.3)

lim

δ→0

lim

ε→0

Ω

[u]



∇P

i=1

∇



dxdt= 0;

(3.4)

lim

δ→0

lim

ε→0

Ω

[u]



∆u

i=1

∆



dxdt= −

Ω

|∇u|

dxdt.

(3.5)

(3.2)¥1˜‘,1o‘,1Ê‘'uδÚε4•(JdeˆÚn‰Ñ.

Ún3.1.(3.2)¥1˜‘÷v

lim

δ→0

lim

ε→0

Ω

[u]

·∂



i=1



dxdt= −

Ω

|u|

dxdt.

DOI:10.12677/pm.2021.115089745nØêÆ

<

y²d½Âª(2.5)•

Ω

[u]

·∂



i=1



dxdt

Ω

[u]

·∂

[u]

dxdt

Ω

∂

|[u]

dxdt

=−

Ω

|[u]

dxdt.

éþª‘,¿-ε→0,δ→0,Šâξ

,η

k.59·K2.3

lim

δ→0

lim

ε→0

Ω



|[u]

−|u|



dxdt= 0.

l,

lim

δ→0

lim

ε→0

Ω

[u]

·∂



i=1



dxdt= −

Ω

|u|

dxdt.

Ún3.1.y..

Ún3.2.(3.2)¥1o‘÷v

lim

δ→0

lim

ε→0

Ω

[u]



div(b⊗b)

i=1

div(

b⊗



dxdt= −

Ω

b·div(b⊗u)dxdt.

y²k©ÜÈ©,k

Ω

[u]



div(b⊗b)

i=1

div(

b⊗



dxdt

=−

Ω

∇η

⊗[u]



(b⊗b)

i=1

(

b⊗



dxdt

−

Ω

∇[u]



(b⊗b)

i=1

(

b⊗



dxdt

−

Ω

[u]



∇ψ

: (b⊗b)

i=1

∇ψ

: (

b⊗



dxdt

:=I

DOI:10.12677/pm.2021.115089746nØêÆ

<

kwI

=−

Ω

∇η

⊗[u]





(b⊗b)

−(b⊗b)



i=1



(

b⊗

−(b⊗b)





dxdt

−

Ω

∇η

⊗([u]

−u) : (b⊗b)dxdt

−

Ω

∇η

⊗u: (b⊗b)dxdt.

^H¨olderØª9(1.4),(1.9),(1.10),(2.4),Ún2.1 9·K2.1,·K2.2,·K2.3,

lim

δ→0

lim

ε→0

= 0.

(3.6)

2wI

=−

Ω

∇[u]





(b⊗b)

−(b⊗b)



i=1



(

b⊗

−(b⊗b)





dxdt

−

Ω

(∇[u]

−∇u) : (b⊗b)dxdt−

Ω

∇u: (b⊗b)dxdt.

dH¨olderØª9(1.4),(1.10),(2.4),·K2.1,·K2.2 Ú·K2.3,

lim

δ→0

lim

ε→0

= −

Ω

∇u: (b⊗b)dxdt= −

Ω

b·div(b⊗u)dxdt.

(3.7)

•wI

=−

Ω

[u]



∇ψ



(b⊗b)

−b⊗b



i=1

∇ψ



(

b⊗

−b⊗b





dxdt

−

Ω

[u]

i=0

∇ψ

: (b⊗b)dxdt

=−

Ω

[u]



∇ψ



(b⊗b)

−b⊗b



i=1

∇ψ



(

b⊗

−b⊗b





dxdt

−

Ω

[u]

∇



i=0



: (b⊗b)dxdt

dH¨olderØª9(1.4),(1.10),(2.4),·K2.1,·K2.2 Ú·K2.3,

lim

δ→0

lim

ε→0

= 0.

(3.8)

DOI:10.12677/pm.2021.115089747nØêÆ

<

Šâ(3.6),(3.7),(3.8)k

lim

δ→0

lim

ε→0

) = −

Ω

b·div(b⊗u)dxdt.

Ún3.2.y..

Ún3.3.(3.2)¥1Ê‘÷v

lim

δ→0

lim

ε→0

Ω

[u]



∇(

|b|

)

i=1

∇(

)



dxdt= 0.

y²k©ÜÈ©,k

Ω

[u]



∇(

|b|

)

i=1

∇(

)



dxdt

=−

Ω

∇η

·[u]



(

|b|

)

i=1

(

)



dxdt

−

Ω

div[u]



(

|b|

)

i=1

(

)



dxdt

−

Ω

[u]



∇ψ

∇(

|b|

)

i=1

∇ψ

∇(

)



dxdt

:=I

kwI

−

Ω

∇η

·[u]



(

|b|

)

i=1

(

)



dxdt

=−

Ω

∇η

·[u]





(

|b|

)

−

|b|



i=1



(

)

−

|b|





dxdt

−

Ω

∇η

·([u]

−u)



|b|

i=1

|b|



dxdt

−

Ω

∇η

·u

|b|

dxdt.

^H¨olderØª9(1.4),(1.9),(1.10),(2.4),Ún2.1 9·K2.1,·K2.2,·K2.3,

lim

δ→0

lim

ε→0

= 0.

(3.9)

DOI:10.12677/pm.2021.115089748nØêÆ

<

2wI

−

Ω

div[u]



(

|b|

)

i=1

(

)



dxdt

=−

Ω

div[u]





(

|b|

)

−

|b|



i=1



(

)

−

|b|





dxdt

−

Ω

(div[u]

−divu)(

|b|

)dxdt−

Ω

divu(

|b|

)dxdt.

dH¨olderØª9(1.4),(1.10),(2.4),·K2.1,·K2.2 Ú·K2.3,

lim

δ→0

lim

ε→0

= −

Ω

divu(

|b|

)dxdt= 0.

(3.10)

•wI

−

Ω

[u]



∇ψ

∇(

|b|

)

i=1

∇ψ

∇(

)



dxdt

=−

Ω

[u]



∇ψ



∇(

|b|

)

−

|b|



i=1

∇ψ



∇(

)

−

|b|





dxdt

−

Ω

[u]

i=0

∇ψ

|b|

dxdt

=−

Ω

[u]



∇ψ



∇(

|b|

)

−

|b|



i=1

∇ψ



∇(

)

−

|b|





dxdt

−

Ω

[u]

·∇(

i=0

)

|b|

dxdt.

dH¨olderØª9(1.4),(1.10),(2.4),·K2.1,·K2.2 Ú·K2.3,

lim

δ→0

lim

ε→0

= 0.

(3.11)

Šâ(3.9),(3.10),(3.11)k

lim

δ→0

lim

ε→0

) = 0.

Ún3.3.y..

d(3.3),(3.4),(3.5),Ún3.1,Ún3.29Ún3.3•,-•§(3.2)¥ε→0,δ→0 

−

Ω

|u|

dxdt+

Ω

b·div(b⊗u)dxdt+µ

Ω

|∇u|

dxdt= 0.

(3.12)

DOI:10.12677/pm.2021.115089749nØêÆ

<

(Ü•§(1.1)

,Ù¥

Ω

b·div(b⊗u)dxdt

Ω

b·div(b⊗u)dxdt−

Ω

b·div(u⊗b)dxdt

Ω

b·



div(b⊗u)−div(u⊗b)



dxdt

Ω

b·∂

bdxdt

=−

Ω

|b|

dxdt,

l,(3.12)z{¤

−

Ω

(|u|

+|b|

)dxdt+µ

Ω

|∇u|

dxdt= 0.

(3.13)

duξ

(t) ∈C

((τ,T−τ)),éu?¿t

>0 ,?¿τ,α,¦τ+α<t

,

(t) =











0,0 ≤t≤τ,

t−τ

,τ<t≤τ+α,

1,τ+α<t≤t

+α−t

<t≤t

+α,

0,t≥t

+α.

(3.14)

ò(3.14)“\(3.13),

−

2α

τ+α

Ω

(|b|

+|u|

)dxdt+

2α

+α

Ω

(|b|

+|u|

)dxdt

+µ

τ+α

t−τ

Ω

|∇u|

dxdt+µ

τ+α

Ω

|∇u|

dxdt

+µ

+α

−t+t

+α

Ω

|∇u|

dxdt= 0.(3.15)

dÈ©ýéëY5,

lim

α→0

τ+α

t−τ

Ω

|∇u|

dxdt= 0;

lim

α→0

+α

−t+t

+α

Ω

|∇u|

dxdt= 0.

DOI:10.12677/pm.2021.115089750nØêÆ

<

E(t) =

Ω

(|b|

+|u|

)dx,

F(t) = µ

Ω

|∇u|

dxds.

äó:

u,b∈C



[0,T];L

(Ω)



.(3.16)

¯¢þ,d(1.1)

,(1.4),(1.8),(1.9),(1.10),•u

∈L



0,T;H

−1

(Ω)



.3Ún2.4.¥,X=

(Ω),Y=H

−1

(Ω),lu ∈C



[0,T];L

(Ω)



.Ón,d(1.1)

,(1.4),(1.9),(1.10) ÚÚn2.4.,

b∈C



[0,T];L

(Ω)



.(ÜUþØª(1.6)ÚÐŠ^‡,u,b∈C



[0,T];L

(Ω)



(E+F)(t

) = (E+F)(τ).

(3.17)

3(3.17)¥,du(3.16),-τ→0

,

(E+F)(t

) = (E+F)(0),t

∈(0,T),

d=(1.6).dt

?¿5,®¤½n1.1.y².

ë•©z

[1]Chen,R.M.,Liang,Z.L.,Wang,D.H.andXu,R.Z.(2020)EnergyEqualityinCompressibleFluidswith

PhysicalBoundaries.SIAMJournalonMathematicalAnalysis,52,1363-1385.

https://doi.org/10.1137/19M1287213

[2]Kufner,A.,John,O.andFuk,S.(1977)FunctionSpaces.Academia,Prague.

[3]Evans,L.C.(1998)PartialDiﬀerentialEquations.AmericanMathematicalSociety,Providence,RI.

[4]Lions, P.L.(1996)Mathematical Topicsin FluidMechanics.Vol.1,Incompressible Models.OxfordUniver-

sityPress,NewYork.

DOI:10.12677/pm.2021.115089751nØêÆ