Integral estimates of magnetohydrodynamics equations

INTEGRAL ESTIMATES

OF

MAGNETOHYDRODYNAMICS EQUATIONS

SANGJEONG KIM

ABSTRACT. Inthis paper, weshow that the weak solutions of the time-dependent Magnetohydrodynamics equations in 3 dimensional periodic domain belong to £0."-.,(0,T;v,.) following the method of Foias-Guillope-Temam for Navier-Stokes equations.

1. Introduction

In this paper, we consider the following magnetohydrodynamics (MHD) equations in the non-dimensional form,

(1.1)

(1.2) (1.3)

{) 1 1 2

[)tU+(U.V)u- ReLlu+SV(2B )-S(B·V)B=f,

{) 1

[)tB

+

(u· V)B - (B· V)u

+

Rmcurl (curIB)

=

0, divu

= divB =

0.

Here we denote u (Ul(X,t),U2(X,t),U3(X,t)) as the velocity of the particle of fluid, B = (B1(x, t), B2(x, t), B3(x, t)) as the magnetic field, f

=

f(x, t) as the volume density force at (x, t). The constantsRe, Rm and M are the Reynolds number, the magnetic Reynolds number and

M2

the Hartan number respectively. DefineS= ReRm

The equations are important ones in the field of plasma physics. The existence of weak and strong solutions and some regularities are estab- lished by M. Sermange and

R.

Temam [3J. More precisely, they proved the weak solution of MHD equations belongs to L⁴(0, T; V) where V is

Received June 7, 1997. Revised February 24, 1998.

1991 Mathematics Subject Classification: 76W05.

Key words and phrases: Magnetohydrodynamics equations, integral estimate.

870 Sangjeong Kirn .

divergence-free subspece ofHI. Now we are interested in this weak so- / lution. We prove that the weak solution belongs to

L2r:'1

(0,

T;

~(Q)) in 3 dimensional periodic domain forT 2: 1. We will follow the method of Foias-Guillope-Temam for Navier-Stokes equations [1].

The organization of this paper is the following: In section 2, we in- troduce function spaces that are used in this paper. In section 3, we recall known results about existence, uniqueness and regularity of weak solutions. In section 4, we prove the main theorem, namely we show that if

f

^E

LOO

(0,

T;

~-1), then the weak solution belongs to

L2r:'1

(0,T;~) for T 2: 1. As a corollary, we prove that the weak solution belongs to

L

¹

(0, T; Loo(Q))

if

f E LOO (0, T; H)

following the idea of L. Tartar

[1].

2. Functionspaces

We supplement the system (1.1) '" (1.3) with following initial and boundary conditions

(2.1) u(x,0) = Uo(x),B(x,O) = Bo(x) for allx E ]R3, (2.2) u(x

+

Lei'

t) =

u(x,t),B(x

+

Lei'

t) =

B(x,t),

for allx E]R3 andt

>

O. HereL is the period and{~H=1an orthonormal basis of the space. But we will regardL to be 21r for notational simplicity.

Let T

>

0 and let X be a Banach space. We shall consider

LP(O, T; X),

1 :S p:S 00,which is the space of functions from [O,T] into X, which are

LP

for the Lebesgue measure

dt.

This is a Banach space for the norm

, I

T -

(ll1u(t)"~dt)

^{p for 1 :Sp}

^<

^{00, esssup}09~T

^lIu(t)lIx

^{for P = 00.}

We denote

L

²

(Q)

as the space of ]R-valued functions on Qwhich are square integrable for the Lebesgue measuredx

=

dXl dX2 dX3. This is a Hilbert space for the scalar product

(u, v)

= l

u(x) . v(x) dx.

We use the same notation also for

VO(

Q),the space of]R3-valued functions which are square integrable on Q.

Using Fourier series expressions,

\tQ(Q)

is identified with the space of functions u satisfying

(2.3)

u

= L

^{uiCt)eij.x, Uj(t) E (:3, U_j}

⁼

^{Uj, for t E [0,}

^T].

jEZ3

For mEN,we also introduce

Vm(Q)

=

{u E

\tQ(Q) I (271")3 L ^{Ijl2m l}

^uj

^{l2 <}

^{00, Uo}

^{= O}}

jEZ3

with inner product

(u, V)Hm =

1 ^L

^{Dku· Dkv dx =}

(271")3 L ^Ijl2m

^{Uj . V_j'}

QIkl=m jEZ3

We will use (',·)Hm also in scalar case if it does not make confusion. Let V-m(Q) be the dual space ofVm(Q). Especially,

V = {u E Vi.(Q)1i .Uj = 0 for allj E

Z3},

H =

{u

E

\tQ(Q)lj .

Uj = 0 for all

j

E

Z3}

andV' is the dual space ofV. We equip V with the scalar product

3 OU

8v

((u, v)) =

L(ox 'ox)

=

L:(271")3IjI2

^{Uj •Vj}

k=l k k. jEZ3

which is also scalar product on Vi.(Q).

V and lH[ are defined as followings:

V = {Cu, B)lu, BE V}, lH[= {Cu, B)lu, BE H}.

We equiplH[ with the following scalar products

(cp,

'1t)

= (u, v) +

(B, C) for allcP

= (u,

B), '1t

= (v,

C) ^E lH[

with associated norm,

Icpl = {(cp, cp)}L

We also equip V with the scalar products

((cp,

'1t))

=

((u, v))

+

((B, C)) with associated norm,

I!cpll = {((cp, cp))}L

We define an operator

A

E

.C(V; V')

so that (Au,v) = ((u,v)) for allu,vE V.

Then we recall that

A

is unbounded operator on H, whose domain is V(A)

=

{u E V,Au

EH} = H

²

n

V.

872 Sangjeong Rim

Since we consider divergence free functions on periodic domain,

A

is actually -.6..

Now we define a trilinear form on £l(Q) x W1,1(Q) X£l(Q) by setting

3

1 0

b(u,v,w)

= L

^{Ui DiVj Wj}

^dx

^{(where D i}

^{= ox),}

i,j=l Q 2 .

whenever the integrals make sense. We know the trilinear form b is continuous on

(H

1(Q))3

[4].

Thus it is natural to define a continuous bilinear operator

B

from

V

x

V

into

V'

so that

(B(u,v),w)

=

b(u,v,w).

3. Known results

Let ^T

>

0 be given and let us assume that

(p,

u,^{B) is} a smooth solution of (1.1) ^rv (2.2).

We multiply (1.1) by a test function v E V and integrate over Q. Then

o

¹

.(3.1)

at

^{(u, v)}

+

Re ((u, v))

+

b(u, u, v) - Sb(B, B, v)

=

(I,v).

We multiply (1.2) by a test function C EV and integrate overQ. Then

o

¹

(3.2)

at

(B, C)

+

Rm ((B, C))

+

b(u, B, C) - b(B, u, C)

=

O.

This suggests the following weak formulation of the problem (1.1) ^{r v} (2.2).

DEFINITION 3.1 (Weak solution). Let

N =

2 or 3,

f

E £2(0,T; V') and q>o

= (uo,

^B

o)

given in lHI:. Then q>

= (u,

^B) is called weak solution of MHD equations if it satisfies

<I> E £2(0, T; V)

n

£00(0, T;lHI), <I>(O)

=

q>o, (3.1) and (3.2) for all 'It

= (v,

C) E V.

DEFINITION 3.2 (Strong solution). Let

N =

2 or 3,

f

E £2(0,

T; H)

and <I>o

= (uo,

^B

o)

given in V. Then<I>

= (u,

^B) is called strong solution of MHD equations if it satisfies .

u,BE£2(0,T; V(A))n£00(0,T; V), (3.1) and (3.2) for all 'It= (v,C) EV.

Byusing operators

A

and

E,

equations (3.1), (3.2) are written as (3.3)

($.4)

QU 1

at

+ Re Au + E(u, u) - SE(B, B) = f,

~R 1

'-;' +

_R~AB

+

B(u, B) - BtB , u)

=

O.

For these equations, M. Sermange and R. Temam established follow- ing results about existence and uniqueness [3].

THEOREM 3.3. For

f,

Uo,Bo given with

(3.5) f

E

L

²

(0, T; V'),

epo

= (Uo, B

_o)^E E,

there exists a weak solution ep

=

(u,B) of MHD equations satisfying (3.6)

Furthermore,

1. if N = 2,ep is unique and satisfies

(3.7) ep' E

L

²

(0, T;

V), ep EC([O,

T];

lHI),

2. if N=3, there is at most one weak solution of MHD equations satisfying

(3.8)

THEOREM 3.4. Let

f, uo,

B

o

^{begiven with}

f

E LOO

(0, T; H),

epo

= (uo, Bo)

E

V,

1. if N

=2,

the strong solution ep

= (u,

B) of MHD equations satisfies (3.9) ep E L²(0, T; D(A»

n

LOO (0, T; V),

2. if N=3, there exists T*

> °

^(dependingon

^{n, f,}

^{"eplUand on [0,T*],}

there exists a unique strong solution ep of MHD equations which satisfies (3.9) with T replaced by T*.

4. Integral estimate

In this section we will prove the following theorem:

874 Sangjeong Kim

THEOREM 4.1. (N=3, space periodic) We assume that

<Po

EH,

f

E

LOO

(0,T;

Vm-d

and that

<P

is a weak solution of the MHD equations.

Then

<P

satisfies

(4.1) iT [<P(t)IU~ldt ~

^Cr

^<

⁰⁰

forr

=

1,··· ,

m +

1, where the constantCr

=

CrUe'R~'Q,

<Po, I).

To prove the theorem, we need some lemmas. From now on, we set

N=3.

LEMMA 4.2. Letu,v E

Hr+l

andwE

HT.

1~k ~r, and1 ~l,m::;

r - 1. Then the following inequalities hold.

(4.2)

I~

u DT+1v DTw

dxl ~ cllul~~lulitHlvlw+llwIHr,

(4.3) l~

Dku DT-k+1v DTw

dxl

l-!£+~ Lir k-l ^I-I£=!

~ c21ulH1r

lul Hr•1

I

^v

lil

IvlHr.r

Iwlw,

I{

^{2 ( T)2}

I ^I

^,2-!

^I

^I!

^{I I}

^£

^{I 1}

^{2- £}

(4.4) lQ

^{u Dv}

dx

~ ^C3U H1r U Hr.1 v Hl V_Hr~l' (4.5)

I 1 (

^Qu D vDu D^{T I T-I}v

dx I

^{~ C4}

^I

^{U H1}

^,2-

^414;1

^{I I}

^UW-l

~;1 I

V HI V Hr...l ,

_14I4~51 1 2- ~;:5 (4.6) l~

Dmu DT-mv Diu DT-1v

dxl

2 21-!?-1 2l-2m-1 2l-2m.l 2 2l_2m...l

~

cslul;

lul.;~-lvIH12r Ivl;,....~, where^Cl>C2, C3, C4,Cs areconstants independent of u, v,w.

Proof. By Gagliardo-Nirenberg inequality,

Il

u DT+1v DTw

dxl~

lulv",lvlw-1IwIHr

~ clul~~lul~r.llvIHr.llwlw.

By Sobolev imbedding, HI <-+ L6(Q), Hi(Q) <-+ L3(Q),

Il

Dku DT-k+lV DTW

dxl ^<

IDkUlpIDT-k+lvIL6IDTWI£2

<

_{cluI Hk}

-!

[V[T-k+2[wlw.

Thus by interpolation inequalities, we infer that

1

1

Dku DT-k+lV DTw

dxl<

_

c

1

lull-~+ir lul~-ir Ivlk;:-llvll-~·;llwl

HI Hr-l HI Hr-I Hr.

Q

Inequalities (4.4) r v (4.6) come from interpolation inequality and the iollowing inequalities:

Il

^u2

^(D

^Tv)2

^dxl::;; clulloolvl~T,

Il

u DTv Diu DT-IV

dxl::;;

cluILoolvlnrIDluIL4IDT-lvIL4,

\~

Dmu DT-mv Diu DT-IV

dxl::;;

cIDmuIL4\DT-mvIL4\DluIL4\DT-lvIL4.

o

LEMMA 4.3. If

<P

is a smooth solution of MHD equations, then for eacht

>

0 and r ~ 1

(4.7)

~1<P(t)I~T ⁺ ~1<P(t)I~T_I ^::;;

^LT(l

⁺ 1<P(t)l~d<P(t)li;:l),

where R = minUe'_R~)' LT

=

LT(Re,Rm, Q,NT-1(f)) and NT-1(f) = IfILOO(o,T;VT_I)' Moreover, wehave .

(4.8)

~1<P(t)I~T ⁺ ~1<P(t)l1-r-1 ^::;; L~(I ⁺ 1<p(t)I~I)2T+l

where_L~ = _L~(Re,Rm, Q,NT-1(f)).

Proof. We take the scalar product of H in (3.3) and (3.4) with ATU

and ATB respectively. Then we obtain that (4.9)

!dd _lul~r +

RI

lul1-r-1 =

(f,u)nr - (B(u,u),ATU )

+

S(B(B,B),AT^{U ),}

2 t e

(4.10)

!dd

IBI1-r

+

RI IBI1-r-1 = (B(B,u),ATB) - (B(u,B),ATB).

2

t

m .

The first term in the right-hand side of (4.9) is majorized by

I

2 { }2

(4.11) If(t)lnr-Ilu(t)lw-1 ::;; 4R1u(t)lw-I

+

R NT-1(f) .

876 Sangjeong Kim

Then by (4.2) and (4.3), other terms in right-hand side of (4.9) and (4.10) are governed by

1<p1~~~I<pI~~I<plw.

Thus we obtain that

1d

I 12

1

2

2"dt UW + Re!u!w-I

(4.12)

< RI "4 UW-I

1

2 + R l{N r-1! ()}2 +

^C4

'14>1 ¹ HI - fr l4>1 W-I ¹

⁺

^fr l4>/ W

< ~ IUI~r-1 ⁺ ~ ^{N

^{r -1}

^{(J)}2 +} ~ 14>I~r-l ⁺ c~I4>I~l/4>lii~1 ^,

1d

I 12

^{1 2 R / 2}

'I 2 _2;~1

(4.13)

2" dt B W + Rm /Blw-l

~

"4 4>/W-I +

~

4>IHd4>/w .

By adding (4.12) and (4.13),

~ ~14>I~r ⁺ lRI4>I~r-l ~ c714>I~d4>lii~1 ^{+ {N r_1(J)}2.}

Thus we obtain (4.7). By interpolation inequality,

4r 4r-2 4(r-l)

1

+

14>(t)l~d4>(t)I}{;1 ~ 1

+

cl4>(t)l;lll4>(t)/;~;

~ 1

+

EI4>(t)l~r-1

+ C€I4>(t)Ii;;2

for all^E

>

0. This yields to (4.8).

n

An immediate consequence of (4.8) isthat

4>

remains inV_r as ,long as

114>11

remains bounded.

LEMMA 4.4. If

4>0

E Vr;

!

E £00(0, T;

v;.-1)

and r

2:

1, then the strong solution

4>

ofMHD equations given byTheorem, 3.4 belongs. to C([O, T.];

v;.).

If

4>0

E V,

!

E £00(0,T;

v;.-1)

and r

2:

1, then

4>

E

C((O, T.]; v;.).

Proof. We first consider the case

4>0

E

v;.,

and we first show that

4>

belongs to £0°(0,

T.; v;.).

For that it suffices to prove that the Galerkin approximation^Um of^U remains bounded in £00(0,

T.; v;.)

as m ~ ^00.

(4.14)

~ 1 .

d;(t) + ReAum(t) + PmB(um(t),um(t)) - SPmB(Bm(t) , Bm(t)) == Pm!,

(4.15)

dBm

1 ( . (

dt(t) + Rm ABm(t) + PmB(Bm t), um(t)) - PmB Um(t) , Bm(t)) =

0,

(4.16)

We take the scalar product in

H

of (4.14) with

Arum = (-It

~rum,and of (4.15) with

ArBm =

(-1

t

~r

Bm.

Since

Pm

is self-adjoint in H and

PmArum

⁼

Arum,

^{we get}

1 d ² 1 ²

2dtlBmlw + RmIBmIW-1

(-lrb(Bm,Um,

^~r

Bm) - (-lrb(um,

^Bm~r

Bm).

This is similar to (4.9) and (4.10). Thus, exactly as in Lemma 4.3, we get the analogue of (4.8),

~1<Pml1-r ⁺ ~1<Pml1-r+1 ^::; £~(1 ^{+ l<p} ml1-1)2r+l.

Since

<Pm E .l?O(O,

T.; V), for

°::;

^{t ::;T.}

l<pm(t)l1-r ::;

c~T.

+ l<p m(O)I1-r,

~ IT*I<Pm(t)l1-r_ldt ::;" c~T* ^{+ l<p m(O)I1-r.}

IP _m <polw ::; I<polw

implies that

<Pm

remains bounded in

£00(0,

T.;

v;.)

and

£2( 0, T.; v;.+l).

Since

luIHl,IBIHI

are uniformly bounded for

°::; ^{t ::;}

^T.,

^B(u,u),

B(u, B), B(B, u)

and

B(B, B)

are in

£2(0,

T.;

v;.-l)

by Lemma 4.2. Fur- thermore,

f

E

£2(0,

T.;

l!,.-l)

and

Au,

AB E

£2(0,

T.;

v;.-l).

Thus it follows that

<p'

^E

£2(0,

T.;

v;.-l).

Therefore

<P

E C([O, T.]; Vr ) [5] (Chap Ill. §1.4).

For

<Po

^E V, we observe that the strong solution of MHD equations belongs to

£2(0, T.; D(A).

Thus

<p(t)

E

D(A) = "V:!

almost every- where on (0,T.), and we can find t₁ that is arbitrarily small so that

<p(td

E

"V:!.

The first part o(the proof shows that

<P

E

C([t},

^T.];

"V:!) n

£2(t

1,T.;V3). Hence

<P(t2)

E

V3

for some

t2

^E

[t},

^T.] arbitrarily close to tt, and

<P E C([t2,

^T.];

V3) n £2(t2,

T.;

\14).

By induction we arrive at

<P E C([tr-

^1,T.];

v;.) n _{£2(tr_}

1,T.;

v;.+l).

Since

tr-

¹is arbitrarily close to

0, the lemma is proved. 0

878 Sangjeong Kim

Let m

2:

1. We say that a solution~of MHD equations is Vm-regular on

(tl, t2)

(0 ::;

tl < t2)

if ~ E

C((tl, t2); Vm(Q)).

We say that

Vm-

regularity interval

(t

l ,

t2)

is maximal if there does not exist an interval of Vm-regularity greater than

(tl, t2)'

The local existence of an interval of Vm-regular solution is given by above lemma: if ~ E

Vm

^and

f

^E

Loo(O, T; Vm- l ),

then there exists an

Vm-

regular solution of the MHD equations defined on some interval (0,

to).

Also, it follows easily by above lemma that if

(tl, t2)

is a maximal interval of Vm-regularity of a solution ~, then

(4.17) limsup

lu(t)lv

_m

=

00.

t--+t2-0

Now we prove the main theorem.

Proof of the theorem Let

(ai,f3i)

be maximal interval of Vm -

regularity of

q,

^{for i} EN. On each interval

(ai, f3i),

the inequality (4.7) is satisfied for r

=

1, .. , , m. We write them in the slightly weaker form (4.18)

~1q,lk ⁺ ~1q,(t)ILl

^::;Lr

^{(1 +} 1~(t)I~J(l ⁺ 1~(t)!~J~.

Then we deduce

d(1

+

1~12

)

dt V^r

<

L (1

+

1~(tW

).

(1

+

1~(t)I~J2;:1

-

^r V1 By integration int from

ai

to

f3i,

we get

2r -1 2r-1

- - - = 1 -

+

1

(1

+

1~(f3i - O)I~J2r-l (1

+

I~(ai

+

O)I~J2r-l

i

^{p· 1~(t)12}

^1{J;

+ R ' V;I2r dt::; L

r (1

+ 1~(t)I~)dt.

0"; (1

+ 14>(t)lv,.)2r-l

^0";

From (4.17), the first term in the left-hand side of the inequality vanishes along a sequence converging to

f3i,

since

(ai, f3i)

is a maximal interval of Vm-regularity. Thus

i P _. .

^1~(t)12

_{Vr;l 2r dt::;}

_---!:.

LiP.

_•(1

₊ 1<p(t)I~I)dt.

0"; (1

+

1~(t)lvJ2r-l

R

^0";

Bysummationofthese relations fori E N, wefind,since~E

L2(O,T;V),

i

^{T 1~(t)l~r_l}

(4.19)

2r dt::;

er for r

=

1, . " , m.

o (1

+

1<p(t)I~)2r-l

The proof of (4.1) is now made by induction. The result is true for

r

= 1. We assume that it is true for 1""

,r,

and prove it for r+ 1(r:::; m). We have

[T 2

lif>f+\12r=I

^A+

Jo "

'"'i:li,_;'''

=

iT [ lcI>(t)I~'_l ^{] 2,:'1} [(1 ^{+ 1cI>(tW )irS] 'Jhi dt}

o (1

+

1cI>(t)I~J2;~1 ^v"

< [iT 1cI>(t) ILl 2, dt] 2':'1 [I ^{T (1 + 1cI>(tW} )~dt] 2;~1.

o (1

+ 1cI>(t)I~J2r=I

^{0 V,}

Therefore (4.1) holds for r

+

1 due to (4.19) and the induction assumption.

Now we obtain the following corollary using Tartar's idea [1].

COROLLARY 4.5. We assume N

=

3, periodic. We assume cI>0 E H and

f E DXl(O,

T; H). Then any weak solution

cI>

ofMHD equations belongs to

£1(0,

T;

DXl(Q)).

Proof By Gagliardo-Nirenberg inequality

Thus we obtain that

By Holder's inequality it follows that

!.T 1<I?(tlIL-(Q) " c(!.T IA<I?(t111dt) 1 (!.T 1I<1?(t)II'dt) I

Thus the right-hand side is finite thanks to (4.1).

o

ACKNOWLEDGMENTS. The author thanks Professor Dongho Chae for suggesting the problem and for many helpful discussions.

880 Sangjeong Kim

References

[1J C. Foias and C. Guillope and R. Temam, New a priori estimates for Navier- Stokes equations in dimension3, Comm.Partial Differential Equa.tions6(1981) no. 3, 329-359.

[2J J. Lions andE. Magenes,Non-Homogeneous Boundary Value Problems and Ap- plications, Springer-Verlag, 1972.

[3J M. Sermange andR.Temam,Some Mathematical Questions Related to the MHD Equations, Comm. Pure Appl. Math.36 (1983), 635-664.

[4] R.Temam,Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS- NSF Regional Conference series in Applied Mathematics, SIAM, 1983.

[5] ,Navier-Stokes Equations: Theory and Numerical Analysis, Elsevier Sci- ence Publishers RV., 1985.

[6J , Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, voL 68, Springer-Verlag, 1988.

Department of Mathematics Seoul National University Seoul 151-742, Korea

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