Cauchy problem for the euler equations of a nonhomogeneous ideal incompressible fluid II

J. Korean Math. Soc. 32 (1995) , No.1 , pp. 41-50

CAUCHY PROBLEM FOR THE EULER EQUATIONS OF A NONHOMOGENEOUS

IDEAL INCOMPRESSIBLE FLUID

S Hl G E HARU ITOH

1. Introduction

Let us consider the system of equations

꾀

1

’

4 /I l‘

、

{ p +U W=o

p[Vt + (v . V)v] + '\1 p = pi ,

div v = 0 ,

in ^QT = R ³ X [0 , T] , subject to the initial conditions

(1. 2)

Here lex , t) , po(x)

vo(x) are given , while the density p(x ,t), the velocity vector v(x ,t) = (v

，t) ， ν2 (X ， t) , v ^3(x , t)) and the pressure p(x , t) are unknowns. The system (1. 1) describes the motion of a non- homogeneous ideal incompressible fluid.

The aim of the present paper is to establish the unique solvability ,

local in time , of the problem (1. 1) and (1. 2). This will be carried out by applying the method of the Galerkin approximations. Furthermore ,

we

that the'assumptions to po (x) are weakened compared with those in the previous paper [2] , in which we proved the similar result by showing the existence of a :fi xed point of some map.

Our theorem is the following.

Received ^Se ptember 8 , 1993.

THEOREM 1.1. Assume that

(1. 3) inf poex)

m>O

suPPo(x)

M<

(1. 4) Vpo(x) E H

(R ³⁾ ,

(1.5) vo(x) E

^Vo = °

and

(1. 6) f(x , t) E L∞(0， T; H 3 (R 3)) _aηd div f E L∞(0， T; L

(R 3)).

Then there exists T* E (0 , T] such that the problem (1.1) and (1.2) has a unique solution(p , v ,p) which satisfies

(1. 7) and (1. 8)

m S; p( x , t) S; M

(Vp , ν ， Vp) EL∞(O，T*;H

(R

xL∞(0， T*; H 3 (R 3)) _{xL∞ (0，} T*; H 3 (R 3)).

In section 2 , we establish

a priori estimate of solutions , and then theorem will be proved in section 3.

2. A priori estimate

Let (p , v , p) be a sufficiently regular solution.

c is the generic constant related to the imbedding theorems

Cl is the one dependent only on m , M and c.

LEMMA 2.1. For p(x , t) , the estimates (2.1)

and

(2.2)

m S; p(x , t ) s; M

옳 ^{/I V p(t)lb s; c IIv(t) 1I3 I1V p(t) 1I}

Cauchy problem for the Euler equations 43

hold , where II- Il ^k = ^1I - II Hk(R3) .

Proof. By means of the classical method of characteristics , we have the representation

(2.3) p(x , t) = ^poe ν(T， x , t)lr:=o)' where y( T , x , t) is the solution of the Cauchy problem

(2 .4)

꺼

$

/이

=

=피ν-T

T

d-d

씨

/--,‘--

It results from this that the estimate (2.1) holds.

Let a be a multi-index. We apply the operator

D

= (옳)

(옳)

(앓)

。n

each side of the first equation of (1. 1).

we multiply the result by

D

p , integrate over R ³ and sum over 1 ::; lal ::; 3 , then we have the

equality

(2.5)

i짧찮 IIπ\lp 얘 쩨p미메빠II

+ 잖a (@) 43 ^{Dβv .} \l(DC>-βp)(DC> ^p)dx]

The first term of the right hand side is zero , as is seen by

integration by parts using div v = O. The second term can be estimated by using the inequalities

(2.6) _IIg hllo _::;

_{IIg112 II}

_。 and

(2.7) _IIg hllo _{::; C IIgll1 II} hll ₁ ,

and then we get

(2.8) 깊 。앓a ^{( ; )} 113 Dβν \7(D ^a -βP)(DQp)dxl

::; C

lI\7 vll

11 \7 pll; ::; cIIvll

11 \7 pll; .

Hence we find that

(2.9) 찮 ^{11 \7 p( t) II; ::; c} II ^V (삐

口

LEMMA 2.2. If we put (2.10)

then the estimates

η(x ， t) = p(x , t)-l ’

(2.11) and (2.12)

M-

<

t) ::; m-

옳 11\7η(t싸 ^::; cllv(t) 1I311\7η(t)lb

(2.13) hold.

Proof. We can easily see that

t) satisfies the equation { ηt +

· \7η =0 ,

ηIt=o = PO(X)-l 三 ηO (x) .

Therefore the estimates directly follow from Lemma2. 1.

LEMMA 2.3. For p(x , t) , the estimate

(2.14) ^lI\7 p(t ) 1I 3 흐

⁽¹ + 11 \7 p(t) 1I 2 + 11 \7까t) 1I2 t

(IIν(t)lIi + IIf(t) 1I

+ II 퍼v ^{f(t)II £l} (R3»)

holds.

Cauchy problem for the Euler equations 45

Proo f. Applying the divergence operator on both sides of the second equation of (1. 1) , we get

(2.15) 퍼v (η\7p) = - ε 파i 악‘ + ^div f _프 E

표 we multiply this equation by p and integrate over R

, then we obtain (2.16)

M- ¹ II\7pll~ ~ ¹ !FIIL6/s(R3) IIp Il L 6( R 3) ~

¹ !FIIL6/s(R3) II \7pllo

~ cllFII짧R3) |lF|la/3 |l%|lo S c(; ||F||L1(R3) + i ||Fl|o) llWl|o Hence we get

(2.17) _lI\7 pllo

c1(IIFII £l(R3) + ¹ \Fllo)·

In order to accomplish our purpose , we use the following inequality (cf.

[3]):

(2.18) II ull2 ~ 많(lI.6.ull o ^{+ IIμ110) for any μ E H 2(R3 ) .}

Noting that (2.15) can be written in the form .6. p = pF -

. \7 p , we get that for a with lal = 2 ,

(2.19)

IID apll2 ~ 많(II ^Da 빠 p\7η ^{\7 p)110 +} II D apllo)

s 끓(IIDa(pF)llo ^{+ IIDa(p \77J .} \7p빠

By the direct calculation , we obtain (2.20)

and (2.21)

II D a(pF) lI o ~ c(M + 11 \7 plb) IIFI12

II

\7η . \7 p)llo _~ c(M + 11 \7 p 1l2) 11 \7η11211 \7p1l2 ,

(2.22)

(2.27)

and then we get

II D aplb ::; cl[(1 + II V' p(t) 1I2 + II V'η(t) lb ) 1 /F1I2 + (1 + lIV' p(t )1I2 +

Now , from the interpolation inequality and Young ’s inequality , we have (2.23) cl(1 + II V' p(t) 1I2 +

^lIV' p ll2

s _￡ l|?pll3 + Cl(1 + |lW(t)|l2 + |l?η(t)lb)3 11쩨10

Therefore we find that

(2.24) lIV' p Il 3 ::; Cl (1 + lIV' p(t )lb +

^F

/Flb)·

On the other hand , it is easy to verify that

(2.25) 1 /F 1I2 + 1 /FII £l ( R 3) ::;

^{+ II div} fll 2 ^{+ II}

Il L l ( R 3») .

Consequently , the desired estimate is established.

LEMMA 2 .4 . For vex , t) , the estimate

(2.26) 옳 IIv이얘 뼈 빼 빼 (“떠 써 삐t) 셰씨)1게I!J굉5잔 달 쇄 쐐 C따떠 데 떼 빠 1나네페 빠 [I데빠II싸삐 Iv벼U이뼈 뼈 빼 (μ써 베 씨t) 씨씨)1게넓II넓|얹g ⁺ (n1써+1 애IIV'야η미(t야씨삐 빠 꽤)1 애싸 빠IIμ 빠12ω2μ) I川IIπ?야 얘 행p 야마 뼈 뼈 (t에 베t)세)1빠 |

holds.

Proof. We rewrite the second equation of (1. 1) in the form

+ (v·

'V )v +

= f. Applying the operator Da on each side of this equa- tion , multiplying the result by Da v , integrating over R

summing over 0 ::; lal ::; 3 , then we have the equality

짧 ^{IIvll; = -} 효 [ l3 ^{(v . V' D a v) .} D짧

lal=O

+ _{ε (α} ) ^{λ (D l3 v· V' D a- l3 v) . Davdx}

0<β::; a 、I / ..,.1.‘

+ L3 ηD ^{a 야} D ^{a η&}

+ ε (~) L~ DβηD

βV'p. ^Davdx]

0<β~a ，，~/ v.H.

+4D@f Daνdx

Cauchy problem for the Euler equations Similarly to the proof of Lemma 2.1 , we obtain

47

(2.28) 43 ^(U ?Dau) Daud$ = o ,

|끓。앓a (@) |43 ^{(Dβv . '\7D 망 D Q -뀌βI3 v 띠)깨DQv샘 짧 U뼈냈 팩 dx썩xl판 I ~얀 의 혜c이셰싸 IIπ|π아 패 폐'\7v 에 에U이에II않 |뎌센 원 원 3월 쌀 웰 즈F 돼 헤c야셰II넓 넓 Iv써떼 U비셰쇄II펴|녀3}

(α원 2.30이) l잃 끓 l않 L3 η써D ^Q '\7p . DQvd 짧 U뼈짧 팩dx 씌z커I ~챔 센 m빠-1깨 1

(2.31 ) 싫。앓Q

IL3 _DβηD

^{참 DQVdX/}

~ Cll\세1211 '\7p Il211 '\7ν112 ~ cll '\7η112 ^lI'\7 plI3 II

and

(2 32) !뚫 IL3 ^D

^{1· DQVdxl} _~ ^{111113 IIvll}

Hence we find that the estimate (2.26) holds.

LEMMA 2.5. There exist T* E (0 , T] such that

(2.33) sup (11 '\7 p(t) ^1I 2 + 11 '\7η(t) 1I 2 + IIv(t)113 + ^lI'\7 p(t ) ^1I 3 ) :::; c.

O<t<T*

口

Here T* and C depend onlyon m , ^M , 1I 111 L∞ (O ，T;H3(R3» ’

II div l l1 L∞ (O，T; £l (R3» ， 11 '\7 Po 112' _{IIν0 113} and the constants ofimbeddings.

Proo f. If we set

(2.34) yet) = 1 + 11 '\7 p(t) 1I 2 + 11 '\7η(t)112 + IIν(t) 1I 3

and

(2.35) K = ¹ +

+ II

’

then , from the above lemmas , we have a differential inequality

(2.36) dY(t)

- - - E C1KY(t)6.

dt Therefore we conclude that (2.37)

Yet) ::; Y(O)(l - 5c1K Y (Ot t )- 1/5 provided t < (5c 1K Y (O)5)- 1 ,

and thus

(2.38) yet) ::; 2Y(O) for

31

t < T* = --- - -= - - 160c 1K Y (O)5

口

3. Proof of theorem 1. 1

Since the proof of uniqueness is standard , we will just show the existence of a solution. Let H;(R

be the closure of J(R

=

e

{C 8"(R ³ ) }3 ; div μ = O} in H ^3(R3 ) . Since H;(~) is separable and J(R

is dense in H;(R

, there exists {¢>i(x)} C J , which is total in H;(R ^{3 ) .} We may assume that ¢>i(x) are orthogonal in H ^{3(R3 ) .}

We apply the Galerkin method with this {¢>l ex )}. Namely , we look

N

for pN(x ,t) , _{νN(X， t)} = 원 af(t)¢>i(x) andpN(x ,t) satisfying (3.1) __ __ __

pi

+ v

V'

= 0 ,

«굉

+ (v N. V' )v N +

1 ^V' pN , ¢>i )) =

^{j =} 1 ,··· , N, y

dive 숨 V'pN ) =- ε1 (νN)~j (νN)t+ 퍼v f ,

pNlt=o = po(x) ,

VNlt=o = PNVO(X) ,

Cauchy problem for the Euler equations 49

where ((e , e» stands for the scalar product in H ^{3(R3 )} and PNis the orthogonal projection in H3 on the space spanned by

, --- ,

The second equations of (3.1) form a system of ordinary differential equations for af ^(t) and the fifth one gives the i피ti려 conditions.

표

we

j = 1, · · · , N, then we obtain the relation

(3.2) ((vf + ^{(v N . V)v N} + ηN VpN， v N » ^{= ((f, v N} » ^,

where ηN = (pN)-l. This coincides with (2.27) replacing ν， p a끄d η by v N , pN 없ld ηN respectively. Therefore , from the results in section 2 , we have

(3.3) ^m ~ ^pN(x , t ) _~ M

and

<- C

、---/

?i

”” ”””

””

/1

,,

、

P N

”” ?

+ ”””

?u

싸

/,

•

‘

、

ν

N

”” ”””

,‘ +

배

nl N

”” ?

+ ”””

’ι

써끼

/l

,

‘

P· N

”” ?

”””

/l

,

‘

*

m )T

‘

-

5K -

찌

”

/t

,

‘

Moreover , since 4>

orthogonal in H

, we deduce from the second equations of (3.1) that

(3.5) _쇄 = PN (f - (v N . V)ηN ηNVpN ).

Hence we get (3.6)

IIνf ^{(t )112} ~ ^{cl [11 f (t ) 1I 2} + ^{IIv N} _(t)ll~ + ⁽¹ + _{11 \7ηN} (t)112) IIVpN (t) ^1I2 J, and with (3 .4) it is easily found that

(3.7) oF밸 IIνf(t)112 ~ c.

<t<T*

These estimates

the unique

of the problem (3.1) on the interval [0 , T*] , and furthermore permit to pass to the limit in the

terms using a standard compactness theorem (cf. [1] , [4] ,

[5]). Hence we

verify the existence of a unique solution of the prob-

lem (1. 1) and (1. 2) on the interval [0 , T*] as well as the applicability

of the inequalities (2.1) and (2.33) to it. This completes the proof.

References

1. S. N. Antontsev , A. V. Kazhikhov and V. N. Monakhov , Boundary value prob- lems in mechanics of

fluids , North-Holland , Amsterdam-New York-Oxford-Tokyo , 1990.

2. S. !t oh ,

problem for the Euler equations of a

ideal incompressible fluid , J. Korean Math. So c. 31 (1994) , 367-373.

3. O. A. Ladyzhenskaya , The mathematical

of viscous

flow ,

Gordon

Breach , 1969.

4. J. L. Lions , Quelques methodes de resolution des problemes

limites non lindires , Dunod-Gauthier-Villars , Paris , 1969.

5. R. Temam , Navier-Stokes equations , North-Holland , Amsterdam-NewYork-Oxf 000 , 1984 .

Department of Mathematics Hirosaki University

Hirosaki 036 , ^Japan

e-mail: sitoh@@fed.hirosaki-u.ac.jp