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

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J. Korean Math. Soc. 31 (1994), No. 3, pp. 367-373

CAUCHY PROBLEM FOR THE EULER EQUATIONS OF A NONHOMOGENEOUS

IDEAL INCOMPRESSIBLE FLUID

SHIGEHARU ITOH

1. Introduction

Let us consider the Cauchy problem pt + v· Vp = 0

p[Vt + (v· V)v] + Vp = pf

div v = ⁰ ^(1.1)

plt=o = po(x) vlt=o = vo(x)

in QT =]R3 X [0, T], where f(x,t),po(x) and vo(x) are given, while the density p(x,t), the velocity vector v(x,t) = (v

¹

(X,t),v

²

(X,t),v

³

(x,t)) and the pressure p(x,t) are unknowns. The equations (1.1)1 - (1.1)a describe the motion of a nonhomogeneous ideal incompressible fluid.

The aim of the present paper is to establish the unique solvability, local in time, of the problem (1.1). For the problem in a bounded domain with the boundary condition v . n = 0, where n is the unit out normal to the boundary, that fact has been proved under appropriate assumptions by many authors, for instance, [1], [2], [3], [7], [8]. On the other hand, as far as we know, there is no investigation in an unbounded domain.

Our theorem is the following.

THEOREM. Assume that

po(x) - P E H ³ (]R3) for some positive constant p, (1.2) inf po(x) == m> 0 and suppo(x) == M <

^00,

(1.3) vo(x) E H 3(]R3) and div Vo = 0 (1.4)

Received April 9, 1993.

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and

f(x, t) E L l (0, T : H 3 _(R 3)) and div f E Ll(QT). (1.5) Then there exists T* E (0, T] such that the problem (1.1) has a unique solution (p, v,p) which satisfies

. , "

(p - p,v, "Vp) ELOO(O,T : H*

³

(R

³))

(1.6) x LOO(O,T: H*

³

(R

³)) X

Ll(O,T: H*

³

(R

³)).

Moreover, if f(x, t) E CO (0, T : H3(R3)) and div f E CO(o, T L

^l

(R

³^)),

then the solution (p,v,p) is classical.

In section 2, we solve three linear problems in sequence and get the estimates. In section 3, we first show the existence of a fixed point of some map, and then it is proved that this becomes the solution of (1.1).

2. Preliminaries

LEMMA

2.1. For a given v

E

Loo(O,T : H

³

(R

³))

with div v = 0, the problem

'{ Pt + v . "V p = °

, plt=o = ^po(x)

has a solution satisfying

m$. p $. M.

Moreover, if we put p = P - p, then

(2.1)

(2.3)

where po = po - p,

^Cl

is a positive constant depending only on imbed- ding theorems and 11· Ilk =

1I·IIHk(1l3).

Proof. It is well-known that, using the classical method of charac- teristics, the solution of (2.1) is given by

p(x,t) = po (y(r,X,t)IT=O)' (2.4)

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Cauchy problem for the Euler equations

where y( r, x, t) is the solution of the Cauchy problem

{* _ylr=t ⁼ ^v(y,r) ₌ _x.

369 (2.5)

(2.6) Thus the first assertion has shown. Next we establish the estimate for

p. It follows from (1.1)1 and (1.1)4 that p satisfies the equation

{ Pt + v . \7 P = ⁰

plt=o = po( x).

Applying the operator DQ (= (8/

^8X1

)Ql (8/

^8X2

)Q2 (8/

^8X3

)Qs) to (2.6h, multiplying the result by D

^Q

p, integrating over]R3 and adding in a with

10'1 (=

0'1

+

^0'2

+

^0'3)

:s: 3, then we have

(2.7)

Hence, by Gronwall's inequality, it is easy to see that (2.3) holds.

Q.E.D.

LEMMA

2.2. Let p and v be in Lemma 2.1. Then the problem div (p-1\7p) = - 2: V~j V~i + div f == F (2.8)

i,j

has a solution satisfying

lI\7p(t)lh (2.9)

9

:S:C2

(2: ^IIp(t)1I0 (IIv(t)lI~ + Ilf(t)1I3 +

¹¹

div f(t)II£1(]iS»),

j=o

where

^C2

is a positive constant depending only on rn, M, imbedding theorems and interpolation inequalities.

Proof. Thanks to (2.2), the solvability easily follows from the general theory of the elliptic partial differential equations of second order (cf.

[5]), and so we can restrict ourselves to get the estimate (2.9).

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Multiplying (2.8) by P and integrating over lRa , then we obtain M-II1VpII~ ~IIFIIL6/5(Ji'.3) IIpIIL6(Ji'.3) (2.10)

~c3lIFll L6/5(Ji'.3) llVpllo

I 2/3 1/3 I

~C4 IF

11£1

(Ji'.3) IIFllo lI\7pl

0

~C4 [(2/3)IIFIILl(l!3) + (1/3)IIFllo] II Vpllo.

Hence we get

IIVpllo _~

C5

(II F lILl(Ji'.3) + IIFllo). (2.11) In order to accomplish our purpose, we use the following inequality (cf. [7]):

lIull2 _~ (3/2)1/2 (IILlullo + ^IIullo) for any u E H

²

(R

³).

(2.12) Noting that (2.8) can be written in the form Llp = pF + p-

¹

V p. Vp, we get that for a with lal = 2

IIDQpll2 _~ (3/2)1/2 (II DQ (pF + p-

¹

Vp. Vp)lIo + IIDQpllo). (2.13) By the direct calculation, we obtain

the right hand of (2.13)

:'S C6 [(M + 11'7plhlllFlh + et. ^{m -;} ^11'7 ^pllO ^II'7PII + ^(2.14)

Now, from the interpolation inequality and Young's inequality, we have

AIIV'plb ~ AIIVpll;/311Vpll~/3 (2.15)

=AIIVpllo + AII\72pll;/311\7pll~/3

~ell\72pll2 + (A + e- 2 A 3 )IIVpllo.

Therefore, we find that

9

IIVpll3 ~

^C7

(2: ^11\7 plI~) (IIFII£1(Ji'.3) + IIFlb)· (2.16)

j=O

On the other hand, it is easy to verify that

11F1I£1(Ji'.3) + 11F1I2 ~

^Cs

⁽¹¹ ^div 1I1Ll(Ji'.3) + 11 div 1112 + lI\7vIlD· (2.17) Consequently, the desired estimate is established. Q.E.D.

Moreover, similar to the above lenmlas, we have

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Cauchy problem for the Euler equations

371 LEMMA

2.3. Let p and v be in Lemma 2.1 and p in Lemma 2.2.

Then the problem

{

Wt

+ (v. V')w = ^_p-1V'p + f

wlt=o = vo(x) has a solution satisfying

(2.18)

3. Proof of Theorem

Let P be the orthogona.l projector in {£2(JR3)} 3 onto H = the closure of J = {u E {Cgo(JR 3) P : ^div u ⁼ o} in {£2 (lR3)}3. Then it is immediately follows from lemmas in section 2 that if 11;50113 ::; A and SUP0StS;T IlV(t)1l3 = Il v(t)lb,oo,T ::; K, then

IIPw(t)lh (3.1)

[ KT

sllw(t)1l3 S Il ^v ol13 + IlfIILJ(o,T:H3(J!3» + ^h(Ae ^C1 ) (J{

²

T + IIfIILl(O,T:H3(li.3» + 11 div fIILJ(QT»] e C1KT ,

where her) =

C2

(L::~=o ^r ^{i )} (L::~=o

^m-(j+l)r

^{i ).}

Therefore if we choose

4K 2:llvoll3 + 11/11£1 (0,T:H3(li.3

»

(3.2)

+ ^h(2A) (1Ifll£l(0,T:H3(J!3» +

¹¹

div fll£l(QT»)

and define S = {v E £00(0, T; H 3(JR3» : Pv = v, Ilv(t)1I3,00,T ::; K},

then, from (3.1), the map F : v -. Pw satisfies F(S) C S for sufficiently

small T, for example T ::; T = min{[4Kh(2A)]-I,* (c

I

I()-110g2}. Let

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{Vj} C S. Then we can verify that {Fvj} is the Cauchy sequence in

£<Xl(O, T; H3(R3 » with few difficulties. Hence, as ^F becomes a compact map, there exists a fixed point v = Pw from Schauder's theorem. We have thus solved the problem

Pt+Pw,Vp=O

Wt + (Pw. V)w = ^_p-l\lp + f

div [p-l\lp + (Pw. \l)Pw - f] = 0 (3.3) plt=o = po(x)

Wlt=o = vo(x) This is equivalent to (1.1) if Pw = w.

Let us show that Qw = (I - P)w = O. Apply the projection Q to (3.3)2, then, due to (3.3)a, we have

(Qw)t + Q[(Pw. V)Qw] = O.

Multiplying (3.4) by Qw and integrating over R3, we get

(3.4)

d d r ^IQwl

²

^dx ⁺ r Q[(Pw. V)Qw] . Qw dx = O. (3.5)

t 1m3 1m3

Furthermore, from the definition of P, we obtain

r Q[(Pw. V)Qw] . Qw dx (3.6)

1m.

³

= r (Pw. V)Qw· Qw dx - r P[(Pw. V)Qw] . Qw dx

1m3 1m3

=.!. r Pw. VIQwl2dx = ^O.

21m.

³

Hence we find that Qw = 0, since Qwlt=o = ^Qvo = ^O.

Uniqueness is proved in [4].

This completes the proof.

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Cauchy problem for the Euler equations References

373

1.

H. Beiriio da Veiga and A. Valli, On the Euler equations for nonhomogeneous fluids (I), Rend. Sem. Mat. Univ. Padova

63

(1980),151-167.

2. H. Beiriio da Veiga and A. Valli, On the Euler equations for nonhomogeneous fluids (II), J. Math. Anal. Appl.

73

(1980),338-350.

3. H. Beiriio da Veiga and A. Valli, Existence of coo solutions of the Euler equa- tions for nonhomogeneous fluids, Comm. in PDE. 5 (1980), 95-107.

4.

D. Gram, Il teorema di unicitd per

i

fluidi incompressibili, perfetti, eterogenei, Rev. Un. Mat. Argentina 17 (1955),73-77.

5. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, (Second edition), Springer-Verlag (1983).

6. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach (1969).

7.

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

J. Korean Math. Soc. 31 (1994), No. 3, pp. 367-373

CAUCHY PROBLEM FOR THE EULER EQUATIONS OF A NONHOMOGENEOUS

IDEAL INCOMPRESSIBLE FLUID

SHIGEHARU ITOH

1. Introduction

Let us consider the Cauchy problem pt + v· Vp = 0

p[Vt + (v· V)v] + Vp = pf

div v = 0 (1.1)

plt=o = po(x) vlt=o = vo(x)

in QT =]R3 X [0, T], where f(x,t),po(x) and vo(x) are given, while the density p(x,t), the velocity vector v(x,t) = (v

(X,t),v

(X,t),v

(x,t)) and the pressure p(x,t) are unknowns. The equations (1.1)1 - (1.1)a describe the motion of a nonhomogeneous ideal incompressible fluid.

Our theorem is the following.

THEOREM. Assume that

po(x) - P E H 3 (]R3) for some positive constant p, (1.2) inf po(x) == m> 0 and suppo(x) == M <

(1.3) vo(x) E H 3(]R3) and div Vo = 0 (1.4)

Received April 9, 1993.

and

f(x, t) E L l (0, T : H 3 (R 3)) and div f E Ll(QT). (1.5) Then there exists T* E (0, T] such that the problem (1.1) has a unique solution (p, v,p) which satisfies

(p - p,v, "Vp) ELOO(O,T* : H

(R

(1.6) x LOO(O,T*: H

(R

Ll(O,T*: H

(R

Moreover, if f(x, t) E CO (0, T : H3(R3)) and div f E CO(o, T L

(R

then the solution (p,v,p) is classical.

In section 2, we solve three linear problems in sequence and get the estimates. In section 3, we first show the existence of a fixed point of some map, and then it is proved that this becomes the solution of (1.1).

2. Preliminaries

2.1. For a given v

Loo(O,T : H

(R

with div v = 0, the problem

'{ Pt + v . "V p = °

, plt=o = po(x)

has a solution satisfying

m$. p $. M.

Moreover, if we put p = P - p, then

(2.1)

(2.3)

where po = po - p,

is a positive constant depending only on imbed- ding theorems and 11· Ilk =

Proof. It is well-known that, using the classical method of charac- teristics, the solution of (2.1) is given by

p(x,t) = po (y(r,X,t)IT=O)' (2.4)

Cauchy problem for the Euler equations

where y( r, x, t) is the solution of the Cauchy problem

{* ylr=t = v(y,r) = x.

369

(2.5)

(2.6) Thus the first assertion has shown. Next we establish the estimate for

p. It follows from (1.1)1 and (1.1)4 that p satisfies the equation

{ Pt + v . \7 P = 0

plt=o = po( x).

Applying the operator DQ (= (8/

)Ql (8/

)Q2 (8/

)Qs) to (2.6h, multiplying the result by D

p, integrating over]R3 and adding in a with

10'1 (=

+

+

:s: 3, then we have

(2.7)

Hence, by Gronwall's inequality, it is easy to see that (2.3) holds.

Q.E.D.

2.2. Let p and v be in Lemma 2.1. Then the problem div (p-1\7p) = - 2: V~j V~i + div f == F (2.8)

has a solution satisfying

lI\7p(t)lh (2.9)

(2: IIp(t)1I0 (IIv(t)lI~ + Ilf(t)1I3 +

div f(t)II£1(]iS»),

j=o

where

is a positive constant depending only on rn, M, imbedding theorems and interpolation inequalities.

Proof. Thanks to (2.2), the solvability easily follows from the general theory of the elliptic partial differential equations of second order (cf.

[5]), and so we can restrict ourselves to get the estimate (2.9).

Multiplying (2.8) by P and integrating over lRa , then we obtain M-II1VpII~ ~IIFIIL6/5(Ji'.3) IIpIIL6(Ji'.3) (2.10)

~c3lIFll L6/5(Ji'.3) llVpllo

div v = ⁰ ^(1.1)

po(x) - P E H ³ (]R3) for some positive constant p, (1.2) inf po(x) == m> 0 and suppo(x) == M <

f(x, t) E L l (0, T : H 3 _(R 3)) and div f E Ll(QT). (1.5) Then there exists T* E (0, T] such that the problem (1.1) has a unique solution (p, v,p) which satisfies

(p - p,v, "Vp) ELOO(O,T : H*

(1.6) x LOO(O,T: H*

Ll(O,T: H*

, plt=o = ^po(x)

{* _ylr=t ⁼ ^v(y,r) ₌ _x.

{ Pt + v . \7 P = ⁰

(2: ^IIp(t)1I0 (IIv(t)lI~ + Ilf(t)1I3 +

IIVpllo _~

lIull2 _~ (3/2)1/2 (IILlullo + ^IIullo) for any u E H

IIDQpll2 _~ (3/2)1/2 (II DQ (pF + p-

:'S C6 [(M + 11'7plhlllFlh + et. ^{m -;} ^11'7 ^pllO ^II'7PII + ^(2.14)

(2: ^11\7 plI~) (IIFII£1(Ji'.3) + IIFlb)· (2.16)

⁽¹¹ ^div 1I1Ll(Ji'.3) + 11 div 1112 + lI\7vIlD· (2.17) Consequently, the desired estimate is established. Q.E.D.

+ (v. V')w = ^_p-1V'p + f

Let P be the orthogona.l projector in {£2(JR3)} 3 onto H = the closure of J = {u E {Cgo(JR 3) P : ^div u ⁼ o} in {£2 (lR3)}3. Then it is immediately follows from lemmas in section 2 that if 11;50113 ::; A and SUP0StS;T IlV(t)1l3 = Il v(t)lb,oo,T ::; K, then

sllw(t)1l3 S Il ^v ol13 + IlfIILJ(o,T:H3(J!3» + ^h(Ae ^C1 ) (J{

(L::~=o ^r ^{i )} (L::~=o

^{i ).}

+ ^h(2A) (1Ifll£l(0,T:H3(J!3» +

small T, for example T ::; T = min{[4Kh(2A)]-I,* (c

£<Xl(O, T; H3(R3 » with few difficulties. Hence, as ^F becomes a compact map, there exists a fixed point v = Pw from Schauder's theorem. We have thus solved the problem

Wt + (Pw. V)w = ^_p-l\lp + f

d d r ^IQwl

^dx ⁺ r Q[(Pw. V)Qw] . Qw dx = O. (3.5)

=.!. r Pw. VIQwl2dx = ^O.