- Email: [email protected]
About the motion of nonhomogeneous ideal incompressible fluids
Nonhnear Ano/ysic. Theory. Printed m Great Britain.
Methods & Applications,
Vol.
12, No.
1. pp. 43-50,
1988. 01988
ABOUT THE MOTION OF NONHOMOGENEOUS INCOMPRESSIBLE FLUIDS
0362-546x/88 53.00+ .oO Pergamon Journals Ltd.
IDEAL
ALBERTO VALLI* Dipartimentodi Matematica, Universita di Trento, 38050 Povo (Trento), Italy
and Institute of Fundamental
WOJCIECH M. ZAJACZKOWSKI* Technological Research, Polish Academy of Sciences, 00-049 Warsaw, Poland
(Received 2 August 1985; received for publication 27 October 1986) Key
words
and phrases:
Nonhomogeneous
incompressible
fluids, Euler
equations,
method
of
characteristics.
1. INTRODUCTION IN
THIS
paper the following
System
of
eqUatiOnS
+ (u - V)u] + Vp = div v = 0 p,+v*vp=o u],,o = 0 PI,=0 = PO(X) bnlan = 0 p[u,
pf
in IO,T[ x n = Qr,
in QT, in QT, inS2, in &, on]O, T[ x as2=xT
(1.1) (1.2) (1.3) (1.4) (1.5)
(1.6)
is considered in a bounded domain &2C UP’with boundary a!2 Here u, = u * n, and n = n(x) is the unit outward vector normal to the boundary. The equations (l.l)-( 1.3) describe the motion of an ideal incompressible nonhomogeneous fluid where u is the velocity, p the pressure, p the density and f the external force field per unit mass. This problem has been considered by many authors: see for instance [14] under some restrictions for f; [4-51 in Holder’s spaces; [6] in Sobolev’s spaces of Hilbert type, and in c”; and some additional references about this problem and others related to it can be found in [4,5]. However we think that the proof presented here is simpler than the others already known; we show the existence of a solution by means of a constructive procedure (by proving that a suitable sequence of successive approximations converges), and moreover one has no difficulties arising from the geometry of the domain B and the dimension m, while the approaches which use as unknown the vorticity o = curl u need some additional procedures if & is not simply connected and X2 is not connected, or if m > 3. The idea of the proof is taken from [17,18], in which some results for the Euler equation for homogeneous fluids were proved. By following this approach, one has to solve two equations for the velocity u and the density p which are of the same type and can be integrated by means of the characteristics (hence they are reduced essentially to ordinary differential * Work partially supported by G.N.A.F.A.
of C.N.R. (Italy). 43
A. VALLI and W. M. ZAJACZKOWSKI
44
equations), and then one needs to consider two Neumann problems for an elliptic equation of second order: the first one gives the projection over the subspace of divergence free vectors tangential to the boundary, and the second is related to the pressure p. We recall that the use of a Neumann problem for controlling the behaviour of the pressure was used also in Temam’s paper [15], whose approach however is different from ours. Finally, we remark that uniqueness is proved by Graffi [lo] (see also [3]). We want to finish this introduction by remarking that if p,,(x) = p* > 0, then problem (l.l)(1.6) reduces to the usual Euler equations for homogeneous incompressible fluids. Hence our approach gives also a method for showing the existence of a (local) solution for these equations. This method is different from the others already known, for instance: the vorticity method (see [ll, 12, 161 for m = 2; [2] for m = 3); the method of finding geodesics over an infinite dimensional Riemannian manifold of diffeomorphisms (see [l, 91); the method which transforms the problem into an ordinary differential equation on a closed set of a Banach space (see [8]); the Galerkin method (see [15]); the recent abstract method of Kato and Lai [13] (in which it is observed that Lai’s method based on successive approximations require “a considerable amount of preparation”. We think on the contrary that our result in this sense is quite simple, however compare our linearization with equation (E’) in [13] and the related linearization in Lai’s thesis). 2. NOTATIONS
In this paper the following notations are used
I141,,r~ lI4lL”(o.T;w!(n)) = lbllLr.-,T7
Ilullw!(n) =
for1 E
N, 1 c r < m. For noninteger
1 we set
IlullWkdQ)
=
Ib\ll,r,dR
.
We shall assume also that O
-i;fp,
Finally, the Einstein convention
S pO(x) S sup pa = M2 < $03. R about summation over repeated indices is assumed.
3. EXISTENCE
(2.1)
OF SOLUTIONS
To prove the existence of solutions of problem (l.l)-( 1.6) we shall use the following method of successive approximations: p;+l + “k. v k+l = 0 inQ,, (3.1) in 9, > Pk+Yr=o = PaLI where vk is treated as a given function, such that div vk = 0, vk . nldR = 0; div(l/#+‘Vpk+‘) I@+’
i!g
= divf-
Djv,“D,v”
in(t) x Sz,
tE [0, ZJ,l
on{f} x as2,
= f - n + vfv,kD3ni
I
(3.2)
where vk and pk+ ’ are treated as given functions, and n is a regular extension of n to 3; ,,,;+I
+ vk . v&+1
= _ -
P Wk+lj,+)
= a(x)
1 k+l VP k+l
+f
inQr, in 9,
(3.3)
Motion of nonhomogeneous
ideal incompressible
45
fluids
where vk, pk+l and pk+’ are treated as given functions. The function wk+l constructed by (3.3) is such that in general div wk’I # 0 and wkcl - n\an # 0. Therefore the method of successive approximations is possible if we introduce the projection n determined by nWk+l
3
Wk+l
-
Vlpk+‘,
(3.4)
where qk’l is the solution of the problem Ag7k+l zz div wk+l drp
in S2
k+l
an
an
= wktl
0naQ
dR
and then we set vk” E zwkil. We assume also that v0 = - a. For getting easily some estimates for l/pk+‘, we can consider also the following problem 17f+’
+ vk
.v
rl
k+l
(3.5)
Tlk+%O = l/PO where rfk+’ E l/pkf’. Assume now that aS2 E C4, vk E C’([O, T]; W:(Q)), I > m, p. E W;(Q) and satisfies (2.1), = 0 on a!& Then the existence of the solutions to ~~~~~~)withdivu=Oin~andu.~l~~ problems (3.1) and (3.3) is proved by means of the method of characteristics. Moreover, by applying D’ to (3.1), (1~1G 2), by multiplying by IDypk+11r-2D)‘pk”1 and by integrating in 5;2 (and repeating the same procedure also for (3.3), and (3~9,) one gets easily
(3 *6)
+ c(lJrlk+‘l12.,t~k+1J13.r + Ilfl12.~)Jlwk+‘Ili,‘~ s ~llvkll*.illWk+‘II~,~ (here and in the sequel each constant will depend at most on 52, m, r, M1 and M2). Moreover, the representation of p k+l by means of the characteristics gives that Ml s pk+l(f, X) c MZ,
M;’ 6 qktl(t, X) G Mi’.
Remark also that pk+ * and vk+’ belong to C”([O, T]; W:(Q))) , since the characteristic Vk(t, s, x) belong to c([O, T] X [0, T] X W:(Q)) ( see, for instance, [8]). The same will be true also for I@+’ since we will see that Vp k+l E L’(0, T, W:(Q)). From the projection (3.4) we obtain also at once that
ll~k+‘l12,rasclJwk+‘I/z,r.
(3.7)
A. VALLIand W. M. ZAJACZKOWSKI
46
Now we shall consider problem (3.2). We write it in the short form div(qVp) =: Fz + 5
&
divf - DiV,D,Ui
= G s f * n + v,v,D,ni
in Q (3.8)
on aB,
and one verifies easily that fQF = _fanG. Hence there exists a solution of problem (3.8) since the necessary compatibility condition is satisfied (see, for instance, [4]). Then we can restrict ourselves to get an a priori estimate. For having the uniqueness of the solution, we assume also that (3.9)
p = 0.
I .Q rp[tiplying (3.8)1 by p, integrating O,2one gets
over Q and using (3.9) for estimating /~]]o.z in terms of (3.10)
IIVP~IO,, =z cWl10.~ + IlGllLZ). Writing the problem (3.8) in the form hp = -pVr).
Vp + pF
in s2, (3.11)
ap
on asi,
G=PG we have, by classical results about the Neumann
problem
l/&r d c(lbF - ~Vrl . VPIIL,+ Il~Gllz-~/r.r.m1 s 4llPVrt - VP!1.r +-~lp~l~,~+ I~PG~~~,~).
(3.12)
The last two terms in the right-hand side of (3.12) are estimated by cl~~~~~.~(l~Fll~.~ + 11G]Iz.,) (recall that W;(B) is an algebra for ST> m). The first term in the right-hand side of (3.12) can be estimated by
s cl~~~i~.rlirll~~.r(~~V~ilo.~ + llD2~hr). ~llPlI~.rll~ll~.rllVPIl1.~ Using the interpolation inequalities of Gaghardo-Nirenberg have that this last expression is estimated by
(see for instance f7], p. 312) we
4lPlI%r+ 4IlPl1I.rlIT1ll2.r)llVPllO,Z~ where
(3.13) Therefore,
from the above considerations
LEMMA 3.1. L’(0,
T;
Let
W#L?)),
as2Ec3, G E L’(0,
we have W:(Q)), r > m. Then
pEL”(O,T; T;
W;(Q)),
qEL”(O,1”;
there
W;(Q)),
exists a unique
FE
solution
Motion of nonhomogeneous
p E L’(0, T; Wj(9))
ideal incompressible
fluids
47
of problem (3.8), (3.9) such that =S~~~Y(lI~II~.~llrlll~,~> + ll~l1dll%
II&,
+ llGhr),
(3.14)
where a( +) is described by (3.13). Hence using the form of F we have proved the following. LEMMA 3.2.
Let &2 E C4, pk+’ E C”([O, 7j; W:(Q)), $+‘E c”([O, T’J; W?(Q)), UkE P([O, T]; WZ(sz)), f E L’(O,T; w:(Q)), r > m. Then there exists a unique solution of the problem (3.2) such that pk+’ E L’(0, T, W:(Q)), jn pk+’ = 0 a.e. in [O, ?‘j and Ilpk+%J g
~~~tllP”+‘Il~,~ll~k+‘ll*,~~ + lI~k+‘ll~,~l~llfl~,~ + Ibkll:.r>-
(3.15)
From (3.6), and (3.6), we have
IIP”+‘M2,r d IJ~dl2,~exptc~lull~,~,~,~), t E IO97%
Ilsk+‘~ot12,‘ s ll~O”112,~ e~P@l141Z,r,m,lh cE lo, II.
(3.16)
Using (3.15) and (3.16) in (3X5), we get (3.17)
where I-I(.) is a nondecreasing function of its arguments. Using (3.7) (for k) and integrating (3.17) with respect to t we get IIw~+~IIz,~ 6 exp(c~~w~~~*,~,~,~ )[W .) i’ (f +
ilflld dt + tl~il~.,]
(3.18)
0
Let cr > 1 and let fl~‘$,,~,r s o$&; then assuming that 1 d fo, to such that the right-hand side of (3.18) is smaller than o$z~~~,,,we obtain from (3.7)
II~k+lll*,r=G41Wk+111Z,r s c4l&,rt for each natural k. To prove the convergence
f E P, hd
of these sequences we have to consider the following problems: Rf+’
+ ok. VRk+’ + Ok- Vpk = 0,
.
(3.20) Rk+llt=O = 0, where
Rk+’
= pk+’
(3.19)
- pk, @kE gk - Uk-1, k
&
1;
E,k+‘+vk.VEk++ek-v?jk=O, Ek+ll,_O = 0,
(3.21) I
whereEk+l==~k+l-~k,k~l; Wf”
+ (vk * V)Wk+’ + (ok. V)Wk = -,p+lVpk
- ,,k+lvpk+l,
(3.22) Wk+’ Itso = 0,
A.
48 where
wktl
s
,,,k+l
_
,+,k
9
VALLI and W. M. ZAJACZKOWSKI
k+l - pk, k 2 1. Moreover we have P + Ek”‘Vpk) = -D,S,“D,u; - L&U;-‘&?;,
pk+l=
div(rlk+‘VPk+’
aPk+l rfk+l- an + p+’
(3.23)
= (efuf + uye#Qzj.
g
At last from (3.4) we get ok+1
=
nwk+le
(3.24)
By proceeding in the same way used for getting (3.6), from problems (3.20), (3.21) and (3.22) we have
where E depends on the right-hand side of (3.19). At last from (3.23) and (3.24) we get Il@%,r s ~llWkllLr.
llP”“%,F g ~(ll~k+~vPkll~,~+ Jv%,A Knowing that t s to, the above inequalities imply
Ilwk+‘(t)\ll,r S I5 1 4 of”/lf(r)11z,, dr) i’ Ilw”(r)lli,~ dr. ( I 0 After standard considerations it follows that the sequence {uk, wk, pk, pk} converges to a solution of the problems (3.1)-(3.3) for the limit functions u, W, p and p, and u = xw. It remains to show that for the limit functions we have u = W, that is div w = 0, w *nldn = 0. Taking the divergence of (3.3) and using (3.2) and (3.4) one obtains (div w), I- u - V div w = -D,u,D,D,g, and taking the normal component (W * FZ), +
By using the properties
of (3.3), projecting U a V(W
*n) =
in Qr
(3.25)
on as2and using (3.2) one has
U~~~~iDj~
(3.26)
on Cr.
of the function sp defined by (3.4), we have easily
$IldivwIlo.2s cll~4l0,&412.2 c cII~ullo.mWiv wlh2+ Ilw44/~.2,dR).
(3.27)
To obtain an estimate for w . r&o we introduce the following curves on aS2 %(C
r, Y) = u(t, V(G t7 Y>)
V(t, T7Y) = Y
Therefore
on CT,
(3.28)
on ask
we can write (3.26) in the form
-$qV(t, V(t,
t, Y)) = U,(f, V(t, t, Y))D~ni(Vt~y t7Y))DiqCfy
= K(f, V(C t, Y>>,
‘ttt ‘T Y)) (3.29)
Motion of nonhomogeneous where
r+ = we n.
ideal incompressible
fluids
49
From (3.29), by setting $(t, y; t) = r,$t, V(t, z, y)) and by remarking that tl@r/~o
s c ex~(c~~ll~ll~.,a.~,)llrcll~.~.~~
we have $11~1~1/?,2,&2s clHI*.r.~J~ exP(cr*fl~ll2.r,~,t,>flvcoli1/2.2.dR s 4412.~.Dc.10 exp(cf,llu112.r,~:,i,)(ltdiv40.2 +
lb - 4l~m~2 1.
(3.30)
On the other hand one has also that III41l12,2,a~ s c exP(ctOllUtjZ.,.~.t~)~~~~~t~*.~.~~ , hence from (3.27) and (3.30), knowing that div ~],,a = diva = 0 in 52 and w . n],=, = a . n = 0 on dft, we get that div w = 0 in R, w. ~&o = 0 on aS2. Therefore we have proved the following. 3.1. Let a& E C*, a~ W;(Q), diva = 0, a +nlao = 0, p. E W?(a), infa po > 0,fE L’(0, T; W:(n)), r 2 m. Then there exist r. E IO, T], to sufficiently small (see 0 E c”([O, toI; Wf(Q 1) n Cl([O, Gil; ante)), (3.19)), P E co(K)> toI; (3.1% such that (p, u, p) is a solution of (1.1) W(W) fl w (0, to; WQ)),P E w)? to; ww (1.6) in Q,,. Moreover one has
THEOREM
0 < itf
p.
If fE CO([O, 7J; Wf(S2)), then (p, u,p) is a classical solution.
c p(t, p E
x) c
sip
po < +m
c”(]O, to]; W:(Q))
V(t, X) E &.
and u E C’([O, &,I; ~~(~)~,
hence
Remark 3.1. This method can be used also for obtaining more regular solutions, for instance
by choosing the space Wi(sL) instead of W:(G),
for I> 1 + y, q > 1 (and &2 E C’+2).
Acknowfedgemenr-The authors would like to thank Professor Hugo Beirlo da Veiga for very fruitful conversations about the subject of this paper. REFERENCES 1. ARNOLD V. I., Sur la geomitrie differentietle des groups de Lie de dimension infmie et ses applications a l’hydrodynamique des fluides parfaits, Annfs Inst. Fourier Univ. Grenoble 16, 319-361 (1966). 2. BARDOSC. & FRISCH U., Finite-time regularity for bounded and unbounded ideal incompressible fluids using Holder estimates, Lecture Nores in Mathematics, 565, l-13, Springer, New York (1975). . 3. BEIRAODA VEIGA H. & VALL~A., On the motion of a non-homogeneous ideal incomnressibfe fluid in an external 1 force field, Rc. Semin. mat. Univ. Padova 59, 117-145 (1978). 4. BEIRAO DA VEIGA H. & VALLI A., On the Euler equations for non-homogeneous fluids (I), Rc. Semin. mat. Univ. Padova 43, 151-158 (1980). 5. BEIRAODA VEIGA H. % VALLI A., On the Euier equations for non-homogeneous Appl. 73, 338-350 (1980).
fluids (II), J. math. Anatysk
6. BEIRAODA VEIGA H. & VALLI A., Existence of C solutions of the Euler equations for non-homogeneous fluids, Communs partial d$$ Eqns 5, 95-107 (1980). 7. BESOV0. V., IL’INV. P. & NIKOL’SKIIS. M., Integral Representations of Functions and Imbedding Theorems, 1, Winston & Sons, Washington (1978); translated from Russian. 8. BOURGUIGNON I. P. & BREZIS H., Remarks on the Euler equation, J. funct. Analysis 15, 341-363 (1974). 9. EBIN D. G. & MARSDENJ. E., Groups of diffeomorphisms and the motion of an incompressible fluid, Arm. Math. 92, 102-163 (1970).
50
A. VALLI and W. M. ZAJACZKOWSKI
10. GRAFFI D., 11 teorema di unicita per i fluidi incompressibili, perfetti, eterogenei, Reutu Un. map argenr., 17, 7% 77 (1955). 11. JUDOVIC V. I., A two-dimensional problem of unsteady flow of an ideal incompressible fluid across a given domain, Am. math. Sot. Transl. 57. 277-304 (1966); in Russian in Mar. Sb. 64, 562-588 (1964). 12. KATO T., On classical solutions of the two-dimensional non-stationary Euler equation. Archs rarion. Mech. Analysis 25, 188-200 (1967). 13. KATO T. & LAI C. Y., Nonlinear evolution equations and the Euler flow, J. funct. Analysis 56. 15-28 (1984). 14. MARSDENJ. E., Well-posedness of the equations of a non-homogeneous perfect fluid. Communs partial diff- Eqns 1, 215-230 (1976). 15. TEMAMR. On the Euler equations of incompressible perfect fluids, J. funcr. Analysis 20. 32-43 (1975). 16. WOLIBNERW., Un theorbme sur I’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendant un temps infiniment long, Math. 2. 37, 698-726 (1933). 17. ZAJACZKOWSKI W. M., Local solvability of nonstationary leakage problem for ideal incompressible fluid, 2, Pacif. J. Math. 113, 229-255 (1984). 18. ZAJACZKOWSKIW. M., Solvability of the leakage problem for the hydrodynamic Euler equations in Sobolev spaces, IFTR Reports 21/1983, Warsaw (1983).
Methods & Applications,
Vol.
12, No.
1. pp. 43-50,
1988. 01988
ABOUT THE MOTION OF NONHOMOGENEOUS INCOMPRESSIBLE FLUIDS
0362-546x/88 53.00+ .oO Pergamon Journals Ltd.
IDEAL
ALBERTO VALLI* Dipartimentodi Matematica, Universita di Trento, 38050 Povo (Trento), Italy
and Institute of Fundamental
WOJCIECH M. ZAJACZKOWSKI* Technological Research, Polish Academy of Sciences, 00-049 Warsaw, Poland
(Received 2 August 1985; received for publication 27 October 1986) Key
words
and phrases:
Nonhomogeneous
incompressible
fluids, Euler
equations,
method
of
characteristics.
1. INTRODUCTION IN
THIS
paper the following
System
of
eqUatiOnS
+ (u - V)u] + Vp = div v = 0 p,+v*vp=o u],,o = 0 PI,=0 = PO(X) bnlan = 0 p[u,
pf
in IO,T[ x n = Qr,
in QT, in QT, inS2, in &, on]O, T[ x as2=xT
(1.1) (1.2) (1.3) (1.4) (1.5)
(1.6)
is considered in a bounded domain &2C UP’with boundary a!2 Here u, = u * n, and n = n(x) is the unit outward vector normal to the boundary. The equations (l.l)-( 1.3) describe the motion of an ideal incompressible nonhomogeneous fluid where u is the velocity, p the pressure, p the density and f the external force field per unit mass. This problem has been considered by many authors: see for instance [14] under some restrictions for f; [4-51 in Holder’s spaces; [6] in Sobolev’s spaces of Hilbert type, and in c”; and some additional references about this problem and others related to it can be found in [4,5]. However we think that the proof presented here is simpler than the others already known; we show the existence of a solution by means of a constructive procedure (by proving that a suitable sequence of successive approximations converges), and moreover one has no difficulties arising from the geometry of the domain B and the dimension m, while the approaches which use as unknown the vorticity o = curl u need some additional procedures if & is not simply connected and X2 is not connected, or if m > 3. The idea of the proof is taken from [17,18], in which some results for the Euler equation for homogeneous fluids were proved. By following this approach, one has to solve two equations for the velocity u and the density p which are of the same type and can be integrated by means of the characteristics (hence they are reduced essentially to ordinary differential * Work partially supported by G.N.A.F.A.
of C.N.R. (Italy). 43
A. VALLI and W. M. ZAJACZKOWSKI
44
equations), and then one needs to consider two Neumann problems for an elliptic equation of second order: the first one gives the projection over the subspace of divergence free vectors tangential to the boundary, and the second is related to the pressure p. We recall that the use of a Neumann problem for controlling the behaviour of the pressure was used also in Temam’s paper [15], whose approach however is different from ours. Finally, we remark that uniqueness is proved by Graffi [lo] (see also [3]). We want to finish this introduction by remarking that if p,,(x) = p* > 0, then problem (l.l)(1.6) reduces to the usual Euler equations for homogeneous incompressible fluids. Hence our approach gives also a method for showing the existence of a (local) solution for these equations. This method is different from the others already known, for instance: the vorticity method (see [ll, 12, 161 for m = 2; [2] for m = 3); the method of finding geodesics over an infinite dimensional Riemannian manifold of diffeomorphisms (see [l, 91); the method which transforms the problem into an ordinary differential equation on a closed set of a Banach space (see [8]); the Galerkin method (see [15]); the recent abstract method of Kato and Lai [13] (in which it is observed that Lai’s method based on successive approximations require “a considerable amount of preparation”. We think on the contrary that our result in this sense is quite simple, however compare our linearization with equation (E’) in [13] and the related linearization in Lai’s thesis). 2. NOTATIONS
In this paper the following notations are used
I141,,r~ lI4lL”(o.T;w!(n)) = lbllLr.-,T7
Ilullw!(n) =
for1 E
N, 1 c r < m. For noninteger
1 we set
IlullWkdQ)
=
Ib\ll,r,dR
.
We shall assume also that O
-i;fp,
Finally, the Einstein convention
S pO(x) S sup pa = M2 < $03. R about summation over repeated indices is assumed.
3. EXISTENCE
(2.1)
OF SOLUTIONS
To prove the existence of solutions of problem (l.l)-( 1.6) we shall use the following method of successive approximations: p;+l + “k. v k+l = 0 inQ,, (3.1) in 9, > Pk+Yr=o = PaLI where vk is treated as a given function, such that div vk = 0, vk . nldR = 0; div(l/#+‘Vpk+‘) I@+’
i!g
= divf-
Djv,“D,v”
in(t) x Sz,
tE [0, ZJ,l
on{f} x as2,
= f - n + vfv,kD3ni
I
(3.2)
where vk and pk+ ’ are treated as given functions, and n is a regular extension of n to 3; ,,,;+I
+ vk . v&+1
= _ -
P Wk+lj,+)
= a(x)
1 k+l VP k+l
+f
inQr, in 9,
(3.3)
Motion of nonhomogeneous
ideal incompressible
45
fluids
where vk, pk+l and pk+’ are treated as given functions. The function wk+l constructed by (3.3) is such that in general div wk’I # 0 and wkcl - n\an # 0. Therefore the method of successive approximations is possible if we introduce the projection n determined by nWk+l
3
Wk+l
-
Vlpk+‘,
(3.4)
where qk’l is the solution of the problem Ag7k+l zz div wk+l drp
in S2
k+l
an
an
= wktl
0naQ
dR
and then we set vk” E zwkil. We assume also that v0 = - a. For getting easily some estimates for l/pk+‘, we can consider also the following problem 17f+’
+ vk
.v
rl
k+l
(3.5)
Tlk+%O = l/PO where rfk+’ E l/pkf’. Assume now that aS2 E C4, vk E C’([O, T]; W:(Q)), I > m, p. E W;(Q) and satisfies (2.1), = 0 on a!& Then the existence of the solutions to ~~~~~~)withdivu=Oin~andu.~l~~ problems (3.1) and (3.3) is proved by means of the method of characteristics. Moreover, by applying D’ to (3.1), (1~1G 2), by multiplying by IDypk+11r-2D)‘pk”1 and by integrating in 5;2 (and repeating the same procedure also for (3.3), and (3~9,) one gets easily
(3 *6)
+ c(lJrlk+‘l12.,t~k+1J13.r + Ilfl12.~)Jlwk+‘Ili,‘~ s ~llvkll*.illWk+‘II~,~ (here and in the sequel each constant will depend at most on 52, m, r, M1 and M2). Moreover, the representation of p k+l by means of the characteristics gives that Ml s pk+l(f, X) c MZ,
M;’ 6 qktl(t, X) G Mi’.
Remark also that pk+ * and vk+’ belong to C”([O, T]; W:(Q))) , since the characteristic Vk(t, s, x) belong to c([O, T] X [0, T] X W:(Q)) ( see, for instance, [8]). The same will be true also for I@+’ since we will see that Vp k+l E L’(0, T, W:(Q)). From the projection (3.4) we obtain also at once that
ll~k+‘l12,rasclJwk+‘I/z,r.
(3.7)
A. VALLIand W. M. ZAJACZKOWSKI
46
Now we shall consider problem (3.2). We write it in the short form div(qVp) =: Fz + 5
&
divf - DiV,D,Ui
= G s f * n + v,v,D,ni
in Q (3.8)
on aB,
and one verifies easily that fQF = _fanG. Hence there exists a solution of problem (3.8) since the necessary compatibility condition is satisfied (see, for instance, [4]). Then we can restrict ourselves to get an a priori estimate. For having the uniqueness of the solution, we assume also that (3.9)
p = 0.
I .Q rp[tiplying (3.8)1 by p, integrating O,2one gets
over Q and using (3.9) for estimating /~]]o.z in terms of (3.10)
IIVP~IO,, =z cWl10.~ + IlGllLZ). Writing the problem (3.8) in the form hp = -pVr).
Vp + pF
in s2, (3.11)
ap
on asi,
G=PG we have, by classical results about the Neumann
problem
l/&r d c(lbF - ~Vrl . VPIIL,+ Il~Gllz-~/r.r.m1 s 4llPVrt - VP!1.r +-~lp~l~,~+ I~PG~~~,~).
(3.12)
The last two terms in the right-hand side of (3.12) are estimated by cl~~~~~.~(l~Fll~.~ + 11G]Iz.,) (recall that W;(B) is an algebra for ST> m). The first term in the right-hand side of (3.12) can be estimated by
s cl~~~i~.rlirll~~.r(~~V~ilo.~ + llD2~hr). ~llPlI~.rll~ll~.rllVPIl1.~ Using the interpolation inequalities of Gaghardo-Nirenberg have that this last expression is estimated by
(see for instance f7], p. 312) we
4lPlI%r+ 4IlPl1I.rlIT1ll2.r)llVPllO,Z~ where
(3.13) Therefore,
from the above considerations
LEMMA 3.1. L’(0,
T;
Let
W#L?)),
as2Ec3, G E L’(0,
we have W:(Q)), r > m. Then
pEL”(O,T; T;
W;(Q)),
qEL”(O,1”;
there
W;(Q)),
exists a unique
FE
solution
Motion of nonhomogeneous
p E L’(0, T; Wj(9))
ideal incompressible
fluids
47
of problem (3.8), (3.9) such that =S~~~Y(lI~II~.~llrlll~,~> + ll~l1dll%
II&,
+ llGhr),
(3.14)
where a( +) is described by (3.13). Hence using the form of F we have proved the following. LEMMA 3.2.
Let &2 E C4, pk+’ E C”([O, 7j; W:(Q)), $+‘E c”([O, T’J; W?(Q)), UkE P([O, T]; WZ(sz)), f E L’(O,T; w:(Q)), r > m. Then there exists a unique solution of the problem (3.2) such that pk+’ E L’(0, T, W:(Q)), jn pk+’ = 0 a.e. in [O, ?‘j and Ilpk+%J g
~~~tllP”+‘Il~,~ll~k+‘ll*,~~ + lI~k+‘ll~,~l~llfl~,~ + Ibkll:.r>-
(3.15)
From (3.6), and (3.6), we have
IIP”+‘M2,r d IJ~dl2,~exptc~lull~,~,~,~), t E IO97%
Ilsk+‘~ot12,‘ s ll~O”112,~ e~P@l141Z,r,m,lh cE lo, II.
(3.16)
Using (3.15) and (3.16) in (3X5), we get (3.17)
where I-I(.) is a nondecreasing function of its arguments. Using (3.7) (for k) and integrating (3.17) with respect to t we get IIw~+~IIz,~ 6 exp(c~~w~~~*,~,~,~ )[W .) i’ (f +
ilflld dt + tl~il~.,]
(3.18)
0
Let cr > 1 and let fl~‘$,,~,r s o$&; then assuming that 1 d fo, to such that the right-hand side of (3.18) is smaller than o$z~~~,,,we obtain from (3.7)
II~k+lll*,r=G41Wk+111Z,r s c4l&,rt for each natural k. To prove the convergence
f E P, hd
of these sequences we have to consider the following problems: Rf+’
+ ok. VRk+’ + Ok- Vpk = 0,
.
(3.20) Rk+llt=O = 0, where
Rk+’
= pk+’
(3.19)
- pk, @kE gk - Uk-1, k
&
1;
E,k+‘+vk.VEk++ek-v?jk=O, Ek+ll,_O = 0,
(3.21) I
whereEk+l==~k+l-~k,k~l; Wf”
+ (vk * V)Wk+’ + (ok. V)Wk = -,p+lVpk
- ,,k+lvpk+l,
(3.22) Wk+’ Itso = 0,
A.
48 where
wktl
s
,,,k+l
_
,+,k
9
VALLI and W. M. ZAJACZKOWSKI
k+l - pk, k 2 1. Moreover we have P + Ek”‘Vpk) = -D,S,“D,u; - L&U;-‘&?;,
pk+l=
div(rlk+‘VPk+’
aPk+l rfk+l- an + p+’
(3.23)
= (efuf + uye#Qzj.
g
At last from (3.4) we get ok+1
=
nwk+le
(3.24)
By proceeding in the same way used for getting (3.6), from problems (3.20), (3.21) and (3.22) we have
where E depends on the right-hand side of (3.19). At last from (3.23) and (3.24) we get Il@%,r s ~llWkllLr.
llP”“%,F g ~(ll~k+~vPkll~,~+ Jv%,A Knowing that t s to, the above inequalities imply
Ilwk+‘(t)\ll,r S I5 1 4 of”/lf(r)11z,, dr) i’ Ilw”(r)lli,~ dr. ( I 0 After standard considerations it follows that the sequence {uk, wk, pk, pk} converges to a solution of the problems (3.1)-(3.3) for the limit functions u, W, p and p, and u = xw. It remains to show that for the limit functions we have u = W, that is div w = 0, w *nldn = 0. Taking the divergence of (3.3) and using (3.2) and (3.4) one obtains (div w), I- u - V div w = -D,u,D,D,g, and taking the normal component (W * FZ), +
By using the properties
of (3.3), projecting U a V(W
*n) =
in Qr
(3.25)
on as2and using (3.2) one has
U~~~~iDj~
(3.26)
on Cr.
of the function sp defined by (3.4), we have easily
$IldivwIlo.2s cll~4l0,&412.2 c cII~ullo.mWiv wlh2+ Ilw44/~.2,dR).
(3.27)
To obtain an estimate for w . r&o we introduce the following curves on aS2 %(C
r, Y) = u(t, V(G t7 Y>)
V(t, T7Y) = Y
Therefore
on CT,
(3.28)
on ask
we can write (3.26) in the form
-$qV(t, V(t,
t, Y)) = U,(f, V(t, t, Y))D~ni(Vt~y t7Y))DiqCfy
= K(f, V(C t, Y>>,
‘ttt ‘T Y)) (3.29)
Motion of nonhomogeneous where
r+ = we n.
ideal incompressible
fluids
49
From (3.29), by setting $(t, y; t) = r,$t, V(t, z, y)) and by remarking that tl@r/~o
s c ex~(c~~ll~ll~.,a.~,)llrcll~.~.~~
we have $11~1~1/?,2,&2s clHI*.r.~J~ exP(cr*fl~ll2.r,~,t,>flvcoli1/2.2.dR s 4412.~.Dc.10 exp(cf,llu112.r,~:,i,)(ltdiv40.2 +
lb - 4l~m~2 1.
(3.30)
On the other hand one has also that III41l12,2,a~ s c exP(ctOllUtjZ.,.~.t~)~~~~~t~*.~.~~ , hence from (3.27) and (3.30), knowing that div ~],,a = diva = 0 in 52 and w . n],=, = a . n = 0 on dft, we get that div w = 0 in R, w. ~&o = 0 on aS2. Therefore we have proved the following. 3.1. Let a& E C*, a~ W;(Q), diva = 0, a +nlao = 0, p. E W?(a), infa po > 0,fE L’(0, T; W:(n)), r 2 m. Then there exist r. E IO, T], to sufficiently small (see 0 E c”([O, toI; Wf(Q 1) n Cl([O, Gil; ante)), (3.19)), P E co(K)> toI; (3.1% such that (p, u, p) is a solution of (1.1) W(W) fl w (0, to; WQ)),P E w)? to; ww (1.6) in Q,,. Moreover one has
THEOREM
0 < itf
p.
If fE CO([O, 7J; Wf(S2)), then (p, u,p) is a classical solution.
c p(t, p E
x) c
sip
po < +m
c”(]O, to]; W:(Q))
V(t, X) E &.
and u E C’([O, &,I; ~~(~)~,
hence
Remark 3.1. This method can be used also for obtaining more regular solutions, for instance
by choosing the space Wi(sL) instead of W:(G),
for I> 1 + y, q > 1 (and &2 E C’+2).
Acknowfedgemenr-The authors would like to thank Professor Hugo Beirlo da Veiga for very fruitful conversations about the subject of this paper. REFERENCES 1. ARNOLD V. I., Sur la geomitrie differentietle des groups de Lie de dimension infmie et ses applications a l’hydrodynamique des fluides parfaits, Annfs Inst. Fourier Univ. Grenoble 16, 319-361 (1966). 2. BARDOSC. & FRISCH U., Finite-time regularity for bounded and unbounded ideal incompressible fluids using Holder estimates, Lecture Nores in Mathematics, 565, l-13, Springer, New York (1975). . 3. BEIRAODA VEIGA H. & VALL~A., On the motion of a non-homogeneous ideal incomnressibfe fluid in an external 1 force field, Rc. Semin. mat. Univ. Padova 59, 117-145 (1978). 4. BEIRAO DA VEIGA H. & VALLI A., On the Euler equations for non-homogeneous fluids (I), Rc. Semin. mat. Univ. Padova 43, 151-158 (1980). 5. BEIRAODA VEIGA H. % VALLI A., On the Euier equations for non-homogeneous Appl. 73, 338-350 (1980).
fluids (II), J. math. Anatysk
6. BEIRAODA VEIGA H. & VALLI A., Existence of C solutions of the Euler equations for non-homogeneous fluids, Communs partial d$$ Eqns 5, 95-107 (1980). 7. BESOV0. V., IL’INV. P. & NIKOL’SKIIS. M., Integral Representations of Functions and Imbedding Theorems, 1, Winston & Sons, Washington (1978); translated from Russian. 8. BOURGUIGNON I. P. & BREZIS H., Remarks on the Euler equation, J. funct. Analysis 15, 341-363 (1974). 9. EBIN D. G. & MARSDENJ. E., Groups of diffeomorphisms and the motion of an incompressible fluid, Arm. Math. 92, 102-163 (1970).
50
A. VALLI and W. M. ZAJACZKOWSKI
10. GRAFFI D., 11 teorema di unicita per i fluidi incompressibili, perfetti, eterogenei, Reutu Un. map argenr., 17, 7% 77 (1955). 11. JUDOVIC V. I., A two-dimensional problem of unsteady flow of an ideal incompressible fluid across a given domain, Am. math. Sot. Transl. 57. 277-304 (1966); in Russian in Mar. Sb. 64, 562-588 (1964). 12. KATO T., On classical solutions of the two-dimensional non-stationary Euler equation. Archs rarion. Mech. Analysis 25, 188-200 (1967). 13. KATO T. & LAI C. Y., Nonlinear evolution equations and the Euler flow, J. funct. Analysis 56. 15-28 (1984). 14. MARSDENJ. E., Well-posedness of the equations of a non-homogeneous perfect fluid. Communs partial diff- Eqns 1, 215-230 (1976). 15. TEMAMR. On the Euler equations of incompressible perfect fluids, J. funcr. Analysis 20. 32-43 (1975). 16. WOLIBNERW., Un theorbme sur I’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendant un temps infiniment long, Math. 2. 37, 698-726 (1933). 17. ZAJACZKOWSKI W. M., Local solvability of nonstationary leakage problem for ideal incompressible fluid, 2, Pacif. J. Math. 113, 229-255 (1984). 18. ZAJACZKOWSKIW. M., Solvability of the leakage problem for the hydrodynamic Euler equations in Sobolev spaces, IFTR Reports 21/1983, Warsaw (1983).