L e t U = { X [ r < a } be the infinite cylinder corresponding to V; we begin b y consider- ing t h e Green functions
1 1
F v = 4 ~ R y v ( X ~ a n d F = 4 - ~ - y ( X ~
(R=[X-X~
of t h e Dirichlet p r o b l e m for the L a p l a c e o p e r a t o r in U a n d V respectively; F v is r e l e v a n t because of (II.1) below. I n a d d i t i o n to t h e usual s y m m e t r y a n d p o s i t i v i t y properties (~(X, X ~ = y ( X ~ X) a n d 0 < y < 1/4z~R on V • V), these functions h a v e t h e following be- haviour.
(a) Fv d e p e n d s only on r, r ~ 0 - 0 ~ a n d z - z ~ it is an e v e n function of 0 - 0 ~ a n d z - z ~ a n d has period 2~ in 0 - 0 %
(b) L e t Jo.1 = 2 . 4 0 ... d e n o t e t h e smallest positive zero of t h e Bessel function J0. T h e r e exists a c o n s t a n t K such t h a t
Fu(X ~ X~ ~ < g e x p
(--)O.l[Z--z~
for [ z - z ~ >~a, a n d similarly for the m o d u l u s of each d e r i v a t i v e of Fv.(c) L e t V ~ = { X [ r < a - 6 } a n d V ~ = { X e V I r < a - 8 } , where 8 > 0 . T h e n Yv is (real) a n a l y t i c on ~" • Us, a n d h a r m o n i c in each variable (A~ = A F v = 0 ) t h e r e .
(d) W i t h (X ~ X) e 17 • l? a n d X ~ 4~X, a n d w i t h the s a m e values of X ~ X ~ X1 a n d X~
implied on each side of t h e equation, we h a v e
F(z ~ z) = Fv(z - z ~ - Fu(Z + z ~ - 2b) + Fv(z - z ~ - 4b) - Fv(z + z ~ - 6b) + . . .
- F v ( z + z ~ 1 7 6 2 4 7 1 7 6 ..., (II. 1)
for it is readily verified t h a t this f o r m u l a m a k e s F = 0 on z = + b. T h e series converges ra- pidly because of (b). W i t h t h e n o t a t i o n X = (r, 0, z), d e n o t e t h e p r i m a r y i m a g e points of X ~ with respect to t h e planes z = + b b y
46 L. E . F R A E N K E L A N D M. S. B E R G E R
X (1) = ( r 0, 0 0, 2 b - z ~ a n d X (2~ --- (r ~ 0 ~ - 2 b - z ~ a n d let R j = [ X - X ( J ) I . I t t h e n follows from (c) a n d (II.1) t h a t
,(1 1 1)
F ( X ~ R1 R-2 + A ( X ~ (I1.2)
where A is analytic on 17 • 17~, a n d h a r m o n i c in each variable there.
(e) W e can also estimate derivatives like @ / a t ~ on V • 17 b y means of t h e m a x i m u m principle, w i t h o u t using (II.1). To this end, we define the pseudo-image points
X TM = (2a - r ~ 0 ~ z~ X (4) = ( 2 a - r ~ 0 ~ 2b - z ~ a n d X (5~ = (2a - r ~ 0 ~ - 2 b - z ~ One can show t h a t
1 < ~ < . 3 a - r ~
a+ r ~ say, for r ~ all z - z ~ a n d for r ~ < a, r ~< a, I z - z ~ I ~> 2 a;
a n d that, for fixed X ~ U, t h e function (BSl4zt)~(llRs)l~r ~ which is a positive h a r m o n i c function of X E U, d o m i n a t e s + &yv/ar ~ on ~U a n d hence (by the m a x i m u m principle) on
U. I n this way, one u l t i m a t e l y obtains the estimate
[ V ~ 1 7 6 + ~ on V x ~ , (II.3)
j = l j=B
where Vj is the gradient o p e r a t o r {~/aX~J)}, i = 1, 2, 3, a n d c is a c o n s t a n t d e p e n d i n g only on b/a, with c = 1 for b ~>a. I t follows i m m e d i a t e l y {since Rj ~> R on V • 17 a n d B ~<3) t h a t
I V ~ 1 7 6 ~ 1 on V • ( I I . 4 )
where the c o n s t a n t depends o~rly on b/a.
o
L ~ ~ M A I I A. I/(yJ, ,~) is a generalized solution, the vector potential a E Wl. ~ (V) and has the representation
~(X ~ = 2 f vF(X ~ X) [ - X2, X 1, 0]/(~F) dX, ( ~ = ~ ( x ) ) . (II.5) Proo]. T h a t e E W l . ~ ( V ) was n o t e d after {2.11). To prove {II.5), we shall choose the test function ~ in (3.1) to be a mollified Green f u n c t i o n for L in D. L e t # E C~(R) be a non- decreasing function such that/~(t) = 0 for t ~< 89 and/~(t) = 1 for t >~ 1, a n d define
A G L O B A L T H E O R Y OF S T E A D Y V O R T E X R I N G S I N A N I D E A L F L U I D 47
F~(X~176176176
1),
( n = l , 2 . . . . ), (II. 6 a)f
~G~(x ~ x) = r~
cos co F , (x ~ x, ~o) deo. (II.6b) For the moment we restrict x 0 to t h a t rectangular subset, say Dn, of D whose distance from aD is2/n.
Then l~n=l ~ ford(x, ~D)< 1/n,
and one can verify t h a t~Gn/ar
and~G,/az
are 0(r) for r-~0, and t h a tG~(x ~ .)
belongs toH(D).
Accordingly, we choose ~0(x) = G,(x ~ x) in (3.1) and multiply b y - s i nOO/r ~
and by cosO~ ~
to obtain, after a little manipulationfv
VPn (X ~ X ) . V[a~, =~, 0]dX = 2 f vP . (X ~ X) [ - X2, X,,
0] l(tF)dX,
where V =
{~/~X~}
and X ~ belongs to the figure of revolution Vn generated b y Dn. In- tegrating b y parts on the left-hand side, and introducing the notation, for X ~ E V,fvAF=(X~
X) a(X) dX,
O~n ( X ~ )
. F [ r . ( z ~
(X~ -z)v[F (XO, X)J [-X*'X,,O]I(tF)dX'
we see that ~ n ( X ~ ~ for X ~ Vn. I t is sufficient to prove that ~ = ~ in L~(V), and with I['H2,v denoting the norm of L2(V) for the moment, we have
Now the kernel - A F n in the definition of ~n is a mollifying (or 'averaging') kernel:
(II.6a) implies, since Ay =0, t h a t - A F ~ vanishes outside the ball {R <
1/n},
and the di- vergence theorem shows that the integral of - A F , over this ball is 1, while the integral of [AF~ I is easily bounded. I t follows from standard theory t h a t[[r and
<ll ,ll .v_v-+O
a s ~ .We easily bound
sup~]~.(xo)]
and supv [~(X ~ 1 7 6 b y means of the Schwarz inequality, since I Fn I~< 1/4~R, F = Fn for R >~ 1/n, and/(tF) = ](tF+) is in L2(V) with a norm t h a t depends only on ], V and ~ (cf. the proof of Theorem 3A). Then ]l~nn,.v_v -~0 andTHEOREM I I B.
The vector potential ,*e@~+~(~) and satisfies
(2.10a, b)pointwise.
Proo].
The bound described in Lemma I I A for supv I~n(X~ I now serves also for~(X~ and, since
y ) = X l a ~ - X , al,
the functions v2, tF and ](tF) are also bounded point-48 L . E . F R A E N K E L A N D M . S . B E R G E R
wise. I t is then easy to use the estimate (II.4) to show that ~ is uniformly Lipschitz con- tinuous on V (indeed, this would follow if/(~F) were merely in L~(V), p > 3). For consider two points X 0 and
X'=XO+h
in F, and writeR'= I X - X ' I ;
bounding IF(X', X ) - F(X ~ X)I forR>~21h I
by means of (II.4), we obtain{=*'X~176
{J:<=l~. ( R + ~ ' ) d X +o (vn<n>=l~l,
~]h{dX}
< const. {hi, (i= l, g), (II.7)
where the constant depends only on the data of the problem.
This result allows us to extend ~ to 17 as a Lipschitz continuous function, and since
o
then belongs to Wl.~ (V) N C(17), it vanishes on ~V. Then ~F is Lipschitz continuous and equal to
- 89
onr=a,
and there exists a number ~ > 0 such that forr>~a-~
we have ~F ~< 0 and hence/(~F) = 0. Therefore we can restrict X to V$ in (II.5) and use the form (II.2) of F. B u t this means t h a t the component ~(X~ say, can be regarded as the sum of three Newtonian potentialsu0 (X ~ = ~ a ~ Xl/(~F) dX,
fv x1/( )dX,
(?= 1, 2),and of a fourth function t h a t is clearly analytic and harmonic on [7 because the kernel A is. Moreover, the density function X1/(~F) of the Newtonian potentials belongs to C ~ ( ~ ) , by (2.7c) and the Lipschitz continuity of LF, and is zero on ~V$. Under these circumstan- ces it is classical that ujEC~+~(Ra), (j =0, 1, 2), that u 0 satisfies the second component of the differential equation (2.10a), and t h a t u I and u~ are harmonic functions of X~ 7 (since X (1) and X (~) are then outside V~).
THEOREM I I C .
The stream /unction ~flEC~§ and satisfies
(2.6a, b)pointwise;also, y)=O(r ~) and yJ~=O(r) /or r~O.
Proo/.
Since ~o(x)=Xla2--X20~ 1 and ~EC2+~(17), it is clear t h a t ~EC~+~(/)) and that v2=0 on r = 0 ; since also g = 0 on aV, y~ vanishes on ~D. To check the differential equa- tion (2.6a) we merely transform (2.10a), recalling that r > 0 in D. Finally,V~2= - - ~ cos0, - - ~ r '
and the condition ~ E Cl(l?) is therefore sufficient to bound
~p/r 2
and~p~/r
uniformly on D.A GLOBAL THEORY OF STEADY VORTEX RINGS I N A N I D E A L FLUID 49 Appendix III. The functional $ on the space H(II)
In this Appendix we show that the functional J is continuous on H(I-[), and hence t h a t the solution ~p for the half-plane H maximizes J over the sphere S(~]) in H(II). Here H ( H ) , S(~/) and J(u) are defined as in sections 2.4 and 3.1, but 1] replaces D; the norm
]].]] is now t h a t of H(II).
LEM~tA I I I A . Let O(x)=r189 where cfeC~~ and IIc?ll2<<.~. There exist numbers K~= K~(W, k, ~) and K = K ( W, ~7) such that
(a) f f p> l, and (b) f f r
>5 (r <'% Q-s" K (III. l a , b) Proo[. (a) L e t / ( t ) =p(Mt) p-1 for t > 0; let ~o be the corresponding solution of (2.6) for a domain D containing the support of ~; and suppose t h a t H~0H ~ =~]. ThenM -lff ",
where t2 is as in (5.12). The last integral is bounded in terms of r., z. and II~c'+]] b y (2.13), and HIF+[[~<~7 by (5.5). If [1~0[12<~7, define ~ =~11/~0/[1~0[]; then J ( ~ ) < J ( ~ ) a n d ]]~H~=~.
( b ) Again we note that e?EH(D) for som e D, and refer to L e m m a hA. Let lr {zl(r, z)eA~}; since the set (0, r)• is a subset of B~ and has y-measure 89 we deduce from (5.6) t h a t
89 <llvll,.
(111.2)Now Ir is a countable union of open disjoint intervals I , , say, of length
I I.I,
such t h a t (I) + = 0 at the end-points of each. B y a well k n o w n inequality ([9], p. 185]a~ J1,,
so t h a t
~,~" ( n J In J 7~ J Iq~, r
b y (III.2). Multiplying b y r and integrating with respect to r over {r>o}, we obtain (IIL1 b), with K =4~a/~zW 4.
T H E O R E ~ I I I B . The /unctional J is uni/ormly continuous on any bounded subset o/
H(H).
Proo/. On any bounded set B c g ( [ I ) we have [[~[[~ ~ ~ for some ~7, and it suffices to prove J uniformly continuous on the set B N C~(H), which is dense in B, L e t ~ and y~ be any two functions in B A C~r (in this proof-~p :is n o t nee6ssarily a solution); then
4 - 7 4 2 9 0 8 A c t a r n a t h e m a t i c a 132. Imprim5 lo 18 Mars 1974
50 L. E. FRAENKEL A N D M. S. BERGER
? ?
Is(v) - g(v/)[ <<. J J n { 1 + ( M e + ) m + (MaF+) ~} ](I )+ - ~F+[
< {IA, u A~lt +
M'(IIr + II~+llrm)}
I1r + -','11~,(iii.a)
b y the Schwarz inequality. Now ]A, U A~[ <<.2~/W 2 b y (5.5), and
II(I)+ll~ and II'F+II~
are bounded b y (III.la). Sine. I r 2 4 7 I < I ~ - ~ l at any point, we a~so have lie § - ~F+II~ < II~- ~ll~.<,<0> + lie § + II'~ "+ll~.<,>~>
< 2-*d II~- ~ll § 2K89 (II1.4)
where the term in r comes from integrating the estimate
f r r. 12 l ' .
~ r:Jo
while the term in 1/Q comes from (111.Xb), Choose Q=const. II~-WlI-~'~; then ( m . 3 ) and (111.4) show that
IJ(r <u]]9-Y~l[ x~a, where u = u([, W,/c, ~/).
T H ~ 0 R E ~ IIIC. The solution ~d de/ined by Lemma 5 F maximizes J(u) over the sphere S(~) in H(H).
Proo/. As in section 5.3, consider an expanding sequence {Dj} of domains tending to rI; let sj(~/)= {u EH(D~)]]Iul[ ~ =~?}; let Y~s be our maximizer of g over S,(~/); and write
supj J(~0j) = a, supuEs~) J(u) = s.
The sequence {J(~pj)} is non-decreasing because v/l, extended to be zero outside Dj, belongs to S~+1(~); therefore J(yzj)~a as j-~ ~ . Since vdj~-*v ? in C1+~(~), we know t h a t J(~os~)-+J(y) ).
Accordingly, J ( v / ) = a and a ~< s because ~0 E S(r/).
On the other hand, there exists a sequence {an} in SO?) such t h a t J(un)-~s as n-~ co, and because of Theorem I I I B, we can approximate each u , b y a function ~0n E S(~/) fl C~ (II) such t h a t J(qDn)~s. B u t each q~ESj(~) for some j = j ( n ) ; therefore J(~=)~<a for each n, and if a < s , then J(q),)+-~s. Hence a = s .
R e f e r e n c e s
[1]. AOMON, S., DOUOLIS, A. & NIRENBERO, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I.
Comm. Pure Appl. Math., 12 (1959), 623-727.
A GLOBAL THEORY OF STEADY VORTEX RINGS I n AN IDEAL FLUID 51
[11]. HELMHOLTZ, H., 0-ber Integrale der hydrodynamischen Gleiehungen, welche d e n Wirbel- wegungen entsprechen. J. Reine Angew. Math, 55 (1858), 25-55.
[12]. HIcKs, W. M., Researches on the theory of vortex r i n g s - - p a r t I I . Philos. Trans. Roy. Soc.
London, A 176 (1885), 725-780.
[13]. HILL, M. J. M., On a spherical vortex. Philos. Trans. Roy. Soe. London, A 185 (1894), 213- 245.
[14]. LADYZttENSKAYA, O. A. & URAL'TSEVA, N. N., Linear and quasilincar elliptic equations.
Academic Press, 1968. International Congress Appl. Mech., University o] Brussels, (1957) 1, 173-176.
[19]. MOSTOW, G. D., Quasi-conformal mappings in n-space a n d the rigidity of hyperbolic space forms. Inst. des Hautcs I~tudcs Sci. Publ. Math., 34 (1968), 53-104.
[20]. NIRENBERG, L., On elliptic partial differential equations. Ann. Seuola Norm. Sup. Pisa, (3) 13 (1959), 115-162.
[21]. ~ORBV~Y, J., A steady vortex ring close to Hill's spherical vortex. Proc. Cambridge Philos.
Soc., 72 (1972), 253-284.
[22]. - - A family of steady vortex rings. J. Fluid Mech., 57 (1973), 417-431.
[23]. PbLYA, G. & SZEGb, G., Isoperimetric inequalities in mathematical physics. Princeton Uni- versity Press, 1951.