On the Uniqueness of Viscous Fluid Motions SALVATORERIONERO & GIOVANNI P. GALDI Communicated by C. TRUESDELL

By using a weighted L2-norm, a new method for studying uniqueness of viscous flows in unbounded regions is given. The method allows improvement of earlier theorems.

1. Introduction D. GRAFFI,in a celebrated paper of 1960 [1], presented a method for proving uniqueness of flows of incompressible viscous fluids filling the exterior of a bounded region. Without assuming prescribed data at infinity, he proved uniqueness for classical solutions of the Navier-Stokes equations under the following assumptions: a) The velocity field v and its gradient grad v are uniformly bounded in the time interval [0, T] (T positive and arbitrarily fixed). b) The pressure p tends to a limit p at infinity, and

[p-~l
r2=x2+y2+z 2.

Subsequently methods of this kind have been reconsidered by many authors [-2], [3], [4] ; they essentially improved assumption b) in the more general context of uniqueness forward and backward in time. D.E. EDMUNDS [2] assumes [ p - ~ [ < O ( r -~-~) (e>0), while J.R. CANNON & G.H. KNIGHTLY [3] assume [p -- ~[ <=O(r - ~) only 1 Recently we have suggested a new method, quite different from GRAFFfS, for establishing uniqueness of flows, of an incompressible viscous fluid in unbounded domains [5]. Our method, founded upon the use of a weighted L2-norm, allowed us to prove for a certain choice of the weight function a uniqueness theorem even when grad v is not bounded. As regards the pressure, we considered two main cases: either p tends to p at infinity with the order assumed in [2] while v remains at least bounded, or p may be unbounded but v converges sufficiently fast at infinity 2. i Weaker assumptions on p than [2] and [3] have been made by P. MURATORI in [4], but in the particular case of the uniqueness of the rest solution. 2 Similar assumptions for uniqueness may be found in [6].

296

S. RIONERO& G. P. GALDI

The purpose of the present paper is to show how our method, by choosing a "better" weight function, permits us to improve the hypotheses made in [3] on the order of LP-Pl. It is needless to say that, as in [5], gradv may be unbounded. Moreover, in the case when p may be unbounded, the order of convergence of v is lower than the one assumed in [5]. Specifically, let Q denote the exterior of a fixed region f2o 3, and let ~ (i = 1, 2) be the classes of the classical solutions of the Navier-Stokes equations such that (for both classes) 1) v and its first derivatives are continuous in f2r=f2 x [0, T] with respect to all variables, and v has second spatial derivatives which are generally continuous; moreover, v is bounded in 12, uniformly in [0, T] and prescribed on such a boundary 0Oo as there may be. 2) p is continuous in f2r, and grad p is generally continuous. 3) gradv may be unbounded, but ~ k , M , ? > 0 : Lgradvl
r>?,

and, moreover, 4) Ip-pl<2(logr)+/r +,

t and r > ? (in i t ) , 2>0, a ~ [ 0 ,3[

5) Lp-pllv-Vl
l>0, a~[0, 89 and r > ? (in J2). +

Let Vo be any prescribed (smooth enough) function in f2 such that div vo = 0. The following uniqueness theorem holds:

If in ~ (i = 1, 2) there is a flow {v, p} with initial data v(P, O)=vo(P )

P6f2,

it is unique. 5

2. Proof of the Uniqueness Theorem For the sake of (formal) simplicity, in this section we shall prove the uniqueness theorem in the case when f2o contains a sphere of unit radius. However, the case f2o = ~ presents no conceptual difficulty and will be briefly treated in the next section. Let {v, p}, {v + u, p + A p} be two elements of ~ (i = 1, 2) with the same initial data. Then the difference motion obeys the equations [5]

0u (1)

~-(v+u). grad u = - u . grad v - l g r a d A p + v A z u , 0t p div u = 0,

3 f2~ may be empty (see Section 3). + Condition 5) is weaker than I p - p l Iv-Vl ~ l r - 2 - ~ ( e > 0 ) which was assumed for the analogous class in [5]. 5 Of course, in the class J2 we compare flows for which pressures, as well as velocities, have the same behavior at infinity.

Uniqueness of Viscous Fluid Motions

297

p being the (constant) mass density and v the kinematical viscosity. Solutions of the equations (1) are subject to

(2)

u (P, 0) = 0,

P e t2

(initial data)

u(P, t)=0,

(P, t)~O~2o x [0, T] (boundary data).

We intend to show that (3)

u(P, t ) = 0

V (P, t)E~T .6

If, per absurdum, (3) did not hold, the continuity of u implies that there would be positive numbers 6, ~ and e such that

~ u2 (p, z) d a~> e

(4)

a6

where ao is the intersection with O of a sphere of radius 6 whose center is in O o. If we show that a non-negative function f(a) exists with l i m f ( a ) = 0 such that ~0

~ uz(P, z) da~ < f (~),

(5)

aO

condition (4) is, of course, impossible; in fact, it suffices to choose 7~]0, 1] in such a way that f(~)<~. To this end, for m > 4 and a~[0, 89 let us put

(6)

fl(a)=e"OTek'T(k2~2 +k3 ~1-2~) f2(oO=e'~TekW(qa+c2 al-2~)

(in ~r (in J2)

where k~ = s u p

mlul, 3mlvl, 2~' re(m+2) +m~=k+m/p, P

m

(7) k3-

4~22m 2p (2a)2~e-2~

c 1 =m/p sup ~ lAp[ [uldf2, t

c2 -

4 rclm

.Qr

(2o-)2~ e-2~

P and Or is the intersection of (2 with the sphere of radius ~ and center in ~0. By multiplying equation (1)1 by gu, where g is any twice differentiable scalar function depending on P alone, we easily get 89~-t gu 2 = - 8 9 grad(guZ)+ 89 9grad g)u 2 - 8 9 (8)

grad u 2

-gu-gradv.u-g/pgraddp.u+vgA2u.u, div u = 0.

6 The relation (3) obviously implies Ap=Oin J1- This is true also in J2 when p is not convergent at infinity, on condition that p is ascribed in at least one point of O.

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S. RIONERO& G. P. GALDI

We shall assume in the sequel g(r)=e . . . .

(m>4)

with y such that fi(y)
grad u 2 = 89

grad g) u 2 - 89div (gu 2 u),

ii) - g u . grad v. u = - d i v [g(u. v)u] + ( u . v)u. g r a d g + g u , gradu .v, 11292 iii) gu 9grad u 9v < g ~ - + ( ~ g / 2 )

grad u 9grad u

(~ >0),

vu 2 V iv) vgu. Aau=~-div[g grad u 2 - u 2 grad g] vg grad u" grad u+2~ - agg, v) - (g/p) grad A p. u = (A p/p) grad g. u - div (g A pu/p). Now, denoting by f2R the intersection of f2 with a sphere of radius R and, moreover, f2Rn0f20+r let us integrate formula (8)1 over f2R. Exploiting i)-v), we obtain ~ { 89 i ddt ~,~ gu2dQ<= ~,~

(9)

.v)u.gradg+ 89

2

/,/2/.)2 Ap Yu 2 + g-~-+~grad g 9u + ~ - d2g

+ (4/2 - v) g grad u" grad u} df~ + ~

{v

~grad(u g ) - v u 2 g r a d g

By the properties of the function e . . . . , we get (10)

Igradgl < m T r ~ - l g , A2g <=(2mr~- 2 + m2r2~- 2) g - 2myr~- l g.

Thus, since ~2o contains the unit sphere, the relations (10) give (*)

Igrad g l --
Taking into account the assumptions made on the elements of ~ (i= 1, 2) and choosing ~=2v, we obtain from equation (9) in the limit as R--, + oo

(11) + rn7

r 7-1

P

} g]Apllu]+rn(m+2)gu 2 dr2.

Uniqueness of Viscous Fluid Motions

299

Putting

=~ ~ gu 2dO, f~

by (7)1 and (11) we get

d m7 d-~g
(12)

In the class J l , from Cauchy's inequality

7rY-llApllul< 89

/'2'1

2

+ 89 2

the inequality (12) gives

d~ ~ " m72 ~ rz~-1~ gap 2 dO dt 5kl +-~p a

(13)

where k 1 is given by (7)1. Now, by assumption,

Ap2<22(lOgr)2~, = -

ae[0, ~-[, 1 r>L

r

so it follows from (13) that

d~
le+-m72-Sl l 2

(14)

mY 2

jo r2(~-t)glAp[2

+

4, we have +~

+ ~ e- ~ (log r) 2a , ~ (2 0-)2a 1 ar~= ~

r2r-le . . . . (l~ r)2"dr < 5

1

1

e - 2 cr

,

rY+

since e-'~ (log r)2-< (20")2o. e-2~22 As a consequence, the differential inequality (14) gives

dS <_kl ~ + k2 72 + k3 yl- 2~=kt g + Fl (7)

(15)

at-

with k 2 and k 3 as in (7)2 and (7)3. Integrating (15) from 0 to z and recalling that ~ (0)= 0, by a well known comparison theorem [7] we get (16)

#(v)= 89

k'r

12

Now

#(*)> S u2( P, ~c)e-"r'daa>__e-"~ y u2(p, r)d0-a t~6

~6

(T>z).

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S. RIONERO &; G. P. GALDI

and so by (16) we obtain (17)

~ u 2 (P, z) daa < e ma Te kl r F1 (7) = f l (7), ~76

and uniqueness is demonstrated in the class J1. In an analogous way, under assumption 5), (12) gives my +my dg <_kg+__~[AplluldO [. [Aplluld~ dt P a, P ~-~ <=kgq-c I y-t-c 2 y t - 2c' : k o~ nt- F2(Y)

where cl and c 2 are given by (7)4 and (7)5. Thus proceeding as in the previous case, we obtain

5 u2(p, z) da~
(18)

O'6

The theorem is therefore completely proved. 3. T h e c a s e I2 o =

The case Qo = ~ may be treated in the following way. For the function g(r) we choose "r" (gl(r)= e . . . . , r>l, y>0 gt I=].g2(r)=ar2+br+c, re[0, 1], where e-m

a=-~-(my-m72

+ m2y2),

b=-e-m(2mT-my c = e-m(1 +-~m y - 8 9

2+m2y2), 2 +-~m 2 72).

It is easily checked that g(r) is twice differentiable and that, moreover, for m > 4 g2(r) satisfies a) g2(r)>0, b) ]grad g21__
References 1. D. GRAFFI, Sul teorema di unicith nella dinamica dei fluidi, Ann. Mat. Pura App. 50 (1960) 379-388. 2. D. EDMUNDS,On the uniqueness of viscous flows, Arch. Rational Mech. Anal. 14 (1963) 171-176.

Uniqueness of Viscous Fluid Motions

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3. J.R. CANNON & G.H. KNIGHTLY,Some continuous dependence theorems for viscous fluid motion, SIAM J. App. Math. 18 0970) 62%640. 4. P. MURATORI, Teoremi di unicit/t per un problema relativo alle equazioni di Navier-Stokes, Bollettino U.M.I. 4 (1971) 592-613. 5. G.P. GALDI tfr RIONERO, A uniqueness theorem for hydrodynamic flows in unbounded domains, Ann. Mat. Pura App., in press. 6. J.R. CANNON ~s G.I-I. KNIGHTLY,A note on the Cauchy problem for the Navier-Stokes equations, SIAM J. App. Math. 18 (1970) 641-644. 7. V. LAKSHMIKANTHAM & S. LEELA, Differential and integral inequalities, vol. 1, Academic Press (1969). Istituto di Matematica dell' Universith di Napoli, Via Mezzocannone 8 80131 Napoli

(Received April 7, 1976)