Turbulence and Navier Stokes Equations Proceedings of the Conference
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565 Turbulence and Navier Stokes Equations Proceedings of the Conference Held at the University of Paris-Sud Orsay June 12-13 1975
Edited by Roger T em am
Springer-Verlag Berlin· Heidelberg· New York 1976
Editor Roger Temam Mathematique Batiment 425 Universite de Paris-Sud Centre d'Orsay 91405 Orsay/France
Library of Con gress Cata log ing in Publication Data
V.ain entry unde r t i tle: Turbulence and Navi er Stok.es equations .
(Leoture notes in mathematics ; 565 ) English and t rench. l.
r:'urbule::lce--Congresees ,
2.
Nayier- Stokes
equations--Congresses. L 'remam) Roger. II. Lecture notes in mathematics ('Berlin) ; 565.
Ql\3 .128 no . 565
[ C) ; we clenote by q; the mapping 1
CJ
From the relations
ow n+1 n " n+1 --;;r+uov(l) One deduces, using once again lemma
and
p+1 oW P ~ p+1 __ Wp+•1"" up --;;r+Uoy(l)
and the estimate (3], the inequality
8
(c3
a nd
·~
4
It\ ,;; T'f_ e)
are unif'ormly bounded f'or
From (21) one deduces easily the inequality
Therefore one can
u~c
7l1 in 7l1 •
!lim is a s tric t contraction or
m large enough
Theref'ore for
u
a fi xed point theore m to prove tha t there exis ts
such that :
u E C(-T*+ e , T*- € ; c •~) solution or the equations 1
()"
(23) V/\.u
= w, ~~+up w =
ll.)j'
u • u( x,O) = u0 (x)
From the r elations (23) one deduces that
VI\.(~~+ uvu ) = 0 and therefore , that
(lu
+ u.v u = - ilp (c . r . M. ZERNER [13]) • This complete the proof' of the existence of the solution for theorem 1 ; (20) gives the estimate (5) when n goes to inf'inity.
~
The uniqueness i s easy a nd left to the reader • To consider the case or an unbounded domain we will need
~:
0
Assume that
3
is an open s et or JR
with smoot h boundary ,
containing the exterior or a ball then there exists a bilinear continuous maps 0 1 (u,v) ~ F(u,v) define d on Ca(O) x c • a(O) with value in C ' a(O) with the fol l owing propcrtico eair (u, v) E c 1,a X G~ F(u,v) 1 (ii) For any pair (u,v) E c •a x 0 1 , a
(i)
For
an~
0
(iii) I f ler eguation
C1 , a
u E C(-T'\ T*
()u ()t + u
\1
u
=-
'Vp
'
CY
n L2)
2 ('ilp E L )
is a !ilradient • ~
is a rinite one has
v • ( v.vu-F (u,v)) =0 •
ener!il~
solution of t he Eu-
- ilp = F(u , u)
.
.!.:.'!!!: :
For ths c;ake of simplicity and to emphasize the importance of tho behaviour a t infinity or u 1 we will give the proof or this lemma only in the case 3 0 = R ' when an I (~} the proof is similar , but relies on the analy~~s of t he Green function of the exte rior Neumann problem.
Taking the divergence of both sides
or the Euler equation one obtains
(24)
And if t he right ha nd side of
(24)
solution (up to a constant) or (24)
(25)
p
i s bounded (in is given by :
1 L (tRn)ror instance) the only
9
l =
we put
= K1 (,)
K(,)
1- 9 and write
a =1 in a neighbourhood of zero ,
e E L(A0)
Now we introduce a smooth function
+ K2 ( . ) (K 1 = eK
1
~
= SK
and we pu t
:
(26)
-2... (vK 1 ( .))
is a function with !ZOmpact support , smooLh
axi
0
such that, for
I U(t)l 1 I U(t)l 5
(22)
n> 1
I U(
