ABSTRACT
R e m a r k 1 -1 . In the special case q = 0 ? this result has been trea ted in (J .L .Lions[l]) . The
case where q is in depen den t o f t , and q G L p(£l), (p > n i f n > 2 a nd p > 2 i f n = 1), was p ro v e d b y V .K om orn ik (V .K om orn ik [l] ) , and the case q G L °°(]0 , T [ x i î ) w as p ro v e d b y E .Zuazua (V .K om orn ikf l] ) . |
T h e follow ing p ro p o s itio n holds
PR O PO SITIO N 1-1. Let T > 0 be g iven , T big enough. There exists C { T ) such tha t
(1” 3) l|y°llHoi(Q) + I ^WlHQ) — ^(^)lb lli~ (0,T ;L2(^))- C ( T ) is a constan t which depends on T . |
P r o o f o f p r o p o s i t i o n 1-1: To p ro v e (1-3) we s ta r t by u sing th e m u ltip lie r techn iques. W e d en o te b y m
th e v ec to r field g iven by m (x ) = x — £o, fixed in Rn . W e m u ltip ly (1-1) by m .V y , we have
(1 -4 ) I y" m .V y d x d t — I A y m .V y d x d t + / q y m .V y d x d t = 0. J q J q J q
Now, using th e G reen fo rm u la an d b o u n d a ry cond itions, we get
(1 -5 ) / y " m .V y d x d t = i y 'm .V y d x + ? / (y ) 2d x d t , J q J q o 2 Jq
J q dxdt.
