151
Proof .-. Those results have been proved by Orenga (1995). We are going to give the
main stages of the proof.
• It was shown with the Dunford-Pettis theorem that if the sequences hn and h.log h.
are bounded into L' (Q) then the sequence hn is in a weak compact set of e ( Q). Then,
there exists a subsequence of h. always denoted hn such as : hn ~ h into L' (Q) and
it is easy to verify that J Q hnE> dxdt -+ J Q hE> dxdt for all E> E L' (0, T; L oo(fl)).
• Finally, the Dunford-Pettis theorem and the following Trudinger-Moser inequalities
lead us to conclude the proof of this lemma.
o Let u E HI (fl)2, fl C R2. There exists a constant a positive such as
f exp (a lu 12 2 ) dx ::; C
in
Iluliv
o Let u E HI (fl)2, then for each k E R, there exists K such as : exp (k~) ::; K(k).
o For each a E R+, b E R+ and C E R+, we have : a b ::; C( alog a + ~exp ( ~)).
Lemma 4 .-. Let hn and Vn be two sequences verifying,'
Then
Vn E L2(0,T;Hm(flf) m?: 3; hn E Loo(O, T;LI(fl)) hn ?: 0
hn and hn log hn bounded into L oo (0, T; L' (fl))
hn,t + div(vn hn) + div (wn hn) = 0
vn~vintoL2(0,T;HI(fl)2); hn~hintoL2(0,T;e(fl))
Vn hn ~ K, into LI (0, T;L'(fl)2)
K, = vh
(I.6.2a)
(I.6.2b)
(1.6.2c)
(1.6.2d)
(I.6.2e)
(1.6.2/)
Proof .-. The result (1.6.2/) can be found in Orenga (1995). We are not going to prove
this results but we will give the main stages of the proof.
Let us consider a regularizing sequence p" and the sequence v~ = Vn * pO and we must
show:
v~hno~oovnhn intoL'(0,T;[l(fl)2) uniformlyinn
v~ hn -+ VO h into 1)1 (Q) when n -+ 00
• To show the first relation, we consider the setA = {(f,x) E Q / Iv~ -vnl > T, T > O}
and we majorize hn Iv~ - vnl onto A and its complementary. This is possible using the
following Trudinger-Moser inequalities. The estimates are independent of n.
Proof .-. Those results have been proved by Orenga (1995). We are going to give the
main stages of the proof.
• It was shown with the Dunford-Pettis theorem that if the sequences hn and h.log h.
are bounded into L' (Q) then the sequence hn is in a weak compact set of e ( Q). Then,
there exists a subsequence of h. always denoted hn such as : hn ~ h into L' (Q) and
it is easy to verify that J Q hnE> dxdt -+ J Q hE> dxdt for all E> E L' (0, T; L oo(fl)).
• Finally, the Dunford-Pettis theorem and the following Trudinger-Moser inequalities
lead us to conclude the proof of this lemma.
o Let u E HI (fl)2, fl C R2. There exists a constant a positive such as
f exp (a lu 12 2 ) dx ::; C
in
Iluliv
o Let u E HI (fl)2, then for each k E R, there exists K such as : exp (k~) ::; K(k).
o For each a E R+, b E R+ and C E R+, we have : a b ::; C( alog a + ~exp ( ~)).
Lemma 4 .-. Let hn and Vn be two sequences verifying,'
Then
Vn E L2(0,T;Hm(flf) m?: 3; hn E Loo(O, T;LI(fl)) hn ?: 0
hn and hn log hn bounded into L oo (0, T; L' (fl))
hn,t + div(vn hn) + div (wn hn) = 0
vn~vintoL2(0,T;HI(fl)2); hn~hintoL2(0,T;e(fl))
Vn hn ~ K, into LI (0, T;L'(fl)2)
K, = vh
(I.6.2a)
(I.6.2b)
(1.6.2c)
(1.6.2d)
(I.6.2e)
(1.6.2/)
Proof .-. The result (1.6.2/) can be found in Orenga (1995). We are not going to prove
this results but we will give the main stages of the proof.
Let us consider a regularizing sequence p" and the sequence v~ = Vn * pO and we must
show:
v~hno~oovnhn intoL'(0,T;[l(fl)2) uniformlyinn
v~ hn -+ VO h into 1)1 (Q) when n -+ 00
• To show the first relation, we consider the setA = {(f,x) E Q / Iv~ -vnl > T, T > O}
and we majorize hn Iv~ - vnl onto A and its complementary. This is possible using the
following Trudinger-Moser inequalities. The estimates are independent of n.