146
(v,divv), and (Vh, v)
The last term will be transformed using the mass equation as following; we can write:
(Vh, v) = (~h'Vh) = (Vlogh,vh) = -(logh,div(vh))
= ~(hlogh,J)+ l(hlog h)G+ L hdivw
As divw E HI (0, T; Loo(n)2), the term If) hdivw can be easily bounded. To turn the
difficulty linked to the term (v 2 , div v), we build a stability space as following.
Using Gagliardo Nirenberg's inequality, we can majorize the term (v 2 , div v),
As we want KIIvII~ - (v 2 ,divv) 2:: 0, we are going to choose data as Kllvll~l(v 2 ,divv)l2:: 0, i.e.
K - Cllv(t)11 2:: 0
'It
Moreover, It is easy to check the first part of the estimate (I.3.1b) by studying the
x 1---4 xlogx function, x 2:: O.
Finally, thanks to the small data (I.2.3b), We easily prove that we have Ilv(t)11 < ~
and then we can obtain the estimation (I.3.1b) and the result (I.3.1c).
To pass to the limit in the mass equation, we need the following result.
1.4. Approximated solutions
Let {VI, ... , vn, ... } be a basis of V, Vn belongs to H3(n)2, let Vn be the set of the
linear combination of the n first elements of the basis. We are looking for (V., hn), where Vn
n
is under the form Vn = L ai(t) Vi(X), solution of the following variational problem (Vn) :
i=!
Find Vn E L2( 0, T; Vn) n L oo (0, T; L2(n)2) and hn E C I (Q) such as
(vn,,,v) +Aa(vn, v) - ~(v~,divv) - (vn wn,divv)
+ (curivna(vn),v) + (curlvn a(wn),v) +D((vn+wn)lvn+wnl,v)
= (hn' div II) + (t, v) - (wn,t, v) + ~(w~, divv) - Aa(wn' v) 'Iv E Vn
hn,t + div (vn hn) + div (wn hn) = 0
hn = Pn E C~CB-)
vn(t = 0) = VOn E Vn
hn(t = 0) = hon E C:(rl)
(v,divv), and (Vh, v)
The last term will be transformed using the mass equation as following; we can write:
(Vh, v) = (~h'Vh) = (Vlogh,vh) = -(logh,div(vh))
= ~(hlogh,J)+ l(hlog h)G+ L hdivw
As divw E HI (0, T; Loo(n)2), the term If) hdivw can be easily bounded. To turn the
difficulty linked to the term (v 2 , div v), we build a stability space as following.
Using Gagliardo Nirenberg's inequality, we can majorize the term (v 2 , div v),
As we want KIIvII~ - (v 2 ,divv) 2:: 0, we are going to choose data as Kllvll~l(v 2 ,divv)l2:: 0, i.e.
K - Cllv(t)11 2:: 0
'It
Moreover, It is easy to check the first part of the estimate (I.3.1b) by studying the
x 1---4 xlogx function, x 2:: O.
Finally, thanks to the small data (I.2.3b), We easily prove that we have Ilv(t)11 < ~
and then we can obtain the estimation (I.3.1b) and the result (I.3.1c).
To pass to the limit in the mass equation, we need the following result.
1.4. Approximated solutions
Let {VI, ... , vn, ... } be a basis of V, Vn belongs to H3(n)2, let Vn be the set of the
linear combination of the n first elements of the basis. We are looking for (V., hn), where Vn
n
is under the form Vn = L ai(t) Vi(X), solution of the following variational problem (Vn) :
i=!
Find Vn E L2( 0, T; Vn) n L oo (0, T; L2(n)2) and hn E C I (Q) such as
(vn,,,v) +Aa(vn, v) - ~(v~,divv) - (vn wn,divv)
+ (curivna(vn),v) + (curlvn a(wn),v) +D((vn+wn)lvn+wnl,v)
= (hn' div II) + (t, v) - (wn,t, v) + ~(w~, divv) - Aa(wn' v) 'Iv E Vn
hn,t + div (vn hn) + div (wn hn) = 0
hn = Pn E C~CB-)
vn(t = 0) = VOn E Vn
hn(t = 0) = hon E C:(rl)