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1.5. A trace theorem
Notations .-. We note:
L~iv(D) ~ {u E LI(D)", divu E LI(D)}
equipped with the graph-norm IluliLI. :
d"
IluIIL~;v ~lluIILI(fl)n + lidiv UIiU(fl)
We also denote by L~,div' the closure of D(D)" in L~iv(D).
Theorem .-. Let D be a bounded open subset of Rn, (n E N*) and, its Lipschitzcontinuous boundary. We denote by 1t the normal extern unit to D on ,.
Then we have :
i) The space D(n)n is dense in L~iv(D).
ii) The map, : u ~ u.1t 1""1 defined onto D(n)n can be extended to a map, always
denoted " linear continuous from L~iv(D) into [WI,COb)t
iii) The kernel of , is the space Lb,div'
To prove this theorem, we use the following result :
If F E (L~iv(D)t Then there existsfo E UX'(Dt andJi E LOO(D) as :
1.6. Passage to the limit
/.6.1 Mass Equation
We present three lemma that allow us to pass to the limit in the mass equation.
Lemma 3 .-. Let hn and Vn be two sequences verifying:
Then :
Vn bounded into L2 (0, T; HI (D)2)
hn bounded into L 00 (0, T; LI (D))
hn loghn bounded into L oo (0, T; LI (D))
We can extract from Vn and hn subsequences such as
~ hn B dxdt --+ ~ h Bdxdt for all BE LI (0, T; LOO(D))
Vn hn bounded into L2 (0, T; Ll (D l)
vnhn ~ "'I into LI(Q)
(1.6.1a)
(1.6.1b)
(J.6.1c)
(1.6.1d)
(J.6.le)
(1.6. If)