143
Classically, we obtain the weak formulation (V) associated to the (P) problem:
(V,,'P) +Aa(v,'P) - ~(V2,diVy) - (vw,div'P) + (curl VQ(v),y)
+ (curl va(w), 'P) - (h, div'P) = (f,y 1+ (D(v + w) Iv + wi, y)
1
.
+ 2(w 2 ,dlV'P) -Aa(w,y) - (w,,'P) V'P E vn H 3 (D)
where we have denoted by a( u, 'P) = [ div u div 'P + [ curl u curl 'P.
Jf!
In
The space V on which we are going to work is the following:
Mass equation will be resolved into [l(O,T; W-I,I(D)) with its boundary condition:
h, + div(vh) + div(wh) = a
hl~- = 11
And we must add the initial conditions :
v(t = O,x) = uo(x) - w(t = a,x) = vo(x) p.p into D
h(t = a,x) = ho(x) 2> a p.p into D
V is equipped with the norm II'PII~ = 11'P11~2 + Iidiv 'P11~2 + Ilcurl 'PII~"
Note that if I = aD is of class el,l or a convex polygon, this space V is algebrically
and topologically included in HI (Il) with equivalent norms.
1.2.2 Theorem
Hypothesis.-. Let A, B, e, ei, e i , K, Kj, .A and 8 be constant as :
ej is the constant defined by
C'j Ilull~2(f!) :s: Ilull~3(H)
B = 2inf(A C D) - .A - Z - 2D - A
, I
2
e is Gagliardo Nirenberg's constant defined by :
Il ulli4(H) :s: e Iluli v lIull
C and KI are the constants defined by :
(I.2.2a)
(I.2.2b)
(I.2.2c)
Ilull:4(f!) :s: ei Ilull~J(f!)
(I.2.2d)
KI = [s~p (ft! [Ilholl - ,h+ GfJ 1
(/.2.2e)
K = B -- 2ei (1Iwll~4(H) + 2D IlwllL'(f!))' B > 2C (1Iwll~4(H) + 2D Ilwllu(H) )(I.2.2j)
(I.2.2f:)