BAIL 2006 Book of Abstracts - Institut für Numerische und ...

M. HAMOUDA, R. TEMAM: Boundary layers for the Navier-Stokes equations :
asymptotic analysis
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We assume that f and u0 are given functions as regular as necessary in the channel
Ω∞, and that U is a given constant; at the price of long technicalities, we can also consider
the case where U is nonconstant everywhere.
Theorem 0.1 For each N ≥ 1, there exists C > 0 and for all k ∈ [0, N] an explicit given
function θ k,ε such that :
�v ε N�
− ε k (v k + θ k,ε �L∞ (0,T ;L2 (Ω)) ≤ C ε N+1 , (0.3)
�v ε −
k=0
N�
k=0
ε k (v k + θ k,ε � L 2 (0,T ;H 1 (Ω)) ≤ C ε N+1/2 , (0.4)
where L 2 (Ω) = (L 2 (Ω)) 3 , H 1 (Ω) = (H 1 (Ω)) 3 , and C denotes a constant which depends
on the data (and N) but not on ε. Here v ε denotes the solution of the linearized problem
of (0.2) and v k the ones of the limit problems (ε = 0) at all orders.
The second result concerns the solution of the full nonlinear problem (0.2) :
Theorem 0.2 For v ε solution of the Navier-Stokes problem (0.2), there exist a time T∗ >
0, correctors function θ 0,ε and θ 1,ε explicitly given, and a constant κ > 0 depending on the
data but not on ε, such that :
�v ε − (v 0 + θ 0,ε ) − ε(v 1 + θ 1,ε )� L ∞ (0,T∗; L 2 (Ω)) ≤ κ ε 2 , (0.5)
�v ε − (v 0 + θ 0,ε ) − ε(v 1 + θ 1,ε )� L 2 (0,T∗; H 1 (Ω)) ≤ κ ε 3/2 . (0.6)
We recall here the limit problem which is the Euler problem and its solution v0 satisfies :
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∂v
⎪⎨
⎪⎩
0
∂t − UD3v 0 + (v 0 .∇) v 0 + ∇p 0 = f, in Ω∞,
div v 0 = 0, in Ω∞,
v 0 3 = 0, on Γ0,
v 0 = 0, on Γh,
v 0 (0.7)
is periodic in the x and y, directions with periods L1, L2.
It is obvious that we can not expect a convergence result between v ε and v 0 (in H 1 (Ω) for
example) and more precisely we are leading with a boundary layer problem close to some
parts of the boundary Γ. This is the main object of our work.
At the following order v 1 is solution of
⎧
⎪⎨
⎪⎩
∂v 1
∂t − UD3v 1 + (v 1 .∇) v 0 + (v 0 .∇) v 1 + ∇p 1
= − ∂ϕ0
∂t + UD3ϕ 0 − (v 0 .∇) ϕ 0 − (ϕ 0 .∇) v 0 + ∆v 0 , in Ω∞,
div v 1 = 0, in Ω∞,
v 1 3 = 0, on Γ0,
v 1 = 0, on Γh,
v 1 is periodic in the x and y, directions with periods L1, L2.
The function ϕ 0 is a known function at this level.
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Speaker: HAMOUDA, M. 89 BAIL 2006
(0.8)
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