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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Boundary layers for the Navier-Stokes equations : asymptotic
analysis.
M. Hamouda ♯ and R. Temam ∗♯
∗ Laboratoire d’Analyse Numérique, Université de Paris–Sud, Orsay, France.
♯ The Institute for Scientific Computing and Applied Mathematics,
Indiana University, Bloomington, IN, USA.
Abstract
In this talk, we consider the asymptotic analysis of the solutions of the Navier-
Stokes problem, when the viscosity goes to zero; we consider the flow in a channel of
R 3 , in the non-characteristic boundary case. More precisely, a complete asymptotic
expansion, at all orders, is given in the linear case. For the full nonlinear Navier-Stokes
solution, we give a convergence theorem up to order 1, thus improving and simplifying
the results of [TW].
MSC : 76D05, 76D10, 35C20.
We consider the Navier-Stokes equations in a channel Ω∞ = R 2 × (0, h) with a permeable
boundary, making the boundaries z = 0, h, non-characteristic. More precisely we
have
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which is equivalent to
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∂u ε
∂t − ε∆uε + (u ε .∇) u ε + ∇p ε = f, in Ω∞,
div u ε = 0, in Ω∞,
u ε = (0, 0, −U), on Γ∞,
u ε is periodic in the x and y directions with periods L1, L2,
u ε | t=0 = u0,
∂v ε
∂t − ε∆vε − UD3v ε + (v ε .∇) v ε + ∇p ε = f, in Ω∞,
div v ε = 0, in Ω∞,
v ε = 0, on Γ∞,
v ε is periodic in the x and y directions with periods L1, L2,
v ε | t=0 = v0.
Here Γ∞ = ∂Ω∞ = R 2 × {0, h} and we introduce also Ω and Γ :
Ω = (0, L1) × (0, L2) × (0, h), Γ = (0, L1) × (0, L2) × {0, h}.
Speaker: HAMOUDA, M. 88 BAIL 2006
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