Partial Differential Equations
MATH 612
A. Bonito
due on May 7 (in class)
Spring 2013
Take home part of the FINAL
Exercise 1
60%
Let Ω ⊂ R3 be open, bounded, connected and with smooth boundary. Define
V := v ∈ H01 (Ω)3 : div(v) = 0 ⊂ H01 (Ω)3 ,
where we equip H01 (Ω)3 with the standard semi-norm. Consider the following formulation of the
steady Navier-Stokes system : Find u ∈ V such that
Z
Z
ν
∇u · ∇v + (u · ∇)u · v = hf, vi,
∀v ∈ V.
(1)
Ω
Ω
Here ν > 0 is the fluid viscosity, f ∈ H −1 (Ω)3 is a given body force and
Z
3 Z
X
(u · ∇)u · v =
uj ∂j ui vi
Ω
i,j=1
Ω
with u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ).
We propose to use the following fixed point theorem to show existence of a weak solution : Let
H be an Hilbert space of finite dimension with norm |.|H and scalar product (., .)H . Let F be a
continuous mapping from H into H satisfying : There exists α > 0 such that for all f ∈ H with
|f |H = α we have
(F (f ), f )H > 0,
Then F has at least one zero in the ball centered at 0 and of radius α.
1. Show that (z · ∇)u ∈ L3/2 (Ω)3 whenever z, u ∈ H01 (Ω)3 and deduce that for u, v ∈ H01 (Ω)3
(u · ∇)u · v ∈ L1 (Ω)
which ensures that the nonlinear term makes sense.
2. Show that V is separable by showing that V is a closed subset of H01 (Ω).
3. Since V is separable, it has a countable basis {wi }∞
i=1 such that for all v ∈ V and all > 0,
there exists N ∈ N and vN ∈ span(w1 , ..., wN ) such that
|v − vN |H 1 6 .
Set
Vm = span(w1 , ..., wm )
and define the mapping Φm
Vm → Vm
v 7→ Φm (v) = φm
where φm ∈ Vm satisfies
Z
Z
Z
∇φm · ∇wk = ν
∇v · ∇wk + (v · ∇)v · wk − hf, wk i
Ω
Ω
Ω
for all 1 6 k 6 m. Show that Φm is well defined, i.e. for each v ∈ Vm there exists a unique
φm ∈ V m .
4. Apply the given fixed point theorem to deduce the existence of at least one v ∈ Vm with
|v|H 1 (Ω) 6
1
kf kH −1
ν
and Φm (v) = 0.
5. For each m define um ∈ Vm to be one of the v’s and deduce that there is a subsequence (still
indexed by m) and ũ ∈ H01 (Ω)3 such that
um * ũ.
Show that ũ ∈ V .
6. Fix k0 ∈ N and consider m > k0 . Show that
Z
Z
ν
∇um · ∇wk0 → ν
∇ũ · ∇wk0 .
Ω
Ω
7. Prove using a density argument that for all u, v, w ∈ H 1 (Ω)3 with at least one in H01 (Ω)3
there holds
Z
Z
Z
(w · ∇)u · v = − (w · ∇)v · u −
div(w)u · v.
Ω
Deduce then that
Ω
Z
Ω
Z
(um · ∇)um · wk0 →
Ω
8. Conclude that ũ is a weak solution.
(ũ · ∇)ũ · wk0 .
Ω