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66 2. Boundary Integral Equations

Σ log |x| + O(1) for n =2,
u(x) = −1 (2.3.18)
O(|x| ) for n =3;
p(x) = O(|x| 1−n )as |x|→∞. (2.3.19)

In the two–dimensional case Σ is a given constant vector. The representation
c
formula for solutions of the Stokes equations (2.3.17) in Ω with the growth
conditions (2.3.18) and (2.3.19) has the form
u(x)= −V τ(x)+ Wϕ(x)+ ω , (2.3.20)
p(x)= −Φτ(x)+ Πϕ(x) (2.3.21)

with the Cauchy data ϕ = u |Γ and τ = T(u) |Γ satisfying

Σ = τds ; (2.3.22)
Γ
and ω is an unknown constant vector which vanishes when n =3.

2.3.2 The Stokes Boundary Value Problems
We consider two boundary value problems for the Stokes system (2.3.17) in
c
Ω as well as in Ω . In the first problem (the Dirichlet problem), the boundary
trace of the velocity
u| Γ = ϕ on Γ (2.3.23)
is specified, and in the second problem (the Neumann problem), the hydro-
dynamic boundary traction
T(u)| Γ = τ on Γ (2.3.24)

is given. As consequences of the incompressible flow equations and the Green
formula for the interior problem, the prescribed Cauchy data need to satisfy,
respectively, the compatibility conditions

ϕ · n ds =0 ,
Γ
(2.3.25)

n 1+2(n−2)
τ · (a + b × x) ds = 0 for all a ∈ IR and b ∈ IR ,
Γ
with b × x := b(x 2 , −x 1 ) for n =2.
For the exterior problem we require the decay conditions (2.3.18) and
(2.3.19). We again solve these problems by using the direct method of bound-
ary integral equations.
Since the pressure p will be completely determined once the Cauchy data
for the velocity are known, in the following, it suffices to consider only the