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2.3 The Stokes Equations 65

It is understood that the representation of p is unique only up to an additive
constant. Also, as was explained before,

T E(x, y):= σ E(x, y), −Q(x, y) n(y) .
y
2.3.1 Hydrodynamic Potentials

The last terms in the representation (2.3.11) and (2.3.12) corresponding to
the body force f define a particular solution (U,P) of the nonhomogeneous
Stokes system (2.3.1). As in elasticity, if we decompose the solution in the
form
u = u c + U, p = p c + P,
then the pair (u c ,p c ) will satisfy the corresponding homogeneous system of
(2.3.1). Hence, in the following, without loss of generality, we shall confine
ourselves only to the homogeneous Stokes system. The solution of the homo-
geneous system now has the representation from (2.3.11) and (2.3.12) with
f = 0, i.e.,
u(x)= V τ(x) − Wϕ(x) , (2.3.13)
p(x)= Φτ(x) − Πϕ(x). (2.3.14)

(The subscript c has been suppressed.) Here the pair (V, Φ) and (W, Π)are
the respective simple– and double layer hydrodynamic potentials defined by

V τ(x):= E(x, y)τ(y)ds y ,
Γ
(2.3.15)

Φτ(x):= Q(x, y) · τ(y)ds y ;
Γ

Wϕ(x):= T (E(x, y)) ϕ(y)ds y ,

y
Γ
∂
(2.3.16)

Πϕ(x):=2µ Q(x, y) · ϕ(y)ds y for x ∈ Γ.
∂n y
Γ
In (2.3.13) and (2.3.14) the boundary charges are the Cauchy data ϕ =
u(x)| Γ and τ(x)= Tu(x)| Γ of the solution to the Stokes equations
−µ∆u + ∇p = 0 ,
(2.3.17)
div u =0 in Ω.
For the exterior problems, the representation formula for u needs to be
modified by taking into account the growth conditions at infinity. Here proper
growth conditions are