
12.12. Stokes Equations for Viscous Compressible Barotropic Fluids 1099
◮ Decomposition based on the Stokes–Helmholtz representation of the mass force.
Let the mass force in Eqs. (14) be represented as the sum f = ∇γ + curl ω. Then the
solution of system (14) admits the representation
w = ∇eϕ + curl
e
Ψ,
where the vector function
e
Ψ satisfies the equation
e
Ψ
t
− ν∆
e
Ψ = ω
and the function eϕ is a solution of the equation
eϕ
tt
− (2ν + κ)∆ eϕ
t
− c
2
∆ eϕ = γ.
◮ Decomposition not requiring the force to split in to components.
The solution of system (14) can be represented in the form
w =
∂
2
t
− (2ν + κ)∂
t
∆ − c
2
∆
[
e
v] + ∇
c
2
+ (ν + κ)∂
t
][div
e
v], (15)
where the vector functi ...