
1244 SYSTEMS OF LINEAR COUPLED PDES. DECOMPOSITION METHODS
where the function Φ is an arbitrary solution of the equation
(M
1
L + M
1
K
1
∆ − σM
2
∆)[Φ] = 0.
3
◦
. Equations (19.1.2.5)–(19.1.2.6) with σ = 0 are invariant under the transformation
u =
e
u + ∇Φ, p = ep + Θ,
where the functions Φ and Θ are arbitrary solutions of the equations
(L + K
1
∆)[Φ] = 0, M
1
[Θ] + M
2
[Φ] = 0.
19.2 First-Order Decompositions. Examples
19.2.1 Systems of Linear PDEs without Mass Forces (f = 0)
◮ Decomposition on the basis of two stream functions.
In the absence of m ass forces (f = 0), it is convenient to set G = 0 in (19.1.2.2) and
(19.1.2.7). In this case, the right-hand side of the ...