
1084 SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
◮ Incomplete symmetric decomposition of the Stokes equations.
Every solution of system (1)–(2) can also be represented in the symmetric form
u = ∇ϕ + v, p = −ϕ
t
+ ν∆ϕ + G, (21)
where the vector function v= (v
1
, v
2
, v
3
) satisfies the independent linear nonhomogeneous
equation
v
t
− ∆v = f − ∇G (22)
and the functions ϕ and v are related by the single equation
∆ϕ + div v = 0. (23)
Formulas (21) and Eqs. (22) include an arbitrary scalar function G = G(x, t).
By setting v
1
= 0 and G = F in (21)–(23), we obtain an asymmetric representation of
the solutions (18)–(20).
12.7.4 G eneral Solution of the Steady-Stat ...