Studies on the Motion of Viscous Flows— III 



Lu = Lu . + Lu, = f * ' 



Bj(u) = Bj(uj) + BjCu^) = fflj 



B^Cu) = B2(u^) + B^Cu^) = a^ . 



In general, therefore, we may write 



Lu = Luj + Luj + ••• + LUj^ = f , " (1.26) 



where Up u^ u^ satisfy different equations and boundary conditions. If 



we recall the precise notation, this should have been written 



Lu = LjUj + L2U2 + ••• + L^u^ = f , (1.27) 



where Li, Lj L^ are differential operators which are formally the same 



as L, but with different boundary conditions. The functions ui, i = l, and n may 

 be required to be the solutions of the operator equations 



LjU^ ~^i i=l, ••■,n (no summation) , (1.28) 



where . , _ ^ 



, ■ I '.= f . - ::, •-.. 



1 = 1 ■■-'"' 



and f i belong to S. The functions f^ are suitably selected, depending on the 

 problem. 



Let us assume that in the case of a system of nonlinear differential equa- 

 tions, in particular the Navier-Stokes equations under consideration, there exists 

 an infinite sum of functions 



of the space-time variables and the flow parameters such that it converges to 

 the solution S, which satisfies the Navier-Stokes equations, the continuity equa- 

 tion, and the boundary conditions completely, in the sense that the partial sums 



Spn = So + S, + S, + S3 + ••■ + S^ 



converge to the solution s; i.e., given e > 0, there exists an integer N(e)such 

 that 



Is- S„„| < e ' i- '"" ■ ■ ■''■ ' 



for all n > N. 



With the above assumption, we may write ' ' > 



531 



