Singular perturbation problems whose degenerate forms have many solutions

摘要

A variety of singular perturbation problems from continuum mechanics may be put into the form: ePu(x) = Lu(x), where e is a small parameter, P a partial differential operator, u a function of several variables with values in a Banach space B, and L is an operator on B which is noninvertible. In the paper, (1) a class of problems of the above sort is delineated and a procedure for obtaining approximate solutions is given, (2) specific examples having to do with the dispersion of a solute in flow through a tube and the diffusion of a medium with several reacting components are dealt with in some detail, and (3) the method is proved to be asymptotically correct for one of the examples, and it is shown how such a proof could be carried out for a wide class of problems