Singular perturbations of parabolic equations without boundary layers

Abstract

Since Ludwig Prandtl's 1904 work, motivated by purely physical considerations, the concept of ‘boundary layer’ has been playing an essential role in singular perturbation problems of ordinary and partial differential equations. It seems surprising that in some problems, where boundary layers are expected, these do not appear. This fact rests on the regularity of solutions, which may present smooth variations, while their gradient or higher derivatives may vary rapidly only in some subdomains. Indeed, a kind of ‘higher order boundary layer’ in such cases may appear. Questions of existence and absence of boundary layers are addressed. The arising phenomena are shown by simple examples, suited to better understand the solutions' regularity. A number of more general results are presented for linear parabolic problems with small coefficients (singular perturbations). Some results for nonlinear Fokker–Planck-type equations encountered in applications are also given.