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.

Keywords:

• singular perturbations
• parabolic equations
• ultraparabolic equations
• Fokker-Planck equations
• boundary layers

AMS Subject Classification::

• 35B25
• 35F15
• 35F30
• 35K20
• 35K70

Acknowledgements

Several keen observations made by Martin Stynes, National University of Ireland, Cork, Ireland, have been useful to clarify the practical meaning of the boundary-layer-free behaviour of the solution to certain classes of parabolic boundary-value problems, described in this article. This work has been supported, in part, by the Italian GNFM-INdAM.