Spectral asymptotic analysis in algebras of generalized functions. (English) Zbl 1163.35303

Delcroix, Antoine; Marti, Jean-André; Oberguggenberger, Michael

Asymptotic Anal. 59, No. 1-2, 83-107 (2008).

Summary: We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.

MSC:

35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
46F30	Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
35A21	Singularity in context of PDEs
35G20	Nonlinear higher-order PDEs
35C20	Asymptotic expansions of solutions to PDEs

Keywords:

presheaf properties; regularizing parameter; singular asymptotic spectrum