Éditeur(s) : HAL CCSD Résumé : 21 pages Publié sous le titre: Spectral asymptotic analysis in algebras of generalized functions dans Asymptot. Anal. 59/1-2, 83-107, 2008. International audience 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. Asymptot. Anal.
Éditeur(s) : HAL CCSD Résumé : We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a "frequential microlocal analysis " and into a "microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations. https://hal.archives-ouvertes.fr/hal-00190006
Éditeur(s) : HAL CCSD Résumé : 18 pages This article adresses questions on localization and microlocalization in the framework of the so called (C,E,P)-algebras of generalized functions. We give new results on nonlinear properties of the asymptotic singular spectrum in the multi-parameter case, a tool to describe the singularities of generalized solutions associated to problems which are classically ill-posed. Wa apply the theory to a transport equation with strongly irregular coefficients and initial data. https://hal.univ-antilles.fr/hal-00761112