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

hal-00761112
https://hal.univ-antilles.fr/hal-00761112
https://hal.univ-antilles.fr/hal-00761112/document
https://hal.univ-antilles.fr/hal-00761112/file/DT2010-03_Devoue_al3.pdf

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

Droits : info:eu-repo/semantics/OpenAccess
hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006/document
https://hal.archives-ouvertes.fr/hal-00190006/file/Regularity-Local-Microlocal-Analysis5.pdf
ARXIV : 0711.3688