Parabolic H-measures. (English) Zbl 1286.35009
Summary: Classical H-measures introduced by L. Tartar [Proc. R. Soc. Edinb., Sect. A, Math. 115, No. 3–4, 193–230 (1990; Zbl 0774.35008)] and independently by P. Gérard [Commun. Partial Differ. Equations 16, No. 11, 1761–1794 (1991; Zbl 0770.35001)] are not well suited for the study of parabolic equations. Recently, several parabolic variants have been proposed, together with a number of applications. We introduce a new parabolic variant (and call it the parabolic H-measure), which is suitable for these known applications. Moreover, for this variant we prove the localisation and propagation principle, establishing a basis for more demanding applications of parabolic H-measures, similarly as it was the case with classical H-measures. In particular, the propagation principle enables us to write down a transport equation satisfied by the parabolic H-measure associated to a sequence of solutions of a Schrödinger type equation. Some applications to specific equations are presented, illustrating the possible use of this new tool. A comparison to similar results for classical H-measures has been made as well.
MSC:
| 35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
| 35K10 | Second-order parabolic equations |
| 35A15 | Variational methods applied to PDEs |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
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