Parabolic H-measures

Classical H-measures introduced by Tartar (1990) and independently by Gérard (1991) 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.