We prove that the sequence of averaged quantities R m un(x, p) ρ(p)dp, is strongly precompact in L 2 loc (R d ), where ρ ∈ L 2 c (R m ), and un ∈ L 2 (R m ; L s (R d )), s ≥ 2, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If s > 2 then the coefficients can be discontinuous with respect to the space variable x ∈ R d , otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the H-measures.1991 Mathematics Subject Classification. 35K70, 42B37, 46G10.
Since their introduction H-measures have been mostly used in problems related to propagation effects for hyperbolic equations and systems. In this study we give an attempt to apply the H-measure theory to other types of equations. Through a number of examples we present how do the differences between parabolic and hyperbolic equations reflect in the properties of H-measures corresponding to the solutions. Secondly, we apply the H-measures to the Schrödinger equation, where we succeed in proving a propagation property. However, our conclusion is that a variant of H-measures should be sought which would be better suited to parabolic problems. We propose such a variant, show some fundamental properties and illustrate its applicability by some examples. In particular, we show that the variant provides new information in a number of situations where the original H-measures did not. Finally, we describe how the new variant can be used in small amplitude homogenisation of parabolic equations.
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 Hmeasures. 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.
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