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Fractional H-measures and transport property

Microlocal defect functionals (H-measures, H- distributions, semiclassical measures etc.) are objects which determine, in some sense, the lack of strong compactness for weakly convergent Lp sequences. H-measures are suitable to treat problems where all partial derivatives are of the same order [4]. Recently, parabolic H- measures are introduced in order to treat 1:2 ratio between orders of partial derivatives [1], and also fractional H-measures which treat arbitrary ratios [2, 3]. We generalise Second commutation lemmas introduced in [1] and [4] to fractional Hmeasures, from which we are able to derive the propagation principle for the following fourth order partial differential equation: iu_t + (a(x)u_xx)_xx = f. References [1] N. Antoni´c, M. Lazar, Parabolic H-measures, Journal of Functional Analysis, 265 (2013) 1190– 1239. [2] M. Erceg, I. Ivec, On generalisation of H-measures, accepted for publication in Filomat, 18 pp. [3] D. Mitrovi´c, I. Ivec, A generalization of H-measures and application on purely fractional scalar conservation laws, Comm. Pure Appl. Analysis, 10 (2011) (6) 1617– 1627. [4] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society of Edinburgh, 115A (1990) 193–230.

H-measures, transport property

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