Abstract
Classical H-measures introduced by Tartar (Proc R Soc Edinb 115A:193–230, 1990) and independently by Gérard (Commun Partial Differ Equ 16:1761–1794, 1991) are essentially suited for hyperbolic equations while parabolic equations fit in the framework of the parabolic H-measures developed by Antonić and Lazar (2007–2013). More recently the study of differential relations with fractional derivatives prompted the extension of the theory to arbitrary ratios, thus the fractional H-measures were introduced and applied to fractional conservation laws by Mitrović and Ivec (Commun Pure Appl Anal 10(6):1617–1627, 2011). In this paper we explore the transport property of fractional H-measures by studying fractional derivatives of commutators of multiplication and Fourier multiplier operators. In particular, we prove the Second commutation lemma suitable for fractional H-measures, comprehending the known hyperbolic and parabolic cases, while allowing for derivation of the corresponding propagation principle for fractional H-measures. At the end, on a model example we present this derivation of the transport equation for the fractional H-measure.
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Aleksić, J., Pilipović, S., Vojnović, I.: H-distributions via Sobolev spaces. Mediterr. J. Math. 13, 3499–3512 (2016)
Antonić, N.: H-measures applied to symmetric systems. Proc. R. Soc. Edinb. 126A, 1133–1155 (1996)
Antonić, N., Erceg, M., Lazar, M.: Localisation principle for one-scale H-measures. J. Funct. Anal. 272, 3410–3454 (2017)
Antonić, N., Mitrović, D.: H-distributions: an extension of H-measures to an \({\rm L}^p-{\rm L}^q\) setting. Abs. Appl. Anal. 2011 Article ID 901084 (2011)
Antonić, N., Lazar, M.: H-measures and variants applied to parabolic equations. J. Math. Anal. Appl. 343, 207–225 (2008)
Antonić, N., Lazar, M.: Parabolic variant of H-measures in homogenisation of a model problem based on Navier-Stokes equation. Nonlinear Anal. B Real World Appl. 11, 4500–4512 (2010)
Antonić, N., Lazar, M.: Parabolic H-measures. J. Funct. Anal. 265, 1190–1239 (2013)
Antonić, N., Mišur, M., Mitrović, D.: On the First commutation lemma (submitted)
Erceg, M., Ivec, I.: On generalisation of H-measures. Filomat (in press)
Francfort, G.A., Murat, F.: Oscillations and energy densities in the wave equation. Commun. Partial Differ. Equ. 17, 1785–1865 (1992)
Gérard, P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)
Golbabai, A., Sayevand, K.: Fractional calculus—a new approach to the analysis of generalized fourth-order diffusion-wave equations. Comput. Math. Appl. 61, 2227–2231 (2011)
Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Springer, Berlin (2005)
Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (2007)
Lazar, M., Mitrović, D.: Velocity averaging—a general framework. Dyn. Partial Differ. Equ. 9, 239–260 (2012)
Mišur, M., Mitrović, D.: On a generalisation of compensated compactness in the \(L^p-L^q\) setting. J. Funct. Anal. 268, 1904–1927 (2015)
Mitrović, D., Ivec, I.: A generalization of H-measures and application on purely fractional scalar conservation laws. Commun. Pure Appl. Anal. 10(6), 1617–1627 (2011)
Panov, E.Y.: Ultra-parabolic equations with rough coefficients: entropy solutions and strong pre-compactness property. J. Math. Sci. 159, 180–228 (2009)
Panov, E.Y.: Ultra-parabolic H-measures and compensated compactness. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 47–62 (2011)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivative: Theory and Applications. Gordon & Breach Sci. Publishers, London (1993)
Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinb. 115A, 193–230 (1990)
Tartar, L.: The General Theory of Homogenization: A Personalized Introduction. Springer, Berlin (2009)
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Dedicated to the memory of professor Todor V. Gramchev.
This work has been supported in part by Croatian Science Foundation under the project 9780 WeConMApp, and by University of Zagreb trough Grant PMF-M02/2016, as well as by the bilateral Croatian–Serbian project Microlocal analysis, partial differential equations and application to heterogeneous materials. The first author has partially been supported by 2014–2017 MIUR-FIR Grant No. RBFR13WAET.
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Erceg, M., Ivec, I. Second commutation lemma for fractional H-measures. J. Pseudo-Differ. Oper. Appl. 9, 589–613 (2018). https://doi.org/10.1007/s11868-017-0207-y
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DOI: https://doi.org/10.1007/s11868-017-0207-y