Divergence & curl with fractional order

Journal de Mathématiques Pures et Appliquées - Tập 165 - Trang 190-231 - 2022
Liguang Liu1, Jie Xiao2
1School of Mathematics, Renmin University of China, Beijing, 100872, PR China
2Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada

Tài liệu tham khảo

Adams, 1981 Adams, 1975, A note on Riesz potentials, Duke Math. J., 42, 765, 10.1215/S0012-7094-75-04265-9 Adams, 2012, Morrey spaces in harmonic analysis, Ark. Mat., 50, 201, 10.1007/s11512-010-0134-0 Bonami, 2007, On the product of functions in BMO and H1, Ann. Inst. Fourier (Grenoble), 57, 1405, 10.5802/aif.2299 Brué, 2021, A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II, C. R. Math. Bucur, 2016, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal., 15, 657, 10.3934/cpaa.2016.15.657 Caffarelli, 2007, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245, 10.1080/03605300600987306 Coifman, 1993, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, 247 Comi, 2019, A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up, J. Funct. Anal., 277, 3373, 10.1016/j.jfa.2019.03.011 Comi Comi, 2022, The fractional variation and the precise representative of BVα,p functions, J. Fract. Calc. Appl., 25, 520, 10.1007/s13540-022-00036-0 Dimitriou, 2007, On the geometry of a steady two dimensional potential flows and its physics, Z. Angew. Math. Phys., 58, 100, 10.1007/s00033-006-0081-6 Dimitriou, 2009, Introducing a geometrical potential theory for two dimensional steady flows, J. Eng. Math., 63, 1, 10.1007/s10665-008-9235-1 Dimitriou, 2016, Quantitative analysis of two-dimensional flow visualizations, J. Eng. Math., 98, 145, 10.1007/s10665-015-9815-9 Dimitriou, 2017, Geometrical interpretations of continuous and complex-lamellar steady flows, Eur. J. Mech. B, Fluids, 61, 86, 10.1016/j.euromechflu.2016.07.001 Essén, 2000, Q space of several real variables, Indiana Univ. Math. J., 49, 575, 10.1512/iumj.2000.49.1732 Fefferman, 1972, Hp(Rn) spaces of several variables, Acta Math., 129, 137, 10.1007/BF02392215 Gala, 2007, A note on divergence-curl lemma, Serdica Math. J., 33, 339 Grafakos, 2014, Modern Fourier Analysis, vol. 250 Grafakos, 2017, Fractional Differentiation: Leibniz Meets Hölder. Excursions in Harmonic Analysis, vol. 5, 17 Grafakos, 2010, Boundedness of paraproduct operators on RD-spaces, Sci. China Math., 53, 2097, 10.1007/s11425-010-4042-3 Iwatsuka, 1986, Magnetic Schrïnder operators with compact resolvent, J. Math. Kyoto Univ. (JMKYAZ), 26, 357 John, 1961, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14, 415, 10.1002/cpa.3160140317 Lieb, 2001, Analysis, vol. 14 Liu, 2021, Fractional Hardy-Sobolev L1-embedding per capacity-duality, Appl. Comput. Harmon. Anal., 51, 17, 10.1016/j.acha.2020.10.001 Liu, 2021, Fractional differential operators and divergence equations, Adv. Anal. Geom., 3, 385 Liu, 2020, Morrey's fractional integrals in Campanato-Sobolev's space and divF=f, J. Math. Pures Appl., 142, 23, 10.1016/j.matpur.2020.08.005 Mazowiecka, 2018, Fractional divergence-curl quantities and applications to nonlocal geometric equations, J. Funct. Anal., 275, 1, 10.1016/j.jfa.2018.03.016 K.T. McDonald, Electrodynamics in 1 and 2 spatial dimensions, preprint, 2019. Meerschaert, 2006, Fractional vector calculus for fractional advection-dispersion, Physica A, 367, 181, 10.1016/j.physa.2005.11.015 Peetre, 1966, On convolution operators leaving Lp,λ spaces invariant, Ann. Mat. Pura Appl., 72, 295, 10.1007/BF02414340 Rosenthal, 2014, Calderón-Zygmund operators in Morrey spaces, Rev. Mat. Complut., 27, 1, 10.1007/s13163-013-0125-3 Rosenthal, 2015, Morrey spaces, their duals and preduals, Rev. Mat. Complut., 28, 1, 10.1007/s13163-013-0145-z Shieh, 2015, On a new class of fractional partial differential equations, Adv. Calc. Var., 8, 321, 10.1515/acv-2014-0009 Šilhavý, 2020, Fractional vector analysis based on invariance requirements (critique of coordinate approaches), Contin. Mech. Thermodyn., 32, 207, 10.1007/s00161-019-00797-9 Silvestre, 2007, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60, 67, 10.1002/cpa.20153 Stein, 1993 Tarasov, 2008, Fractional vector calculus and fractional Maxiwell's equations, Ann. Phys., 323, 2756, 10.1016/j.aop.2008.04.005 Weber, 2021, Optimal control of the two-dimensional Vlasov-Maxwell system, ESAIM Control Optim. Calc. Var., 27, 10.1051/cocv/2020069 Xiao, 2006, A sharp Sobolev trace inequality for the fractional-order derivatives, Bull. Sci. Math., 130, 87, 10.1016/j.bulsci.2005.07.002 Xiao, 2019, Qα Analysis on Euclidean Spaces, vol. 1