Xue, Liutang; Ye, Zhuan On the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion. (English) Zbl 1379.35049 Pac. J. Math. 293, No. 2, 471-510 (2018). Summary: We investigate the differentiability property of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical- or critical-type cases. Under the suitable conditions on the velocity field and the forcing term in terms of the spatial Hölder regularity, and for the initial data without regularity assumption, we show the a priori differentiability estimates for any positive time. If additionally the velocity field is divergence-free, we also prove that the vanishing viscosity weak solution is differentiable with some Hölder continuous derivatives for any positive time. Cited in 7 Documents MSC: 35B65 Smoothness and regularity of solutions to PDEs 35Q35 PDEs in connection with fluid mechanics 35R11 Fractional partial differential equations 35B45 A priori estimates in context of PDEs 35R09 Integro-partial differential equations Keywords:Lévy-type operator; fractional Laplacian operator; vanishing viscosity weak solution PDFBibTeX XMLCite \textit{L. Xue} and \textit{Z. Ye}, Pac. J. Math. 293, No. 2, 471--510 (2018; Zbl 1379.35049) Full Text: DOI arXiv References: [1] 10.1007/978-3-642-16830-7 · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7 [2] 10.1080/03605300600987306 · Zbl 1143.26002 · doi:10.1080/03605300600987306 [3] 10.1007/s00205-010-0336-4 · Zbl 1231.35284 · doi:10.1007/s00205-010-0336-4 [4] 10.4007/annals.2010.171.1903 · Zbl 1204.35063 · doi:10.4007/annals.2010.171.1903 [5] 10.1090/S0894-0347-2011-00698-X · Zbl 1223.35098 · doi:10.1090/S0894-0347-2011-00698-X [6] 10.1016/j.aim.2012.04.004 · Zbl 1248.35156 · doi:10.1016/j.aim.2012.04.004 [7] 10.4171/RMI/705 · Zbl 1256.35191 · doi:10.4171/RMI/705 [8] 10.4171/RMI/925 · Zbl 1365.35203 · doi:10.4171/RMI/925 [9] 10.1007/s00220-007-0193-7 · Zbl 1142.35069 · doi:10.1007/s00220-007-0193-7 [10] 10.1007/s00039-012-0172-9 · Zbl 1256.35078 · doi:10.1007/s00039-012-0172-9 [11] 10.1016/j.anihpc.2007.10.001 · Zbl 1149.76052 · doi:10.1016/j.anihpc.2007.10.001 [12] 10.1007/s00220-004-1055-1 · Zbl 1309.76026 · doi:10.1007/s00220-004-1055-1 [13] 10.2140/apde.2014.7.43 · Zbl 1294.35092 · doi:10.2140/apde.2014.7.43 [14] 10.2140/apde.2011.4.247 · Zbl 1264.35173 · doi:10.2140/apde.2011.4.247 [15] 10.1007/s11118-007-9060-6 · Zbl 1128.60071 · doi:10.1007/s11118-007-9060-6 [16] 10.1142/9781860947155 · doi:10.1142/9781860947155 [17] 10.1007/s00526-008-0173-6 · Zbl 1158.35019 · doi:10.1007/s00526-008-0173-6 [18] ; Kiselev, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov., 370, 58 (2009) [19] 10.1007/s00222-006-0020-3 · Zbl 1121.35115 · doi:10.1007/s00222-006-0020-3 [20] ; Komatsu, Osaka J. Math., 32, 833 (1995) · Zbl 0867.35123 [21] 10.1016/j.aim.2013.07.011 · Zbl 1284.35208 · doi:10.1016/j.aim.2013.07.011 [22] ; Sato, Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68 (1999) · Zbl 0973.60001 [23] 10.2422/2036-2145.201009_004 · Zbl 1263.35056 · doi:10.2422/2036-2145.201009_004 [24] 10.1512/iumj.2012.61.4568 · Zbl 1308.35042 · doi:10.1512/iumj.2012.61.4568 [25] 10.1007/s00205-012-0579-3 · Zbl 1264.35077 · doi:10.1007/s00205-012-0579-3 [26] 10.2140/apde.2009.2.361 · Zbl 1190.35177 · doi:10.2140/apde.2009.2.361 [27] 10.1007/s00220-010-1144-2 · Zbl 1248.35211 · doi:10.1007/s00220-010-1144-2 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.