×

zbMATH — the first resource for mathematics

On the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion. (English) Zbl 1379.35049
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.

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 Integral partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 10.1007/978-3-642-16830-7 · Zbl 1227.35004
[2] 10.1080/03605300600987306 · Zbl 1143.26002
[3] 10.1007/s00205-010-0336-4 · Zbl 1231.35284
[4] 10.4007/annals.2010.171.1903 · Zbl 1204.35063
[5] 10.1090/S0894-0347-2011-00698-X · Zbl 1223.35098
[6] 10.1016/j.aim.2012.04.004 · Zbl 1248.35156
[7] 10.4171/RMI/705 · Zbl 1256.35191
[8] 10.4171/RMI/925 · Zbl 1365.35203
[9] 10.1007/s00220-007-0193-7 · Zbl 1142.35069
[10] 10.1007/s00039-012-0172-9 · Zbl 1256.35078
[11] 10.1016/j.anihpc.2007.10.001 · Zbl 1149.76052
[12] 10.1007/s00220-004-1055-1 · Zbl 1309.76026
[13] 10.2140/apde.2014.7.43 · Zbl 1294.35092
[14] 10.2140/apde.2011.4.247 · Zbl 1264.35173
[15] 10.1007/s11118-007-9060-6 · Zbl 1128.60071
[16] 10.1142/9781860947155
[17] 10.1007/s00526-008-0173-6 · Zbl 1158.35019
[18] ; Kiselev, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov., 370, 58, (2009)
[19] 10.1007/s00222-006-0020-3 · Zbl 1121.35115
[20] ; Komatsu, Osaka J. Math., 32, 833, (1995)
[21] 10.1016/j.aim.2013.07.011 · Zbl 1284.35208
[22] ; Sato, Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68, (1999)
[23] 10.2422/2036-2145.201009_004 · Zbl 1263.35056
[24] 10.1512/iumj.2012.61.4568 · Zbl 1308.35042
[25] 10.1007/s00205-012-0579-3 · Zbl 1264.35077
[26] 10.2140/apde.2009.2.361 · Zbl 1190.35177
[27] 10.1007/s00220-010-1144-2 · Zbl 1248.35211
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.