×

Dynamics of one-fold symmetric patches for the aggregation equation and collapse to singular measure. (English) Zbl 1433.35422

Summary: We are concerned with the dynamics of one-fold symmetric patches for the two-dimensional aggregation equation associated to the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows us to analyze the concentration phenomenon of the aggregation patches near the blow-up time. In particular, we prove that the patch collapses to a collection of disjoint segments and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B44 Blow-up in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
92D25 Population dynamics (general)
76B47 Vortex flows for incompressible inviscid fluids
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] 10.1002/cpa.20223 · Zbl 1171.35005
[2] 10.1137/100804504 · Zbl 1255.35012
[3] 10.4310/CMS.2010.v8.n1.a4 · Zbl 1197.35061
[4] 10.1007/BF02097055 · Zbl 0771.76014
[5] 10.1007/s00220-007-0288-1 · Zbl 1132.35392
[6] 10.1088/0951-7715/22/3/009 · Zbl 1194.35053
[7] 10.1002/cpa.20334 · Zbl 1218.35075
[8] 10.1142/S0218202511400057 · Zbl 1241.35153
[9] 10.1137/15M1033125 · Zbl 1355.35110
[10] 10.1016/j.jde.2005.07.025 · Zbl 1089.45002
[11] 10.1016/S0362-546X(99)00399-5 · Zbl 1011.92053
[12] 10.2307/1930099
[13] 10.4171/077-1/1
[14] 10.1007/s00205-005-0386-1 · Zbl 1082.76105
[15] 10.1215/00127094-2010-211 · Zbl 1215.35045
[16] 10.24033/asens.1679 · Zbl 0779.76011
[17] 10.2307/2007065 · Zbl 0497.42012
[18] 10.4171/RMI/276 · Zbl 1158.35404
[19] 10.2307/2939269 · Zbl 0780.35073
[20] 10.1137/110820427 · Zbl 1235.35064
[21] 10.1137/S0036141002408009 · Zbl 1044.35085
[22] 10.1016/j.physd.2012.11.004 · Zbl 1286.35017
[23] 10.1088/0951-7715/24/10/002 · Zbl 1288.92031
[24] 10.1109/TAC.2003.809765 · Zbl 1365.92143
[25] 10.1016/j.physd.2006.07.010 · Zbl 1125.82021
[26] 10.1016/0022-5193(70)90092-5 · Zbl 1170.92306
[27] 10.1080/03605300701318955 · Zbl 1132.35088
[28] 10.1016/j.aim.2008.10.016 · Zbl 1168.35037
[29] 10.1016/j.anihpc.2004.07.002 · Zbl 1070.35036
[30] 10.1007/s002850050158 · Zbl 0940.92032
[31] 10.1007/s00285-004-0279-1 · Zbl 1055.92046
[32] 10.1007/s002050100139 · Zbl 1038.82068
[33] 10.4310/MAA.2002.v9.n4.a4 · Zbl 1166.35363
[34] 10.1137/S0036139903437424 · Zbl 1071.92048
[35] 10.2307/2000515 · Zbl 0596.42005
[36] ; Yudovich, Zh. Vychisl. Mat. Mat. Fiz., 3, 1032 (1963)
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.