Blow-up in multidimensional aggregation equations with mildly singular interaction kernels
File(s)
Accepted version
Author(s)
Bertozzi, Andrea L
Carrillo, Jose A
Laurent, Thomas
Type
Journal Article
Abstract
We consider the multidimensional aggregation equation u(t) - del. (u del K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717-35), whereas for C(2) kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K( x) = vertical bar x vertical bar and some open problems.
Date Issued
2009-03-01
Citation
NONLINEARITY, 22, pp.{683-710}-{683-710}
ISSN
0951-7715
Publisher
IOP PUBLISHING LTD
Start Page
{683-710}
End Page
{683-710}
Journal / Book Title
NONLINEARITY
Volume
22
Issue
3
Copyright Statement
©2009 IOP Publishing Ltd.
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000263259400009&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Publication Status
Published
Article Number
3