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L p theory for the multidimensional aggregation equation
Author(s) -
Bertozzi Andrea L.,
Laurent Thomas,
Rosado Jesús
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20334
Subject(s) - uniqueness , order (exchange) , exponent , mathematics , singularity , combinatorics , mathematical physics , physics , mathematical analysis , linguistics , philosophy , finance , economics
Abstract We consider well‐posedness of the aggregation equation ∂ t u + div( uv ) = 0, v = −▿ K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order | x | α , α > 2 − d , and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < p s . In the special case of K ( x ) = | x |, the exponent p s = d /( d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < p s . We also give an Osgood condition on the potential K ( x ) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc.