Lp theory for the multidimensional aggregation equation

✍ Scribed by Andrea L. Bertozzi; Thomas Laurent; Jesús Rosado

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 322 KB
• Volume: 64
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20334

✦ Synopsis

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.