Lp Theory for the Multidimensional Aggregation Equation Communications on Pure and Applied Mathematics
Document Type
Article
Publication Date
2011
Abstract
We consider well-posedness of the aggregation equation ∂tu + 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 < ps. In the special case of K(x) = |x|, the exponent ps = 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 < ps. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$.
Digital Commons @ LMU & LLS Citation
Bertozzi, A.L., Laurent, T. and Rosado, J. (2011), Lp theory for the multidimensional aggregation equation. Comm. Pure Appl. Math., 64: 45-83. https://doi.org/10.1002/cpa.20334
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