Lp theory for the multidimensional aggregation equation |
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Authors: | Andrea L Bertozzi Thomas Laurent Jesús Rosado |
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Institution: | 1. University of California, Los Angeles, Department of Mathematics, Box 951555, Los Angeles, CA 90095‐1555;2. Universitat Autònoma de Barcelona, Departament de Matemátiques, 08193 Bellaterra (Barcelona), SPAIN |
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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 ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc. |
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