Lp theory for the multidimensional aggregation equation

We consider well‐posedness of the aggregation equation ∂tu + div(uv) = 0, v = −▿K * u with initial data in amssym P₂ ( Rᵈ ) ∩ Lᵖ ( Rᵈ ) 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 amssym P₂ ( Rᵈ ) ∩ Lᵖ ( Rᵈ ) 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 amssym P₂ ( Rᵈ ) ∩ Lᵖ ( Rᵈ ) with p < ps. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in amssym P₂ ( Rᵈ ) ∩ Lᵖ ( Rᵈ ).