L 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 P2(R)∩L(R), in dimensions two and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|, α > 2−d, and prove local well-posedness in P2(R)∩L(R) 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 P2(R)∩L(R) with p < ps. We also give an Osgood condition on the potential K(x) which guaranties global existence and uniqueness in P2(R) ∩ L(R).