  This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a
flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for
$2-d\leq \alpha<2$ for which the smooth densities are known to develop
singularities in finite time. For this range This paper studies the transport
of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We
focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the
smooth densities are known to develop singularities in finite time. For this
range we prove the existence for all time of radially symmetric measure
solutions that are monotone decreasing as a function of the radius, thus
allowing for continuation of the solution past the blowup time. The monotone
constraint on the data is consistent with the typical blowup profiles observed
in recent numerical studies of these singularities. We prove monotonicity is
preserved for all time, even after blowup, in contrast to the case $\alpha >2$
where radially symmetric solutions are known to lose monotonicity. In the case
of the Newtonian potential ($\alpha=2-d$), under the assumption of radial
symmetry the equation can be transformed into the inviscid Burgers equation on
a half line. This enables us to prove preservation of monotonicity using the
classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at
the critical exponent $p$ we exhibit initial data in $L^p$ for which the
solution immediately develops a Dirac mass singularity. This extends recent
work on the local ill-posedness of solutions at the critical exponent.
