arXiv2012
This paper studies the transport of a mass $μ$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*μ$. We focus on kernels $K=|x|^α/ α$ for $2-d\leq α<2$ for which the smooth densities are known to develop singularities in finite time. For this range This paper studies the transport of a mass $μ$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*μ$. We focus on kernels $K=|x|^α/ α$ for $2-d\leq α<2$ 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 $α>2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($α=2-d$), 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 $2 -d < α< 2$ 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.