Vortex formation for a non-local interaction model with Newtonian
repulsion and superlinear mobility

We consider density solutions for gradient flow equations of the form $u_t =
\nabla \cdot ( \gamma(u) \nabla \mathrm N(u))$, where $\mathrm N$ is the
Newtonian repulsive potential in the whole space $\mathbb R^d$ with the
nonlinear convex mobility $\gamma(u)=u^\alpha$, and $\alpha>1$. We show that
solutions corresponding to compactly supported initial data remain compactly
supported for all times leading to moving free boundaries as in the linear
mobility case $\gamma(u)=u$. For linear mobility it was shown that there is a
special solution in the form of a disk vortex of constant intensity in space
$u=c_1t^{-1}$ supported in a ball that spreads in time like $c_2t^{1/d}$, thus
showing a discontinuous leading front or shock. Our present results are in
sharp contrast with the case of concave mobilities of the form
$\gamma(u)=u^\alpha$, with $0<\alpha<1$ studied in [9]. There, we developed a
well-posedness theory of viscosity solutions that are positive everywhere and
moreover display a fat tail at infinity. Here, we also develop a well-posedness
theory of viscosity solutions that in the radial case leads to a very detail
analysis allowing us to show a waiting time phenomena. This is a typical
behavior for nonlinear degenerate diffusion equations such as the porous medium
equation. We will also construct explicit self-similar solutions exhibiting
similar vortex-like behaviour characterizing the long time asymptotics of
general radial solutions under certain assumptions. Convergent numerical
schemes based on the viscosity solution theory are proposed analysing their
rate of convergence. We complement our analytical results with numerical
simulations ilustrating the proven results and showcasing some open problems.