| Čoahkkáigeassu: | We consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla
\mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the
Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the
mobility. For linear mobility, $\gamma(u)=u$, the equation and some variations
have been proposed as a model for superconductivity or superfluidity. In that
case the theory leads to uniqueness of bounded weak solutions having the
property of compact space support, and in particular 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.
In this paper we propose the model with sublinear mobility
$\gamma(u)=u^\alpha$, with $0<\alpha<1$, and prove that nonnegative solutions
recover positivity everywhere, and moreover display a fat tail at infinity. The
model acts in many ways as a regularization of the previous one. In particular,
we find that the equivalent of the previous vortex is an explicit self-similar
solution decaying in time like $u=O(t^{-1/\alpha})$ with a space tail with size
$u=O(|x|^{- d/(1-\alpha)})$. We restrict the analysis to radial solutions and
construct solutions by the method of characteristics. We introduce the mass
function, which solves an unusual variation of Burger's equation, and plays an
important role in the analysis. We show well-posedness in the sense of
viscosity solutions. We also construct numerical finite-difference convergent
schemes.
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