Stable ground states and self-similar blow-up solutions for the
  gravitational Vlasov-Manev system
  In this work, we study the orbital stability of steady states and the
existence of blow-up self-similar solutions to the so-called Vlasov-Manev (VM)
system. This system is a kinetic model which has a similar Vlasov structure as
the classical Vlasov-Poisson system, but is coupled to a potential in $-1/r-
1/r^2$ (Manev potential) instead of the usual gravitational potential in
$-1/r$, and in particular the potential field does not satisfy a Poisson
equation but a fractional-Laplacian equation. We first prove the orbital
stability of the ground states type solutions which are constructed as
minimizers of the Hamiltonian, following the classical strategy: compactness of
the minimizing sequences and the rigidity of the flow. However, in driving this
analysis, there are two mathematical obstacles: the first one is related to the
possible blow-up of solutions to the VM system, which we overcome by imposing a
sub-critical condition on the constraints of the variational problem. The
second difficulty (and the most important) is related to the nature of the
Euler-Lagrange equations (fractional-Laplacian equations) to which classical
results for the Poisson equation do not extend. We overcome this difficulty by
proving the uniqueness of the minimizer under equimeasurabilty constraints,
using only the regularity of the potential and not the fractional-Laplacian
Euler-Lagrange equations itself. In the second part of this work, we prove the
existence of exact self-similar blow-up solutions to the Vlasov-Manev equation,
with initial data arbitrarily close to ground states. This construction is
based on a suitable variational problem with equimeasurability constraint.
