  The Ginzburg-Landau equations play a key role in superconductivity and
particle physics. They inspired many imitations in other areas of physics.
These equations have two remarkable classes of solutions -- vortices and
(Abrikosov) vortex lattices. For the standard cylindrical geometry, the
existence theory for these solutions, as well as the stability theory of
vortices are well developed. The latter is done within the context of the
time-dependent Ginzburg-Landau equations -- the Gorkov-Eliashberg-Schmid
equations of superconductivity -- and the abelian Higgs model of particle
physics.
  We study stability of Abrikosov vortex lattices under finite energy
perturbations satisfying a natural parity condition (both defined precisely in
the text) for the dynamics given by the Gorkov-Eliashberg-Schmid equations. For
magnetic fields close to the second critical magnetic field and for arbitrary
lattice shapes, we prove that there exist two functions on the space of
lattices, such that Abrikosov vortex lattice solutions are asymptotically
stable, provided the superconductor is of Type II and these functions are
positive, and unstable, for superconductors of Type I, or if one of these
functions is negative.
