  We show a stability result for the Schwarzschild singularity (inside the
black hole region) for the Einstein vacuum equations. The result is proven in
the class of polarized axial symmetry, under perturbations of the Schwarzschild
data induced on a hypersurface $\{r=\e\}$, $\e<<2M$. Our result is only partly
a stability result, in that we show that while a (space-like) singularity
persists under perturbations as above, the behaviour of the metric approaching
the singularity is much more involved than for the Schwarzschild solution.
Indeed, we find that the solution displays
asymptocially-velocity-term-dominated dynamics and approaches a different
Kasner solution at each point of the singularity. These Kasner-type asymptotics
are very far from isotropic, since (as in Schwarzschild) there are two
contracting directions and one expanding one.
  Our proof relies on energy methods and on a new approach to the EVE in axial
symmetry, which we believe has wider applicability: In this symmetry class and
under a suitable geodesic gauge, the EVE can be studied as a free wave coupled
to (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-time
to the free wave. The fact that the nonlinear part of the Einstein equations is
described by ODEs lies at the heart of how one can overcome a certain linear
instability exhibited by the singularity.
