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We show a stability result for the Schwarzschild singularity (inside theblack hole region) for the Einstein vacuum equations. The result is proven inthe class of polarized axial symmetry, under perturbations of the Schwarzschilddata induced on a hypersurface $\{r=\e\}$, $\e<<2M$. Our result is only partlya stability result, in that we show that while a (space-like) singularitypersists under perturbations as above, the behaviour of the metric approachingthe singularity is much more involved than for the Schwarzschild solution.Indeed, we find that the solution displaysasymptocially-velocity-term-dominated dynamics and approaches a differentKasner solution at each point of the singularity. These Kasner-type asymptoticsare very far from isotropic, since (as in Schwarzschild) there are twocontracting directions and one expanding one.  Our proof relies on energy methods and on a new approach to the EVE in axialsymmetry, which we believe has wider applicability: In this symmetry class andunder a suitable geodesic gauge, the EVE can be studied as a free wave coupledto (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-timeto the free wave. The fact that the nonlinear part of the Einstein equations isdescribed by ODEs lies at the heart of how one can overcome a certain linearinstability exhibited by the singularity.

Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors