Mass inflation from rough initial data for the spherically symmetric Einstein-Maxwell-scalar field system with $\Lambda$

Recent rigorous results on black hole interiors clearly suggest that the strong cosmic censorship conjecture fails in its most fundamental, i.e. weak, formulation: violations are expected for a class of spherically symmetric charged black holes in the presence of a positive cosmological constant near extremality. These results require sufficiently regular solutions. Conversely, when non-smooth, finite-energy initial data are prescribed for linear waves propagating on a fixed black hole background belonging to the aforementioned family, it was shown that the local energy of these linear waves blows up at the Cauchy horizon, hence hinting that non-smooth initial data may suppress the possible violations of the $H^1$ formulation of strong cosmic censorship. In line with this intuition, we prove that rough initial data can also trigger an instability at the Cauchy horizon in the non-linear setting, via mass inflation. In particular, we analyse a characteristic initial value problem for the spherically symmetric Einstein-Maxwell-real scalar field system describing the interior of a black hole. Our results show that, when prescribing 1) initial data asymptotically approaching those of a sub-extremal Reissner-Nordstr\"om-de Sitter solution, and 2) initial data belonging to $W^{1, 2}\setminus W^{1, q}$, for every $q > 2$, along the initial ingoing compact segment; then the Hawking mass diverges at the Cauchy horizon of the black hole solution we construct, for every parameter choice of the reference black hole. In this larger family of configurations, we prove that the smooth data suggesting violations of strong cosmic censorship are non-generic in a ``positive co-dimension'' sense, conditionally to the validity of the expected Price law bounds. Moreover, we illustrate the transition between smooth and rough initial data.