In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner–Nordström family of black holes in the spherically symmetric Einstein–Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one “submanifold” ${\mathfrak{M}}_{\mathrm{stab}}$ of the moduli space of spherically symmetric characteristic data for the Einstein–Maxwell-scalar field system lying close to the extremal Reissner–Nordström family, such that any data in ${\mathfrak{M}}_{\mathrm{stab}}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner–Nordström family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$ , (iv) for “generic” data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit nondecay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the “stability” of the extremal Reissner–Nordström family and points (iii) and (iv) verify the presence of the celebrated Aretakis instability [11] for the linear wave equation on extremal Reissner–Nordström black holes in the full nonlinear Einstein–Maxwell-scalar field model.

In this paper, we give a complete description of the black hole threshold, locally near the Reissner-Nordström family, in the infinite-dimensional moduli space 𝔐 of dynamical spherically symmetric solutions to the Einstein-Maxwell-neutral scalar field system. In a neighborhood of the full Reissner-Nordström family in 𝔐, we prove the following: (i) Any solution that forms a black hole eventually decays to a Reissner-Nordström black hole. (ii) Any solution that fails to collapse into a black hole eventually becomes superextremal along null infinity and exists globally in the domain of dependence of the bifurcate characteristic initial data. (iii) The subset of this neighborhood consisting of black hole solutions admits a C^1 foliation by hypersurfaces of constant final charge-to-mass ratio, up to and including extremality. (iv) The mutual boundary between the set of black hole solutions and noncollapsing solutions, i.e., the black hole threshold, is the extremal leaf of the foliation. Black holes which are not on the threshold are asymptotically subextremal. Our quantitative control of near-threshold solutions allows us to prove "universal" scaling laws for the location of the event horizon and its final area and temperature (surface gravity), with critical exponent 1/2. Moreover, we show that the celebrated Aretakis instability is activated for an open and dense set of threshold solutions and that generic near-threshold subextremal black holes experience a transient horizon instability on the timescale of their inverse final temperature.

We study scattering solutions ϕ of the linear wave equation on extremal Reissner-Nordström spacetimes, satisfying the following properties: i) ϕ attains a prescribed radiation field ψ_ℐ through future null infinity, which decays at an inverse polynomial rate; ii) ϕ is regular in the exterior region up to and including the future event horizon, i.e. ϕ∈ C^N, where N≫1 is independent of the decay rate of ψ_ℐ. We prove that such solutions exist for arbitrary N, and that they are not unique. The proof consists of: 1) finding an approximate solution ϕ_app with fast decaying error; 2) the use of backwards energy estimates in order to correct ϕ_app to an exact solution. Extremality is used only in the second step. The methods of the linear case described above are then used to show the same results for semilinear equations where the nonlinearity satisfies the null condition, as well as to geometries describing the hyperbolic orbit of multiple extremal Reissner-Nordström black holes.

We revisit global existence and decay for small-data solutions of semilinear wave equations on extremal Reissner–Nordström black hole backgrounds satisfying the classical null condition, a problem which was previously addressed by the first author in joint work with Angelopoulos et al. (Ann PDE 6(2):12, 2020). In this paper, we develop a new approach based on propagating a significantly weaker set of estimates, which allows for a simpler and more streamlined proof. Our proof does not require tracking sharp estimates for the solution in the near-horizon region, which means that it is compatible with, but does not imply, the non-decay and growth hierarchy of derivatives of the solution along the event horizon expected from the Aretakis instability. In particular, this approach is in principle compatible with other settings where stronger horizon instabilities are expected, such as nonlinear charged scalar fields on extremal Reissner–Nordström, or nonlinear waves on extremal Kerr. We also sketch how our proof applies to semilinear problems on spacetimes settling down to extremal Reissner–Nordström, such as those constructed in our joint work with Angelopoulos et al. (Nonlinear stability of extremal Reissner–Nordström black holes in spherical symmetry, 2024. arXiv:2410.16234 ).

We consider a supersonic flow past an airfoil in the context of the steady, isentropic and irrotational compressible Euler equations. We show that for appropriate data, either a shock forms, in which case certain derivatives of the solution blow up, or a sonic line forms, which translates to the fact that the hyperbolic character of the equations degenerates.

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies $ > 1 and $ > 2. The $dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting ("Price's law"). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing $-dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for addressing conflicting results on the numerology of Kerr late-time asymptotics appearing in the numerics literature and definitively determining the correct numerology. (c) 2023 Elsevier Inc. All rights reserved.

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies $\ell\geq 1$ and $\ell\geq 2$. The $\ell$-dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting ("Price's law"). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing $\ell$-dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for resolving the conflicting numerology in Kerr late-time asymptotics that appears in the numerics literature.

In this paper, we prove precise late-time asymptotics for solutions to the wave equation supported on angular frequencies greater or equal to ℓ on the domain of outer communications of subextremal Reissner–Nordström spacetimes up to and including the event horizon. Our asymptotics yield, in particular, sharp upper and lower decay rates which are consistent with Price’s law on such backgrounds. We present a theory for inverting the time operator and derive an explicit representation of the leading-order asymptotic coefficient in terms of the Newman–Penrose charges at null infinity associated with the time integrals. Our method is based on purely physical space techniques. For each angular frequency ℓ , we establish a sharp hierarchy of r-weighted radially commuted estimates with length 2ℓ +5 . We complement this hierarchy with a novel hierarchy of weighted elliptic estimates of length ℓ +1 .

In this short note we present an instability result for transonic flows with respect to perturbations of the Mach number at infinity. More specifically we show that a perturbation of a transonic solution in the context of a Cauchy problem for the 2-dimensional steady, isentropic and irrotational Euler equations, which is a solution that solves an elliptic equation in certain parts of its domain and a hyperbolic equation in other parts, can only take place in the analytic category if the elliptic part of the domain is affected.

We present the first rigorous study of nonlinear wave equations on extremal black hole spacetimes without any symmetry assumptions on the solution. Specifically, we prove global existence with asymptotic blow-up for solutions to nonlinear wave equations satisfying the null condition on extremal Reissner–Nordström backgrounds. This result shows that the extremal horizon instability persists in model nonlinear theories. Our proof crucially relies on a new vector field method that allows us to obtain almost sharp decay estimates.

We construct semi-global $(1+3)$-dimensional Lorentzian spacetimes satisfying the Einstein vacuum equations that contain curvature singularities that are propagated all the way up to future null infinity. Special cases of our constructions are semi-global spacetimes containing the interaction of two impulsive gravitational waves. The spacetimes that we construct can be considered as the semi-global analogues of the local spacetimes containing weak null singularities constructed by Luk, and of the local spacetimes containing the interaction of impulsive gravitational waves constructed by Luk and Rodnianski.

It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner–Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.

We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner–Nordström black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti–Murata–Reall–Tanahashi, Ori–Sela and Blaksley–Burko.

We consider discrete analogues of two well-known open problems regarding invariant measures for dispersive PDE, namely, the invariance of the Gibbs measure for the continuum (classical) Heisenberg model and the invariance of white noise under focusing cubic nonlinear Schrödinger equation. These continuum models are completely integrable and connected by the Hasimoto transform; correspondingly, we focus our attention on discretizations that are also completely integrable and also connected by a discrete Hasimoto transform. We consider these models on the infinite lattice ${\mathbb{Z}}$.Concretely, for a completely integrable variant of the classical Heisenberg spin chain model (introduced independently by Haldane, Ishimori, and Sklyanin) we prove the existence and uniqueness of solutions for initial data following a Gibbs law (which we show is unique) and show that the Gibbs measure is preserved under these dynamics. In the setting of the focusing Ablowitz--Ladik system, we prove invariance of a measure that we will show is the appropriate discrete analogue of white noise.We also include a thorough discussion of the Poisson geometry associated to the discrete Hasimoto transform introduced by Ishimori that connects the two models studied in this article.

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner-Nordstrom families of black holes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections were heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman-Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for compactly supported initial data (and hence with a vanishing Newman-Penrose constant) explicitly obtained here appear to be new.

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordström black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.

In our previous work [Y. Angelopoulos, S. Aretakis, and D. Gajic, Late-time asymptotics for the wave equation on spherically symmetric stationary backgrounds, in Advances in Mathematics 323 (2018), 529-621] we showed that the coefficient in the precise leading-order late-time asymptotics for solutions to the wave equation with smooth, compactly supported initial data on Schwarzschild backgrounds is proportional to the time-inverted Newman-Penrose constant (TINP), that is the Newman-Penrose constant of the associated time integral. The time integral (and hence the TINP constant) is canonically defined in the domain of dependence of any Cauchy hypersurface along which the stationary Killing field is non-vanishing. As a result, an explicit expression of the late-time polynomial tails was obtained in terms of initial data on Cauchy hypersurfaces intersecting the future event horizon to the future of the bifurcation sphere. In this paper, we extend the above result to Cauchy hypersurfaces intersecting the bifurcation sphere via a novel geometric interpretation of the TINP constant in terms of a modified gradient flux on Cauchy hypersurfaces. We show, without appealing to the time integral construction, that a general conservation law holds for these gradient fluxes. This allows us to express the TINP constant in terms of initial data on Cauchy hypersurfaces for which the time integral construction breaks down.

It is shown that the conserved charges on the event horizon and the Cauchy horizon associated with scalar perturbations on extremal black holes are externally measurable from null infinity. This suggests that these charges have the potential to serve as an observational signature. The proof of this result is based on obtaining precise late-time asymptotics for the radiation field of outgoing perturbations.

We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner-Nordstrom families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman-Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

We show that degenerate horizons exhibit a new trapping effect . Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner–Nordström up to and including the future event horizon. We show that such an estimate requires (1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and (2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a new L^2 concentration phenomenon (namely, a global trapping effect, in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces. Our methods and results play a crucial role in our upcoming works on linear and nonlinear wave equations on extremal black hole backgrounds.