arXiv subjects

Istvan Kadar

Publications and source records attributed to Istvan Kadar.

9 recordsLinked to original sources

arXiv2026

Smooth finite time singularity formation without quantization

We revisit the finite time singularity formation of Krieger-Schlag-Tataru [KST09] for the focusing energy critical wave equation in $\mathbb{R}^{3+1}$ from a geometric singular-analytic point of view, following Hintz [Hintz23]. We construct $C^{\nu/2-}$ regular approximate solutions that settle down to multiple solitons, shrinking at a rate $t^{\nu}$ with $\nu>1$, and approaching the origin on different geodesics $\{x=zt\}$. By fine tuning the velocities, sizes and signs of the solitons, we are able to construct smooth ans\"atze with any $\nu>8$. Using robust energy estimates, the ans\"atze are corrected to exact solutions.

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arXiv2026

Matching conditions for scattering solutions of scalar wave equations on extremal Reissner-Nordstr\"{o}m spacetimes

We study scattering solutions $\phi$ of the linear wave equation on extremal Reissner-Nordstr\"{o}m spacetimes, satisfying the following properties: i) $\phi$ attains a prescribed radiation field $\psi_{\mathcal{I}}$ through future null infinity, which decays at an inverse polynomial rate; ii) $\phi$ is regular in the exterior region up to and including the future event horizon, i.e. $\phi\in C^N$, where $N\gg1$ is independent of the decay rate of $\psi_{\mathcal{I}}$. We prove that such solutions exist for arbitrary $N$, and that they are not unique. The proof consists of: 1) finding an approximate solution $\phi_{\mathrm{app}}$ with fast decaying error; 2) the use of backwards energy estimates in order to correct $\phi_{\mathrm{app}}$ 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-Nordstr\"{o}m black holes.

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arXiv2026

A note on exterior stability of isolated singularity formation for nonlinear wave equations

We study the stability of the exterior of Type I and Type II singularity formation for the wave maps equation in $\mathbb{R}^{d+1}$ with $d\geq2$ and the power nonlinear wave equation in $\mathbb{R}^{d+1}$ with $d\geq3$:Given characteristic initial data on the backwards lightcone of the singularity $\mathcal{C}=\{t+r=0\}$ converging to the singular background solution along with suitable data on an outgoing cone, we establish existence in a region $\{t+r\in(0,v_1),t-r\in(-1,0)\}$ for some suitably small $v_1$, i.e. all the way to the Cauchy horizon. Our result hinges on a particular set of assumptions on the regularity properties of these initial data, which conjecturally can be recovered by a more detailed stability analysis of the behaviour inside the past light cone; indeed, in certain settings, this was achieved in [BDS21,KAD26], and we strongly expect they can be proved in many other settings as well. The proof goes via a suitable change of coordinates and an application of the scattering result of [KK25], which, in particular, also applies to scaling-critical potentials. While no symmetry assumption is made for the power nonlinear wave equation, we only provide the proof in the corotational symmetry class for the wave maps equation, but we also sketch how to lift this restriction.

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arXiv2026

Scattering and stability for ODE-type blow-up surfaces for focusing nonlinear wave equations

We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for solutions exhibiting ODE type blow-up on spacelike hypersurfaces, with the blow-up at each point modelled by the explicit solution $\phi_{\mathrm{model}} = c_p t^{-\alpha_p}$. Given a sufficiently regular spacelike hypersurface $\Sigma_f$, together with auxiliary scattering data $\psi$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $\Sigma_f$ attaining $\psi$ as scattering data. Conversely, we show that such ODE type singularities are (locally) stable to suitably regular perturbations away from the singularity, and that the blow-up surface and scattering data remain regular, in a continuously dependent manner, following such perturbations.

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arXiv2025

Scattering, Polyhomogeneity and Asymptotics for Quasilinear Wave Equations From Past to Future Null Infinity

We present a general construction of semiglobal scattering solutions to quasilinear wave equations in a neighbourhood of spacelike infinity including past and future null infinity, where the scattering data are posed on an ingoing null cone and along past null infinity. More precisely, we prove weighted, optimal-in-decay energy estimates and propagation of polyhomogeneity statements from past to future null infinity for these solutions, we provide an algorithmic procedure how to compute the precise coefficients in the arising polyhomogeneous expansions, and we apply this procedure to various examples. As a corollary, our results directly imply the summability in the spherical harmonic number $\ell$ of the estimates proved for fixed spherical harmonic modes in the papers [Keh22b,KM24] from the series "The Case Against Smooth Null Infinity". Our (physical space) methods are based on weighted energy estimates near spacelike infinity similar to those of [HV23], commutations with (modified) scaling vector fields to remove leading order terms in the relevant expansions, time inversions, as well as the Minkowskian conservation laws: $$ \partial_u(r^{-2\ell}\partial_v(r^2\partial_v)^{\ell}(r\phi_{\ell}))=0, $$ which are satisfied if $\Box_\eta\phi=0$. Our scattering constructions apply to systems of equations as well and go beyond the usual class of finite energy solutions. We use this to also derive a scattering theory and prove propagation of polyhomogeneity for the Einstein vacuum equations in a harmonic gauge. In the process, we also need to introduce a novel ansatz accounting for the stronger-than-Schwarzschildean divergence of the light cones, which, in particular, extends existing exterior stability of Minkowski statements in harmonic gauge to allow for slowly decaying data as considered in [Bie10].

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arXiv2024

Construction of multi-soliton solutions for the energy critical wave equation in dimension 3

We study the energy-critical wave equation in three dimensions, focusing on its ground state soliton, denoted by $W$. Using the Poincar\'e symmetry inherent in the equation, boosting $W$ along any timelike geodesic yields another solution. The slow decay behavior of $W$, $W\sim r^{-1}$, indicates a strong interaction among potential multi-soliton solutions. In this paper, for arbitrary $N\geq0$, we provide an algorithmic procedure to construct approximate solutions to the energy critical wave equation that: (1) converge to a superposition of solitons, (2) have no outgoing radiation, (3) their error to solve the equation decays like $(t-r)^{-N}$. Then, we show that this approximate solution can be corrected to a real solution.

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arXiv2024

A scattering theory construction of dynamical solitons in 3d

We study the energy critical wave equation in 3 dimensions around a single soliton. We obtain energy boundedness (modulo unstable modes) for the linearised problem. We use this to construct scattering solutions in a neighbourhood of timelike infinity ($i_+$), provided the data on null infinity ($\scri$) decay polynomially. Moreover, the solutions we construct are conormal on a blow-up of Minkowski space. The methods of proof also extend to some energy supercritical modifications of the equation.

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arXiv2023

Small data nonlinear wave equation numerology: The role of asymptotics

Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to the our classification. Furthermore, we given two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [DP18].

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arXiv2022

On global behaviour of classical effective field theories

We continue the rigorous study of classical effective field theories (EFTs) that was recently initiated in the work of Reall and Warnick [RW22]. We study a system with one light and one heavy field with cubic coupling and prove global existence (of the UV solution) under an effective norm in the high mass limit. Furthermore, we prove that the global solution linearly scatters (in Lax-Philips sense) and that this final state has an expansion in inverse powers of the mass, and can be recovered from the EFT equation alone.

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