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A. D. Ionescu

Publications and source records attributed to A. D. Ionescu.

12 recordsLinked to original sources
arXiv2024

Nonlinear Landau damping and wave operators in sharp Gevrey spaces

We prove nonlinear Landau damping in optimal weighted Gevrey-3 spaces for solutions of the confined Vlasov-Poisson system on $\T^d\times\R^d$ which are small perturbations of homogeneous Penrose-stable equilibria. We also prove the existence of nonlinear scattering operators associated to the confined Vlasov-Poisson evolution, as well as suitable injectivity properties and Lipschitz estimates (also in weighted Gevrey-3 spaces) on these operators. Our results give definitive answers to two well-known open problems in the field, both of them stated in the recent review of Bedrossian [4, Section 6].

math.AP↗
arXiv2019

Long-time existence for multi-dimensional periodic water waves

We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of $\varepsilon^{-5/3+}$, for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of the equations which gives additional smoothing close to the resonant hypersurfaces, (2) another structural property, connected to time-reversibility, that allows us to handle "trivial" cubic resonances, (3) sharp small divisors lower bounds on three and four-waves modulation functions based on counting arguments, and (4) partial normal form transformations and symmetrization arguments in the Fourier space. Our theorem appears to be the first extended lifespan result for quasilinear equations with non-trivial resonances on a multi-dimensional torus.

math.AP↗
arXiv2018

Global solutions of the gravity-capillary water-wave system in three dimensions

In this paper we prove global regularity for the full water waves system in 3 dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water wave models. To construct global solutions we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than $|t|^{-5/6}$) and the presence of a large, codimension 1, set of quadratic time-resonances. To deal with such a situation we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the nonlinearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions we use Harmonic Analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.

math.AP↗
arXiv2014

On the global stability of the wave-map equation in Kerr spaces with small angular momentum

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $Φ$ defined from a fixed Kerr solution $\KK(M,a)$, $0\le a < M $, with values in the two dimensional hyperbolic space $\HHH^2$. A particular such wave map is given by the complex Ernst potential associated to the axial Killing vectorfield $\Z$ of $\KK(M,a)$. We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of $\KK(M,a)$, for all $0\le a<M$ and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.

math.AP↗
arXiv2013

Rigidity of stationary black holes with small angular momentum on the horizon

We prove a black hole rigidity result for slowly rotating stationary solutions of the Einstein vacuum equations. More precisely, we prove that the domain of outer communications of a regular stationary vacuum is isometric to the domain of outer communications of a Kerr solution, provided that the stationary Killing vector-field $\T$ is small on the bifurcation sphere.

gr-qc↗
arXiv2009

Hawking's local rigidity theorem without analyticity

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein space-time. We do not assume analyticity of the space-time. This result will be applied in a second paper to prove a perturbative version of the uniqueness of smooth, stationary black holes in vacuum.

gr-qc↗
arXiv2007

Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper.

math.AP↗