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Alexandru D. Ionescu

Publications and source records attributed to Alexandru D. Ionescu.

At least 19 recordsLinked to original sources
arXiv2026

Finite-time blow-up for the real-valued defocusing energy-supercritical quintic NLW

We consider the defocusing quintic wave equation $\partial_t^2u-Δu+u^5=0$ for real-valued functions on $\mathbb{R}^{1+10}$, which is energy-supercritical. We construct a discretely self-similar solution in a backward light cone, of the form $$u(t,x)=(T-t)^{-1/2}W^*\big(ω^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$ where the nonzero profile $W^*$ is $2π$-periodic in its first variable and real-analytic up to and across the light cone. Cutting off its initial data gives smooth, compactly supported, radial data whose solution coincides with the discretely self-similar solution in the backward light cone, is smooth on $[0,T]\times\mathbb{R}^{10}$ except at the vertex $(T,0)$ of the cone, and blows up there at the self-similar rate. The solutions we construct here appear to be the first real-valued blow-up solutions of defocusing energy-supercritical NLW. Existence of the real profile is proved by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients, with rigorous error control. We find the approximate profile by numerically continuing rotating self-similar profiles of the complex-valued equation. This step is not essential for the blow-up proof, so it is left as non-rigorous numerics. The main analytic ingredients are an exact description of the mode operators for the linear part, which are hypergeometric and upper triangular in the Chebyshev basis, and a bound for their inverses that is uniform at high frequencies. All computer-assisted proofs, including the requisite codes and the approximate objects, are accessible on GitHub at \cite{code}.

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arXiv2026

Nonlinear Landau damping for the unconfined Vlasov-Poisson system in 2D

We prove nonlinear Landau damping for the two-dimensional Vlasov--Poisson system in the unconfined Euclidean setting near the homogeneous Poisson equilibrium \[ μ(v)=\frac{1}{2π}(1+|v|^2)^{-3/2}. \] For small, smooth, localized, and neutral perturbations we establish global existence, quantitative pointwise decay estimates for the electric field, and scattering to free transport. The result we prove here appears to be the first nonlinear Landau damping result for solution of the unconfined 2D Vlasov--Poisson system. The uniform Penrose stability bound degenerates at low spatial frequencies, giving rise to weakly damped plasma oscillations. We separate a faster-decaying regular field from oscillatory components with temporal factors $\cos(t)$ and $\sin(t)$. The proof combines a Volterra representation of the density, weighted estimates for the characteristic flow and nonlinear moments, and integration by parts in time. Neutrality supplies the spatial cancellation needed in the critical two-dimensional estimates.

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arXiv2026

On the non-uniqueness of solutions of the axi-symmetric swirl-free Navier-Stokes equations, I

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and Šverák and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.

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arXiv2025

Quasiperiodic solutions of the generalized SQG equation

This monograph addresses an important problem in mathematical fluid dynamics: constructing stable, long-term solutions to certain quasilinear evolution equations. We implement an elaborate scheme for building global quasiperiodic solutions without relying on external parameters. Instead of relying on artificial external parameters, we exploit the natural structure of initial data to generate families of stable solutions. This approach offers a more robust framework for studying global solutions of quasilinear PDEs. The book combines techniques from KAM theory, a Nash-Moser iteration scheme, and pseudo-differential calculus, and provides tools that extend beyond the specific SQG context and may prove useful for other evolution equations. Specifically, we establish the existence of quasiperiodic patch solutions for the generalized Surface Quasi-Geostrophic equation (SQG) across the parameter range $α\in (1,2)$, in a neighborhood of the disk solution. These solutions exist globally in time without developing singularities, addressing an important question about the behavior of geophysical fluid models. This work provides new insights into global dynamics in a mathematically challenging regime where standard perturbative methods are insufficient. The techniques developed here offer potential applications to other evolution equations in mathematical physics, making this a valuable resource for researchers in partial differential equations, fluid dynamics, and related fields.

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arXiv2025

On the wave turbulence theory of 2D gravity waves, I: deterministic energy estimates

Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKE) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as semilinear Schrödinger equations or multi-dimensional KdV-type equations. However, our situation here is different since the water waves equations are quasilinear and the solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue, in the context of 2D gravity waves.

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arXiv2023

On the stability of homogeneous equilibria in the Vlasov-Poisson system on $\mathbb{R}^3$

The goal of this article is twofold. First, we investigate the linearized Vlasov-Poisson system around a family of spatially homogeneous equilibria in $\mathbb{R}^3$ (the unconfined setting). Our analysis follows classical strategies from physics and their subsequent mathematical extensions. The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green's functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work on the full global nonlinear asymptotic stability of the Poisson equilibrium in $\mathbb{R}^3$.

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arXiv2022

Polynomial sequences in discrete nilpotent groups of step 2

We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular, we present what we call a "nilpotent circle method" that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

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arXiv2022

Polynomial averages and pointwise ergodic theorems on nilpotent groups

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $σ$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $ρ$-variational inequalities on $L^2$ for $2<ρ<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

math.DS↗
arXiv2021

Linear vortex symmetrization: the spectral density function

We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand the smoothness of what we call the spectral density function associated with the linearized operator, which we hope will be a step towards proving full nonlinear asymptotic stability of radially decreasing vortices. The motivation for considering the spectral density function is that it is not possible to describe the vorticity or the stream function in terms of one modulated profile. There are in fact two profiles, both at the level of the physical vorticity and at the level of the stream function. The spectral density function allows us to identify these profiles, and its smoothness leads to pointwise decay of the stream function which is consistent with the decay estimates first proved in Bedrossian-Coti Zelati-Vicol in [5].

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arXiv2020

The Einstein-Klein-Gordon coupled system: global stability of the Minkowski solution

We prove definitive results on the global stability of the flat space among solutions of the Einstein-Klein-Gordon system. Our main theorems in this monograph include: (1) A proof of global regularity (in wave coordinates) of solutions of the Einstein-Klein-Gordon coupled system, in the case of small, smooth, and localized perturbations of the stationary Minkowski solution; (2) Precise asymptotics of the metric components and the Klein-Gordon field as the time goes to infinity, including the construction of modified (nonlinear) scattering profiles and quantitative bounds for convergence; (3) Classical estimates on the solutions at null and timelike infinity, such as bounds on the metric components, weak peeling estimates of the Riemann curvature tensor, ADM and Bondi energy identities and estimates, and asymptotic description of null and timelike geodesics.

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arXiv2020

Nonlinear inviscid damping near monotonic shear flows

We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $\mathbb{T}\times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.

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arXiv2019

On the global regularity for a Wave-Klein-Gordon coupled system

We consider a coupled Wave-Klein-Gordon system in 3D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch-Ma as a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields.

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arXiv2018

Optimal local well-posedness theory for the kinetic wave equation

We prove local existence and uniqueness results for the (space-homogeneous) 4-wave kinetic equation in wave turbulence theory. We consider collision operators defined by radial, but general dispersion relations satisfying suitable bounds, and we prove two local well-posedness theorems in nearly critical weighted spaces.

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arXiv2017

Recent advances on the global regularity for irrotational water waves

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach, as in [7, 1, 2]. We then describe in a unified framework, using the Eulerian formulation, global existence results for three dimensional and two dimensional gravity waves, see [70, 146, 145, 87, 5, 6, 79, 80, 136], and our joint result with Deng and Pausader [60] on global regularity for the 3D gravity-capillary model. We conclude this review with a short discussion about the formation of singularities, and give a few additional references to other interesting topics in the theory.

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arXiv2017

Global solutions for the generalized SQG patch equation

We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter $α\in (1,2)$. The cases $α= 0$ and $α= 1$ correspond to 2d Euler and SQG respectively, and our choice of the parameter $α$ results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.

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arXiv2017

Global analysis of a model for capillary water waves in 2D

In this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension two (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time-resonance structure and modified scattering. It is slightly simpler, however, at the technical level and our goal here is to present our method in this simplified situation. The full system is the subject of a forthcoming paper.

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