arXiv subjects

Jeremie Szeftel

Publications and source records attributed to Jeremie Szeftel.

At least 19 recordsLinked to original sources

arXiv2022

Brief introduction to the nonlinear stability of Kerr

This a brief introduction to the sequence of works \cite{KS:Kerr}, \cite{GKS-2022}, \cite{KS-GCM1}, \cite{KS-GCM2} and \cite{Shen} which establish the nonlinear stability of Kerr black holes with small angular momentum. We are delighted to dedicate this article to Demetrios Christodoulou for whom we both have great admiration. The first author would also like to thank Demetrios for the magic moments of friendship, discussions and collaboration he enjoyed together with him.

math.AP↗

arXiv2022

Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes

This is the last part of our proof of the nonlinear stability of the Kerr family for small angular momentum, i.e $|a|/m\ll 1$, in which we deal with the nonlinear wave type estimates needed to complete the project. More precisely we provide complete proofs for Theorems M1 and M2 as well the curvature estimates of Theorem M8, which were stated without proof in sections 3.7.1 and 9.4.7 of \cite{KS:Kerr}. Our procedure is based on a new general interest formalism (detailed in Part I of this work), which extends the one used in the stability of Minkowski space. Together with \cite{KS:Kerr} and the GCM papers \cite{KS-GCM1}, \cite{KS-GCM2}, \cite{Shen}, this work completes proof of the Main Theorem stated in Section 3.4 of \cite{KS:Kerr}.

math.AP↗

arXiv2021

Kerr stability for small angular momentum

This is our main paper in a series in which we prove the full, unconditional, nonlinear stability of the Kerr family $Kerr(a, m)$ for small angular momentum, i.e. $|a|/m\ll 1$, in the context of asymptotically flat solutions of the Einstein vacuum equations (EVE). Three papers in the series, \cite{KS-GCM1} and \cite{KS-GCM2} and \cite{GKS1} have already been released. We expect that the remaining ones \cite{GKS2}, \cite{KS:Kerr-B} and \cite{Shen} will appear shortly. Our work extends the strategy developed in \cite{KS}, in which only axial polarized perturbations of Schwarzschild were treated, by developing new geometric and analytic ideas on how to deal with with general perturbations of Kerr. We note that the restriction to small angular momentum appears only in connection to Morawetz type estimates in \cite{GKS2} and \cite{KS:Kerr-B}

math.AP↗

arXiv2020

On the implosion of a three dimensional compressible fluid

We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. Two essential steps of the analysis are the existence of $\mathcal C^\infty$ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow which are addressed in the companion papers \cite{MRRSprofile, MRRSdefoc}. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

math.AP↗

arXiv2019

Effective results on uniformization and intrinsic GCM spheres in perturbations of Kerr

This is a follow-up of our paper \cite{KS-Kerr1} on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of $\ell=1$ modes on a $2$-sphere embedded in a $1+3$ vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr.

math.AP↗

arXiv2019

On smooth self similar solutions to the compressible Euler equations

We consider the barotropic Euler equations in dimension d>1 with decaying density at spatial infinity. The phase portrait of the nonlinear ode governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated light (acoustic) cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic C^\infty self-similar solutions with suitable decay at infinity. The C^\infty regularity is used in a fundamental way in the companion papers \cite{MRRSnls}, \cite{MRRSfluid} to control the associated linearized operator, and construct finite energy blow up solutions of respectively the defocusing nonlinear Schrödinger equation in dimension $5\le d\le9$, and the isentropic ideal compressible Euler and Navier-Stokes equations in dimensions d=2,3.

math.AP↗

arXiv2019

On blow up for the energy super critical defocusing non linear Schrödinger equations

We consider the energy supercritical defocusing nonlinear Schrödinger equation $i\partial_tu+Δu-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal C^\infty$ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression in the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $\mathcal C^\infty$ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper \cite{MRRSprofile}.

math.AP↗

arXiv2019

Constructions of GCM spheres in perturbations of Kerr

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for $|a|\ll m$. The paper builds on the strategy laid out in \cite{KS} in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of \cite{KS} was the introduction and construction of generally covariant modulated (GCM) hypersurfaces on which specific geometric quantities take Schwarzschildian values. This was made possible by taking into account the full general covariance of the Einstein vacuum equations. The goal of this paper is to get rid of the symmetry restriction in the construction of GCM spheres and thus remove an essential obstruction in extending the result of \cite{KS} to a full stability proof of the Kerr family.

math.AP↗

arXiv2018

Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

We prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, i.e. solutions of the Einstein vacuum equations for asymptotically flat $1+3$ dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield with closed orbits. While building on the remarkable advances made in last 15 years on establishing quantitative linear stability, the paper introduces a series of new ideas among which we emphasize the general covariant modulation (GCM) procedure which allows us to construct, dynamically, the center of mass frame of the final state. The mass of the final state itself is tracked using the well known Hawking mass relative to a well adapted foliation itself connected to the center of mass frame. Our work here is the first to prove the nonlinear stability of Schwarzschild in a restricted class of nontrivial perturbations. To a large extent, the restriction to this class of perturbations is only needed to ensure that the final state of evolution is another Schwarzschild space. We are thus confident that our procedure may apply in a more general setting.

gr-qc↗

arXiv2017

On strongly anisotropic type I blow up

We consider the energy super critical 4 dimensional semilinear heat equation $$\partial_tu=Δu+|u|^{p-1}u, \ \ x\in \Bbb R^4, \ \ p>5.$$ Let $Φ(r)$ be a three dimensional radial self similar solution for the three supercritical probmem as exhibited and studied in \cite{CRS}. We show the finite codimensional transversal stability of the corresponding blow up solution by exhibiting a manifold of finite energy blow up solutions of the four dimensional problem with cylindrical symmetry which blows up as $$u(t,x)\sim \frac{1}{(T-t)^{\frac{1}{p-1}}}U(t,Y), \ \ Y=\frac{x}{\sqrt{T-t}}$$ with the profile $U$ given to leading order by $$U(t,Y)\sim\frac{1}{(1+b(t)z^2)^{\frac 1{p-1}}}Φ\left(\frac{r}{\sqrt{1+b(t)z^2}}\right), \ \ Y=(r,z), \ \ b(t)=\frac{c}{|\log(T-t)|}$$ corresponding to a constant profile $Φ(r)$ in the $z$ direction reconnected to zero along the moving free boundary $|z(t)|\sim \frac{1}{\sqrt{b}}\sim \sqrt{|\log (T-t)|}.$ Our analysis revisits the stability analysis of the self similar ODE blow up \cite{BK, MZduke,MZgaffa} and combines it with the study of the Type I self similar blow up \cite{CRS}. This provides a robust canonical framework for the construction of strongly anisotropic blow up bubbles.

math.AP↗

arXiv2017

Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System

In this paper we consider the equivariant 2+1 dimensional Einstein-wave map system and show that if the target satisfies the so called Grillakis condition, then global existence holds. In view of the fact that the 3+1 vacuum Einstein equations with a spacelike translational Killing field reduce to a 2+1 dimensional Einstein-wave map system with target the hyperbolic plane, which in particular satisfies the Grillakis condition, this work proves global existence for the equivariant class of such spacetimes.

math.AP↗

arXiv2016

On the stability of type I blow up for the energy super critical heat equation

We consider the energy super critical semilinear heat equation $$\partial_t u=Δu+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.$$ We first revisit the construction of radially symmetric backward self similar solutions and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.

math.AP↗

arXiv2015

Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain symmetry class the existence, in the neighborhood of the catenoid initial data, of a co-dimension 1 Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid.

math.AP↗

arXiv2014

Near soliton dynamics and singularity formation for $L^2$ critical problems

This survey reviews the state of the art concerning the singularity formation for two canonical dispersive problems: the mass critical non linear Schrödinger equation and the mass critical generalized KdV equation. In particular, we address the question of the classification of the flow for initial data near the soliton.

math.AP↗

arXiv2014

The Bounded L2 Curvature Conjecture

This is the main paper in a sequence in which we give a complete proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, $L^2$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1+1 (based on Strichartz estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear $so(3,1)$-valued Yang-Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of \textit{null structure} compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including $L^2$ error bounds which is carried out in [Sz1]-[Sz4], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in \cite{Sz5}.

math.AP↗

arXiv2014

Variants of the focusing NLS equation. Derivation, justification and open problems related to filamentation

The focusing cubic NLS is a canonical model for the propagation of laser beams. In dimensions 2 and 3, it is known that a large class of initial data leads to finite time blow-up. Now, physical experiments suggest that this blow-up does not always occur. This might be explained by the fact that some physical phenomena neglected by the standard NLS model become relevant at large intensities of the beam. Many ad hoc variants of the focusing NLS equation have been proposed to capture such effects. In this paper, we derive some of these variants from Maxwell's equations and propose some new ones. We also provide rigorous error estimates for all the models considered. Finally, we discuss some open problems related to these modified NLS equations.

math.AP↗

arXiv2013

Overview of the proof of the Bounded $L^2$ Curvature Conjecture

This memoir contains an overview of the proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to another, more subtle, scaling tied to its causal geometry. Indeed, $L^2$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1+1 (based on Strichartz estimates) were obtained in \cite{Ba-Ch1}, \cite{Ba-Ch2}, \cite{Ta1}, \cite{Ta2}, \cite{Kl-R1} and optimized in \cite{Kl-R2}, \cite{Sm-Ta}, the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. The entire proof is obtained in a sequence of 6 papers.

math.AP↗

arXiv2013

Sharp Strichartz estimates for the wave equation on a rough background

In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\ggϕ=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$ curvature conjecture proposed in [3], and solved by S. Klainerman, I. Rodnianski and the author in [8], which also relies on the sequence of papers [16][17][18][19]. Obtaining such estimates is at the core of the low regularity well-posedness theory for quasilinear wave equations. The difficulty is intimately connected to the regularity of the Eikonal equation $\gg^{\a\b}\pr_\a u\pr_\b u=0$ for a rough metric $\gg$. In order to be consistent with the final goal of proving the bounded $L^2$ curvature conjecture, we prove Strichartz estimates for all admissible Strichartz pairs under minimal regularity assumptions on the solutions of the Eikonal equation.

math.AP↗