arXiv subjects

Sandrine Grellier

Publications and source records attributed to Sandrine Grellier.

At least 19 recordsLinked to original sources

arXiv2026

Soliton and breather resolution for the cubic Szeg\"o flow on the line

We investigate the long time behaviour of the solutions of the cubic Szeg}o equation on the line in the Sobolev space H^{1/2}(R). We prove that, for every datum of which the Lax operator has simple positive spectrum, the solution asymptotically decouples as an infinite sum of traveling quasi-periodic breather solutions. Under an additional generic condition on the data, we prove that these traveling breather solutions are in fact soliton solutions, leading to a soliton resolution theorem.

math.AP↗

arXiv2025

Stein's theorem in the upper-half plane and Bergman spaces with weights

This is a companion paper to our previous one, Avatars of Stein's Theorem in the complex setting. In this previous paper, we gave a sufficient condition for an integrable function in the upper-half plane to have an integrable Bergman projection. Here we push forward methods and establish in particular a converse statement. This naturally leads us to study a family of weighted Bergman spaces for logarithmic weights (1 + ln + (1/___m(z)) + ln + (|z|)) k , which have the same kind of behavior respectively at the boundary and at infinity. We introduce their duals, which are logarithmic Bloch type spaces and interest ourselves in multipliers, pointwise products and Hankel operators.

math.CA↗

arXiv2022

Global Stein Theorem on Hardy spaces

Let f be an integrable function which has integral 0 on R n. What is the largest condition on |f | that guarantees that f is in the Hardy space H 1 (R n)? When f is compactly supported, it is well-known that it is necessary and sufficient that |f | belongs to L log L(R n). We are interested here in conditions at $\infty$. We do so for H 1 (R n), as well as for the Hardy space H log (R n) which appears in the study of pointwise products of functions in H 1 (R n) and in its dual BMO.

math.CA↗

arXiv2021

Turbulent cascades for a family of damped Szegö equations

In this paper, we study the transfer of energy from low to high frequencies for a family of damped Szegö equations. The cubic Szegö equation has been introduced as a toy model for a totally non-dispersive degenerate Hamiltonian equation. It is a completely integrable system which develops growth of high Sobolev norms, detecting transfer of energy and hence cascades phenomena. Here, we consider a two-parameter family of variants of the cubic Szegö equation and prove that adding a damping term unexpectedly promotes the existence of turbulent cascades. Furthermore, we give a panorama of the dynamics for such equations on a six-dimensional submanifold.

math.AP↗

arXiv2019

On a damped Szego equation (with an appendix in collaboration with Christian Klein)

We investigate how damping the lowest Fourier mode modifies the dynamics of the cubic Szeg{ö} equation. We show that there is a nonempty open subset of initial data generating trajec-tories with high Sobolev norms tending to infinity. In addition, we give a complete picture of this phenomenon on a reduced phase space of dimension 6. An appendix is devoted to numerical simulations supporting the generalisation of this picture to more general initial data.

math.AP↗

arXiv2017

Generic colourful tori and inverse spectral transform for Hankel operators

This paper explores the regularity properties of an inverse spectral transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral transform plays the role of action-angles variables for an integrable infinite dimensional Hamiltonian system -- the cubic Szegö equation. We investigate the regularity of functions on the tori supporting the dynamics of this system, in connection with some wave turbulence phenomenon, discovered in a previous work and due to relative small gaps between the actions. We revisit this phenomenon by proving that generic smooth functions and a G $δ$ dense set of irregular functions do coexist on the same torus. On the other hand, we establish some uniform analytic regularity for tori corresponding to rapidly decreasing actions which satisfy some specific property ruling out the phenomenon of small gaps.

math.AP↗

arXiv2016

Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak-Orlicz type. These spaces are natural generalizations of weighted Hardy-Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BM O. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log. Namely, a closed form in H log is the infinite sum of the wedge product between an exact form in the Hardy space H 1 and an exact form in BM O. The converse result, which generalizes the classical div-curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued H log space can be weakly factorized.

math.CA↗

arXiv2015

The cubic szego equation and hankel operators

This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1.It is devoted to the dynamics on Sobolev spaces of the cubic Szeg{ö} equation on the circle ${\mathbb S} ^1$,$$ i\partial \_t u=Π(\vert u\vert ^2u)\ .$$Here $Π$ denotes the orthogonal projector from $L^2({\mathbb S} ^1)$ onto the subspace $L^2\_+({\mathbb S} ^1)$ of functions with nonnegative Fourier modes.We construct a nonlinear Fourier transformation on $H^{1/2}({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$ allowing to describe explicitly the solutions of this equationwith data in $H^{1/2}({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$. This explicit description implies almost-periodicity of every solution in $H^{\frac 12}\_+$. Furthermore, it allows to display the following turbulence phenomenon. For a dense $G\_δ$ subset of initial data in $C^\infty ({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$, the solutions tend to infinity in $H^s$ for every $s\textgreater{}\frac 12$ with super--polynomial growth on some sequence of times, while they go back to their initial data on another sequence of times tending to infinity. This transformation is defined by solving a general inverse spectral problem involving singular values of a Hilbert--Schmidt Hankel operator and of its shifted Hankel operator.

math.AP↗

arXiv2014

Multiple singular values of Hankel operators

The goal of this paper is to construct a nonlinear Fourier transformation on the space of symbols of compact Hankel operators on the circle. This transformation allows to solve a general inverse spectral problem involving singular values of a compact Hankel operator, with arbitrary multiplicities. The formulation of this result requires the introduction of the pair made with a Hankel operator and its shifted Hankel operator. As an application, we prove that the space of symbols of compact Hankel operators on the circle admits a singular foliation made of tori of finite or infinite dimensions, on which the flow of the cubic Szegö equation acts. In particular, we infer that arbitrary solutions of the cubic Szegö equation on the circle with finite momentum are almost periodic with values in H^{1/2}(S ^1).

math.AP↗

arXiv2013

An explicit formula for the cubic Szegö equation

We derive an explicit formula for the general solution of the cubic Szegö equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel operators are quasiperiodic.

math.AP↗

arXiv2012

Spectral inverse problems for compact Hankel operators

Given two arbitrary sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$ of real numbers satisfying $$|λ_1|>|μ_1|>|λ_2|>|μ_2|>...>| λ_j| >| μ_j| \to 0\ ,$$ we prove that there exists a unique sequence $c=(c_n)_{n\in\Z_+}$, real valued, such that the Hankel operators $Γ_c$ and $Γ_{\tilde c}$ of symbols $c=(c_{n})_{n\ge 0}$ and $\tilde c=(c_{n+1})_{n\ge 0}$ respectively, are selfadjoint compact operators on $\ell^2(\Z_+)$ and have the sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Γ_c$ and of $Γ_{\tilde c}$ in terms of the sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$. More generally, given two arbitrary sequences $(ρ_j)_{j\ge 1}$ and $(σ_j)_{j\ge 1}$ of positive numbers satisfying $$ρ_1>σ_1>ρ_2>σ_2>...> ρ_j> σ_j \to 0\ ,$$ we describe the set of sequences $c=(c_n)_{n\in\Z_+}$ of complex numbers such that the Hankel operators $Γ_c$ and $Γ_{\tilde c}$ are compact on $\ell ^2(\Z_+)$ and have sequences $(ρ_j)_{j\ge 1}$ and $(σ_j)_{j\ge 1}$ respectively as non zero singular values.

math.AP↗

arXiv2011

Effective integrable dynamics for some nonlinear wave equation

We consider the following degenerate half wave equation on the one dimensional torus $$\quad i\partial_t u-|D|u=|u|^2u, \; u(0,\cdot)=u_0. $$ We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system-- the cubic Szegö equation. As a consequence, we prove an instability result for large $H^s$ norms of solutions of this wave equation.

math.AP↗

arXiv2011

Paraproducts and Products of functions in $BMO(\mathbb R^n)$ and $H^1(\mathbb R^n)$ through wavelets

In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal Mf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space $\H^{\log}(\bR^n)$ for the $÷$-$\curl$ lemma.

math.CA↗

arXiv2010

Invariant tori for the cubic Szegö equation

We continue the study of the following Hamiltonian equation on the Hardy space of the circle, $$i\partial _tu=Π(|u|^2u)\ ,$$ where $Π$ denotes the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. In a previous work, we proved that this equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish the stability of the corresponding invariant tori. Furthermore, from the explicit formulae, we deduce the classification of orbitally stable and unstable traveling waves.

math.CV↗

arXiv2009

The Szegö Cubic Equation

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial_tu=Π(|u|^2u) ,$$ where $Π$ is the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.

math.CV↗

arXiv2009

Endpoint for the div-curl lemma in Hardy spaces

We give a div-curl type lemma for the wedge product of closed differential forms on R^n when they have coefficients respectively in a Hardy space and L^infinity or BMO. In this last case, the wedge product belongs to an appropriate Hardy-Orlicz space.

math.CA↗

arXiv2009

Hankel Operators and Weak Factorization for Hardy-Orlicz Spaces

We study the holomorphic Hardy-Orlicz spaces H^Φ(Ω), where Ωis the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in Cn . The function Φis in particular such that H ^1(Ω) \subset H^Φ(Ω) \subset H ^p (Ω) for some p > 0. We develop for them maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Omega). As a consequence, we characterize those Hankel operators which are bounded from H ^Φ(Ω) into H^1 (Ω).

math.CV↗

arXiv2004

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

math.CV↗