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Anne-Sophie de Suzzoni

Publications and source records attributed to Anne-Sophie de Suzzoni.

31 records · Page 2Linked to original sources
arXiv2016

On Gibbs measure and weak flow for the cubic NLS with non-localised initial data

In this paper we prove the existence of an invariant measure for the cubic NLS $$i\partial_t u + \bigtriangleup u - |u|^2 u = 0$$ on the real line in the sense that we prove the existence of a measure $ρ$ supported by non-localised functions such that there exists random variables $X(t)$ whose laws are $ρ$ (thus independent of $t$) and such that $t\mapsto X(t)$ is a solution to the cubic NLS. Our strategy for the proof is inspired by \cite{burqtzv} and relies on the application of Prokhorov and Skorokhod Theorems to a sequence of measures which are invariant under some approximating flows, as we proved in our previous \cite{lastbaby}. However, the work by Bourgain, \cite{B00} provides a stronger result than this one, as it gives almost sure strong solutions for the cubic NLS and the invariance of the measure can be deduced from it.

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arXiv2016

An equation on random variables and systems of fermions

In this paper, we consider an equation on random variables which can be reduced to the equation which describes the evolution of systems of fermions. We give some results of well-posedness for this equation on the spheres and torus of dimension 2 and 3 and on the Euclidean space. We give results of scattering and blow-up on the Euclidean depending on if the equation is defocusing or focusing. We interpret the results in terms of the evolution of fermions.

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arXiv2014

Invariant measure for the Klein-Gordon equation in a non periodic setting

In this paper, we build a Gibbs measure for the 1d cubic Klein-Gordon equation on $\mathbb R$ with a decreasing non linearity, in the sense that the non linearity $f^3$ is multiplied by $χ$ where $χ$ is a sufficiently integrable non negative function. We prove that this equation is almost surely globally well-posed in $H^{1/2-}_{\textrm{loc}}$ with respect to this measure and that the measure is invariant under the flow of the equation.

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arXiv2014

Invariant measure for the Schrödinger equation on the real line

In this paper, we build a Gibbs measure for the cubic defocusing Schrödinger equation on the real line with a decreasing interaction potential, in the sense that the non linearity $|u|^2u$ is multiplied by a function $χ$ which we assume integrable and smooth enough. We prove that this equation is globally well-posed in the support of this measure and that the measure is invariant under the flow of the equation. What is more, the support of the measure (the set of initial data) is disjoint from $L^2$.

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arXiv2013

On the use of normal forms in the propagation of random waves

We consider the evolution of the correlations between the Fourier coeficients of a solution of the Kamdostev-Petviavshvili II equation when these coefficients are initially independent random variables. We use the structure of normal forms of the equation to prove that those correlations remain small until times of order $\varepsilon^{-5/3}$ or $\varepsilon^{-2}$ depending on the quantity considered.

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arXiv2013

On the propagation of weakly nonlinear random dispersive waves

We study several basic dispersive models with random periodic initial data such that the different Fourier modes are independent random variables. Motivated by the vast Physics literature on related topics, we then study how much the Fourier modes of the solution at later times remain decorrelated. Our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data.

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arXiv2012

Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidian space

In this paper, the almost sure global well-posedness of the cubic non linear wave equation on the sphere is studied when the initial datum is a random variable with values in low regularity spaces. The domain is first the 3D sphere, thanks to the existence of a uniformly bounded in $L^p$ basis of $L^2(S^3)$ and then the result is extended to $\R^3$ thanks to the Penrose transform.

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arXiv2012

Large data low regularity scattering results for the wave equation on the Euclidian space

We will consider the resolution of the 3D non linear wave equation under the assumption of spherical symmetry on the Euclidian space. For this purpose, we will build a non trivial measure on distributions such that there exists a set of full measurement onto which the flow is globally defined. We will then discuss different properties of the solutions.

math.AP↗
arXiv2011

Invariant measure for the cubic wave equation on the unit ball of $R^3$

This paper deals with the invariance of a measure on Sobolev spaces of low regularity under the flow of the cubic non linear wave equation on the unit ball of 3 under the assumption of spherical symmetry. It presents two aspects, an analytic one which includes the treatment of local properties of the flow, and a probabilistic one, which is mainly related to the global extension of the flow and the invariance of the measure.

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