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Mouhamadou Sy

Publications and source records attributed to Mouhamadou Sy.

12 recordsLinked to original sources
arXiv2026

Regularity and Pathwise bounds for probabilistic solutions of PDEs

In this paper, we build a procedure that allows to establish regularity and controls in time for probabilistic solutions to PDEs. Probabilistic approaches to global wellposedness problems usually provide ensemble bounds on the solutions. These bounds are the main tools to ensure convergence procedures yielding the existence and uniqueness of global solutions. A question of interest consists in transforming such ensemble bounds into individual controls on the flow ; this, among other uses, gives valuable information on the long-time behavior of the solutions. Toward such question of bounds transformation, Bourgain initiated a successful procedure that exploited the local wellposedness of the PDE, with an estimate of the time of size-doubling. In this note, we construct an estimation procedure which relies on a different local requirement. It turns out that this substitute is flexible enough to be possible to fulfill with the help of the ensemble bound itself. For applications of the procedure, we are able to provide new pathwise controls on solutions to NLS equations.

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arXiv2026

NLS with exponential nonlinearity on compact surfaces

In this paper, we establish a probabilistic global theory in $H^1$ for the NLS with a Moser-Trudinger nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical Schr\"odinger equations \cite{CollianderIbrahimMajdoubMasmoudi2009}. The authors of \cite{CollianderIbrahimMajdoubMasmoudi2009} also identified a trichotomy around the criticality of the equation based on the size of the total energy. In particular, for supercritical regimes (large energy), the equation is known to exhibit instabilities : the (uniform) continuity of the flow fails to hold. Large data distributional non unique probabilistic solutions have been obtained in \cite{CasterasMonsaingeon2024}. The setting of \cite{CasterasMonsaingeon2024} does not handle the uniqueness issue for the $H^1$-data and therefore could not define a flow for this regularity. Our main focus here is to build a single probabilistic framework that provides both existence, uniqueness, and continuity with respect to the initial data in $H^1$. Our uniqueness and continuity are based on the so-called Yudowich argument \cite{Judovic1963}, and the probabilistic estimates are derived through the IID limit procedure \cite{Sy2019}. Beyond the difficulties related to the borderline nature of the context, the major challenge resides in the need to satisfy two features that tend to play against each other : obtaining both continuity property of the flow and large data in the support of the reference measure. This made the design of the dissipation operator inherent in the method, as well as the analysis of the resulting quantities, particularly difficult. Regarding the supercritical regime, we show that a modified energy, with regularity similar to the original total energy, admits values as high as desired, suggesting that the constructed set of data contains supercritical ones.

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arXiv2025

On Properties of Statistically Stationary Solutions to the One-Dimensional Schr\"odinger Map Equation

We investigate further qualitative properties of statistically stationary solutions to the Schr\"odinger map equation (SME) and the Binormal Curvature Flow (BCF), continuing the work initiated by E. G., M. Hofmanov\'a. Concerning the statistically stationary solutions to the SME, we show that the laws of some relevant observables (such as the space average and the energy) are absolutely continuous with respect to the Lebesgue measure, with a Gaussian decay property for the energy. We further prove that the law $\mu$ of the statistically stationary solution has dimension of at least two: this means that any compact set of Hausdorff dimension smaller than two has $\mu$-measure zero. These properties, with appropriate modifications of the norms, pass directly to the statistically stationary solutions to the BCF.

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arXiv2025

Probabilistic global-wellposedness for the energy-supercritical Schr\"odinger equations on compact manifolds

We consider the nonlinear Schr\"odinger equations with a general nonlinearity power in all dimensions. We construct invariant measures concentrated on Sobolev spaces $H^s$ of singular orders, $s\leq\frac{d}{2}$. We prove almost sure global wellposedness and bounds on the growth in time of the solutions via invariant measure arguments. Our setting includes a generic compact Riemannian manifold; we specify the cases of the torus and Zoll manifolds.

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arXiv2022

Global well-posedness for the cubic fractional NLS on the unit disk

In this paper, we prove that the cubic nonlinear Schrödinger equation with the fractional Laplacian on the unit disk is globally well-posed for certain radial initial data below the energy space. The result is proved by extending the I-method in the fractional nonlinear Schrödinger equation setting.

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arXiv2021

Almost sure global well-posedness for the energy supercritical Schrödinger equations

We consider the Schrödinger equations with arbitrary (large) power non-linearity on the three-dimensional torus. We construct non-trivial probability measures supported on Sobolev spaces and show that the equations are globally well-posed on the supports of these measures, respectively. Moreover, these measures are invariant under the flows that are constructed. Therefore, the constructed solutions are recurrent in time.\\ Also, we show \textit{slow growth} control on the time evolution of the solutions. A generalization to any dimension is given. Our proof relies on a new approach combining the fluctuation-dissipation method and some features of the Gibbs measures theory for Hamiltonian PDEs. The strategy of the paper applies to other contexts

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arXiv2021

Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$

In this paper, we present two almost sure global well-posedness (GWP) results for the energy supercritical nonlinear Schrödinger equations (NLS) on the unit ball of $\Bbb R^3$ using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions. This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinearities with critical and supercritical initial conditions, we prove the almost sure global well-posedness, and the invariance of the measure under the solution flows. This global result is built on a compactness argument and the Skorokhod representation theorem.

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arXiv2021

Global well-posedness and long-time behavior of the fractional NLS

In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension $\geq 2$. In the high regularity result, an {\it Inviscid - Infinite dimensional (IID) limit} is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an independent approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss the adaptation to the periodic settings, in any dimension, for smooth regularities.

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arXiv2021

Invariant measure and long time behavior of regular solutions of the Benjamin-Ono equation

The Benjamin-Ono equation describes the propagation of internal waves in a stratified fluid. In the present work, we study large time dynamics of its regular solutions via some probabilistic point of view. We prove the existence of an invariant measure concentrated on $C^\infty(\T)$ and establish some qualitative properties of this measure. We then deduce a recurrence property of regular solutions and other corollaries using ergodic theorems. The approach used in this paper applies to other equations with infinitely many conservation laws, such as the KdV and cubic Schrödinger equations in 1D. It uses the fluctuation-dissipation-limit approach and relies on a \textit{uniform} smoothing lemma for stationary solutions to the damped-driven Benjamin-Ono equation.

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arXiv2020

Invariant measures and global well posedness for SQG equation

We construct an invariant measure $μ$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $μ$ are initial conditions of global, unique solutions of SQG, that depend continuously on the initial data. In addition, we show that the support of $μ$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measures $μ$ is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.

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arXiv2017

Invariant measure and large time dynamics of the cubic Klein-Gordon equation in $3D$

In this paper we construct an invariant probability measure concentrated on $H^2(K)\times H^1(K)$ for a general cubic Klein-Gordon equation (including the case of the wave equation). Here $K$ represents both the $3$-dimensional torus or a bounded domain with smooth boundary in $\Bbb R^3.$ That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic sense. We also establish qualitative properties of the constructed measure. This work extends the Fluctuation-Dissipation-Limit (FDL) approach to PDEs having only one (coercive) conservation law.

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