arXiv subjects

Oussama Landoulsi

Publications and source records attributed to Oussama Landoulsi.

8 recordsLinked to original sources

arXiv2025

Continuous data assimilation for 2D stochastic Navier-Stokes equations

Continuous data assimilation methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT) [2], are known to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this work, we extend this framework to a stochastic regime by considering the two-dimensional incompressible Navier-Stokes equations subject to either additive or multiplicative noise. We establish sufficient conditions on the nudging parameter and the spatial observation scale that guarantee convergence of the nudged solution to the true stochastic flow. In the case of multiplicative noise, convergence holds in expectation, with exponential or polynomial rates depending on the growth of the noise covariance. For additive noise, we obtain the exponential convergence both in expectation and pathwise. These results yield a stochastic generalization of the AOT theory, demonstrating how the interplay between random forcing, viscous dissipation and feedback control governs synchronization in stochastic fluid systems.

math.PR↗

arXiv2025

The distorted Fourier transform for the linearized Gross-Pitaevskii equation in the Hyperbolic plane

Motivated by the stability problem for Ginzburg-Landau vortices on the hyperbolic plane, we develop the distorted Fourier transform for a general class of radial non-self-adjoint matrix Schrödinger operators on the hyperbolic plane. This applies in particular to the operator obtained by linearizing the equivariant Ginzburg-Landau equation on the hyperbolic plane around the degree one vortex. We systematically construct the distorted Fourier transform by writing the Stone formula for complex energies and taking the limit as the energy tends to the spectrum of the operator on the real line. This approach entails a careful analysis of the resolvent for complex energies in a neighborhood of the real line. It is the analogue of the approaches used in \cite{KS,ES2, LSS25}, where the limiting operator as $r\to\infty$ is not self-adjoint and which we carry out for all energies. Our analysis serves as the starting point for the study of the stability of the Ginzburg-Landau vortex under equivariant perturbations.

math.AP↗

arXiv2025

Construction of multi-solitary waves solution to the focusing nonlinear Schrödinger equation outside an obstacle in the $L^2$-subcritical case

We consider the focusing $L^2$-subcritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle $Θ\subset \mathbb{R}^d$. We construct a solution that, for large times, behaves asymptotically as a finite sum of solitary waves on $\mathbb{R}^d$, each traveling with sufficiently large and distinct velocities, and satisfying Dirichlet boundary conditions. The construction is achieved via a compactness argument similar to that introduced by F.Merle in 1990 for constructing solutions of the NLS equation that blow up at several points, combined with modulation theory, the coercivity property of the linearized operator, and localized energy estimates.

math.AP↗

arXiv2025

Equivariant stability of vortices in Manton's Chern-Simons-Schrödinger system on the hyperbolic plane

In this work we study magnetic vortices on the hyperbolic plane for a Chern-Simons-Schrödinger system introduced by Manton. The model can be thought of as the Schrödinger analogue of the Abalian-Higgs model. It consists of a system of partial differential equations, where the complex Higgs field $Φ$ evolves according to a nonlinear Schrödinger equation coupled to an electromagnetic field $A$. We restrict attention to the self-dual (Bogomolny) case under equivariance symmetry. For each $m\geq 1$ we prove the asymptotic stability of the equivariant vortex of degree $m$. The main novelties are unraveling the favorable structure of the equations after a nonlinear Darboux transform, and the analysis of the elliptic operator relating the original and the transformed variables.

math.AP↗

arXiv2022

Interaction with an obstacle in the 2d focusing nonlinear Schrödinger equation

We present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling toward the obstacle at different angles and with different velocities directions. We introduce a new concept of weak and strong interactions of the solutions with the obstacle. Next, we study the existence of blow-up solutions depending on the type of the interaction and show how the presence of the obstacle changes the overall behavior of solutions (e.g., from blow-up to global existence), especially in the strong interaction case, as well as how it affects the shape of solutions compared to their initial data, (e.g., splitting into transmitted and reflected parts). We also investigate the influence of the size of the obstacle on the eventual existence of blow-up solutions in the strong interaction case in terms of the transmitted and the reflected parts of the mass. Moreover, we show that the sharp threshold for global existence vs. finite time blow-up solutions in the mass critical case in the presence of the obstacle is the same as the one given by Weinstein for {\rm{NLS}} in the whole Euclidean space $\R^d$. Finally, we construct new Wall-type initial data that blows up in finite time after a strong interaction with an obstacle and having a very distinct dynamics compared with all other blow-up scenarios and dynamics for the {\rm{NLS}} in the whole Euclidean space $\R^d$.

math.AP↗

arXiv2020

On Blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle

In this paper, we consider the Schrödinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on $\R^{d}$. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obstacle because of the appearance of boundary terms with an unfavorable sign in the second derivative of the variance. The main idea of our proof is to introduce a new modified variance which is bounded from below and strictly concave for the solutions that we consider.

math.AP↗

arXiv2020

Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle

We study the dynamics of the focusing $3d$ cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_Ω[u_0] M_Ω[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(Ω)} \left\|u_0\right\|_{L^{2}(Ω)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(Ω)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on the energy-critical problem) have proved the existence of a specific global solution that scatters for negative times and converges to the soliton in positive times. We prove that these heteroclinic orbits do not exist for the problem in the exterior domain and that all solutions at the threshold are globally defined and scatter. The main difficulty is the control of the space translation parameter, since the Galilean transformation is not available.

math.AP↗

arXiv2019

Construction of solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle for the $L^2$-supercritical case

We consider the focusing $L^2$-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R.Killip, M.Visan and X.Zhang which is the same as the one on whole Euclidean space.

math.AP↗