arXiv subjects

Hakima Bessaih

Publications and source records attributed to Hakima Bessaih.

At least 19 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↗

arXiv2024

Rate of convergence of a semi-implicit time Euler scheme for a 2D Bénard-Boussinesq model

We prove that a semi-implicit time Euler scheme for the two-dimensional Bénard-Boussinesq model on the torus D converges. The rate of convergence in probability is almost 1/2 for a multiplicative noise; this relies on moment estimates in various norms for the processes and the scheme. In case of an additive noise, due to the coupling of the equations, provided that the difference on temperature between the top and bottom parts of the torus is not too big compared to the viscosity and thermal diffusivity, a strong polynomial rate of convergence (almost 1/2) is proven in $(L^2(D))^2$ for the velocity and in $L^2(D)$ for the temperature. It depends on exponential moments of the scheme; due to linear terms involving the other quantity in both evolution equations, the proof has to be done simultaneaously for both the velocity and the temperature. These rates in both cases are similar to that obtained for the Navier-Stokes equation.

math.NA↗

arXiv2022

Speed of convergence of time Euler schemes for a stochastic 2D Boussinesq model

We prove that an implicit time Euler scheme for the 2D-Boussinesq model on the torus $D$ converges. Various moment of the $W^{1,2}$-norms of the velocity and temperature, as well as their discretizations, are computed. We obtain the optimal speed of convergence in probability, and a logarithmic speed of convergence in $L^2(Ω)$. These results are deduced from a time regularity of the solution both in $L^2(D)$ and $W^{1,2}(D)$, and from an $L^2(Ω)$ convergence restricted to a subset where the $W^{1,2}$-noms of the solutions are bounded.

math.PR↗

arXiv2022

Continuous Data Assimilation for Displacement in a Porous Medium

In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm.

math.NA↗

arXiv2022

Strong L2 convergence of time Euler schemes for stochastic 3D Brinkman-Forchheimer-Navier-Stokes equations

We prove that some time Euler schemes for the 3D Navier-Stokes equations modified by adding a Brinkman-Forchheimer term and a random perturbation converge in $L^2(Ω)$. This extends previous results concerning the strong rate of convergence of some time discretization schemes for the 2D Navier Stokes equations. Unlike the 2D case, our proposed 3D model with the Brinkman-Forchheimer term allows for a strong rate of convergence of order almost 1/2, that is independent of the viscosity parameter.

math.NA↗

arXiv2021

Strong rates of convergence of space-time discretization schemes for the 2D Navier-Stokes equations with additive noise

We consider the strong solution of the 2D Navier-Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the rate of convergence of the schemes is $η\in[0,1/2)$ in time and 1 in space. Let us mention that the coefficient $η$ is equal to the time regularity of the solution with values in $\LL^2$. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Our main idea is to use a discrete Gronwall lemma for the error estimate without any localization.

math.NA↗

arXiv2020

Space-time Euler discretization schemes for the stochastic 2D Navier-Stokes equations

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in $L^2(Ω)$, and describe the rate of convergence for an $H^1$-valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the $L^2(Ω)$-convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott-Vogelius mixed elements and for an additive noise, the convergence is polynomial.

math.PR↗

arXiv2020

Invariant measures for stochastic damped 2D Euler equations

We study the two-dimensional Euler equations, damped by a linear term and driven by an additive noise. The existence of weak solutions has already been studied; pathwise uniqueness is known for solutions that have vorticity in $L^\infty$. In this paper, we prove the Markov property and then the existence of an invariant measure in the space $L^\infty$ by means of a Krylov-Bogoliubov's type method, working with the weak$\star$ and the bounded weak$\star$ topologies in $L^\infty$.

math.PR↗

arXiv2020

Stochastic homogenization of multicontinuum heterogeneous flows

We consider a multicontinuum model in porous media applications, which is described as a system of coupled flow equations. The coupling between different continua depends on many factors and its modeling is important for porous media applications. The coefficients depend on particle deposition that is described in term of a stochastic process solution of an SDE. The stochastic process is considered to be faster than the flow motion and we introduce time-space scales to model the problem. Our goal is to pass to the limit in time and space and to find an associated averaged system. This is an averaging-homogenization problem, where the averages are computed in terms of the invariant measure associated to the fast motion and the spatial variable. We use the techniques developed in our previous paper to model the interactions between the continua and derive the averaged model problem that can be used in many applications.

math.PR↗

arXiv2019

Longtime behavior for 3D Navier-Stokes equations with constant delays

This paper investigates the longtime behavior of delayed 3D Navier-Stokes equations in terms of attractors. The study will strongly rely on the investigation of the linearized Navier-Stokes system, and the relationship between the discrete dynamical flow for the linearized system and the continuous flow associated to the original system. Assuming the viscosity to be sufficiently large, there exists a unique local attractor for the delayed 3D Navier-Stokes equations. Moreover, the local attractor reduces to a singleton set.

math.DS↗

arXiv2018

Stochastic homogenization for a diffusion-reaction model

In this paper, we study stochastic homogenization of a coupled diffusion-reaction system. The diffusion-reaction system is coupled to stochastic differential equations, which govern the changes in the media properties. Though homogenization with changing media properties has been studied in previous findings, there is little research on homogenization when the media properties change due to stochastic differential equations. Such processes occur in many applications, where the changes in media properties are due to particle deposition. In the paper, we investigate the well-posedness of the nonlinear fine-grid (resolved) problem and derive limiting equations. We formulate the cell problems and derive the limiting equations, which are deterministic with nonlinear reaction terms. The limiting equations involve the invariant measures corresponding to stochastic differential equations. These obtained results can play an important role for modeling in porous media and allow the use of simplified and deterministic limiting equations.

math.PR↗

arXiv2018

Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

The present paper is devoted to the numerical approximation of an abstract stochastic nonlinear evolution equation in a separable Hilbert space {$\mathrm{H}$}. Examples of equations which fall into our framework include the GOY and Sabra shell models and { a class of nonlinear heat equations.} The space-time numerical scheme is defined in terms of a Galerkin approximation in space and a { semi-implicit Euler--Maruyama scheme in time}. {We prove the convergence in probability of our scheme by means of an estimate of the error on a localized set of arbitrary large probability.} Our error estimate is shown to hold in a more regular space $\mathrm{V}_β\subset \mathrm{H}$ with $β\in [0,\frac14)$ and { that the explicit rate of convergence of our scheme depends on this parameter $β$. }

math.NA↗

arXiv2018

On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations

We prove that some discretization schemes for the 2D Navier-Stokes equations subject to a random perturbation converge in $L^2(Ω)$. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic NS equations and convergence of a localized scheme, we can prove strong convergence of fully implicit and semi-implicit time Euler discretizations, and of a splitting scheme. The speed of the $L^2(Ω)$-convergence depends on the diffusion coefficient and on the viscosity parameter.

math.PR↗

arXiv2017

On stochastic modified 3d navier-stokes equations with anisotropic viscosity

Navier-Stokes equations in the whole space R^3 subject to an anisotropic viscosity and a random perturbation of multiplicative type is described. By adding a term of Brinkman-Forchheimer type to the model, existence and uniqueness of global weak solutions in the PDE sense are proved. These are strong solutions in the probability sense. The convective term given in terms of the Brinkman-Forchheirmer provides some extra regularity in the space L^{2$α$+2} (R^3), with $α$ > 1. As a consequence, the nonlinear term has better properties which allows to prove uniqueness. The proof of existence is performed through a control method. A Large Deviations Principle is given and proven at the end of the paper.

math.PR↗

arXiv2017

Mean field limit of interacting filaments for 3D Euler equations

The 3D Euler equations, precisely local smooth solutions of class $H^s$ with $s>5/2$, are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments, described by means of the concept of $1$-currents. This work is a continuation of the a previous work, where a preliminary result in this direction was obtained, with the true Euler equations replaced by a vector valued non linear PDE with a mollified Biot-Savart relation.

math.AP↗

arXiv2017

On 3D Navier-Stokes equations: regularization and uniqueness by delays

A modified version of the three dimensional Navier-Stokes equations is considered with periodic boundary conditions. A bounded constant delay is introduced into the convective term, that produces a regularizing effect on the solution. In fact, by assuming appropriate regularity on the initial data, the solutions of the delayed equations are proved to be regular and, as a consequence, existence and also uniqueness of a global weak solution is obtained. Moreover, the associated flow is constructed and the continuity of the semigroup is proved. Finally, we investigate the passage to the limit on the delay, obtaining that the limit is a weak solution of the Navier-Stokes equations.

math.AP↗

arXiv2016

The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order $α$ in the nonlinear term and a $β$-fractional Laplacian; we consider the critical case $α+β=\frac{5}{4}$ and we assume $\frac 12 <β<\frac 54$. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order $α$. We prove global well posedness when the initial velocity is in $H^r$ and the initial temperature is in $H^{r-β}$ for $r>\max(2β,β+1)$. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of the solutions on the initial conditions.

math.AP↗

arXiv2016

Stochastic lattice dynamical systems with fractional noise

This article is devoted to study stochastic lattice dynamical systems driven by a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. First of all, we investigate the existence and uniqueness of pathwise mild solutions to such systems by the Young integration setting and prove that the solution generates a random dynamical system. Further, we analyze the exponential stability of the trivial solution.

math.AP↗