arXiv subjects

Yassine Tahraoui

Publications and source records attributed to Yassine Tahraoui.

14 recordsLinked to original sources

arXiv2026

Turbulent stretching of FENE dumbbell polymer model via special stochastic scaling and singular limits

We investigate the stretching mechanism of Finitely Extensible Nonlinear Elastic (FENE) model of polymers in a random turbulent flow. The turbulent model includes a dominant space-scale $\ell\sim N^{-1}$, a dominant time-scale $τ$, and is white in time. Under suitable scaling assumption, the polymer density equation, initially a stochastic Fokker-Planck equation in the presence of transport-stretching noise, converges weakly as $N\uparrow \infty$ to a limit deterministic equation with a new extra term, a second order operator. This operator, whose shape has been predicted in the physical literature by other arguments, express a sort of average `turbulent stretching' effect. With respect to other derivation of this effective model, the main novelty of our approach is that the deterministic limit is obtained pathwise, without having to take averages with respect to different realizations of the random flow. Next, we consider the limit as $τ\downarrow 0$ and we identify the stationary distribution of the polymer length. The analysis is carried out in appropriate weighted spaces, which take into account the singularity of the FENE force near the boundary and the no-flux boundary condition, and combines stochastic scaling limit and singular limit techniques.

math.PR↗

arXiv2025

Small-scale turbulence limit of Fokker-Planck equation for polymers in turbulent flow

We study the singular limit of Fokker-Planck equation of polymers density as the dominant time-scale of small scale component of turbulent flow goes to zero. Here, we complete the study of Flandoli-Tahraoui[arXiv:2410.00520] about scaling limit as the space-scale of small scale component of turbulent flow goes to zero by using stochastic modeling of turbulence. Depending on certain parameters related to turbulence modeling, the limit density has generalized Cauchy distribution for the end-to-end vector. We discuss also the limit when we don't have a probability density limit. Our approach is based on the derivation of an appropriate estimates on $L^2$ with appropriate weight and investigate the convergence.

math.AP↗

arXiv2025

Hasegawa-Mima equation in bounded domain with singular density

We investigate the well-posedness of Hasegawa-Mima equation (HME) in bounded domain with Dirichlet boundary condition and singular density under different regularity assumptions on the data. Our approach relies on the coupling of a fourth-order regularization of the HME, the spectral Galerkin method and an appropriate regularization of the singular density. Uniqueness holds in the class of solutions with very mild singularities à la Yudovich or with bounded densities.

math.AP↗

arXiv2025

Stochastic pseudomonotone parabolic obstacle problem: well-posedness $\&$ Lewy-Stampacchia's inequalities

We consider obstacle problems for nonlinear stochastic evolution equations. More precisely, the leading operator in our equation is a nonlinear, second order pseudomonotone operator of Leray-Lions type. The multiplicative noise term is given by a stochastic integral with respect to a Q-Wiener process. We show well-posedness of the associated initial value problem for random initial data on a bounded domain with a homogeneous Dirichlet boundary condition. First, we consider a singular perturbation of our problem by a higher order operator. Through the a priori estimates for the approximate solutions of the singular perturbation, only weak convergence is obtained. This convergence is not compatible with the nonlinearities in the equation. Therefore we use the theorems of Prokhorov and Skorokhod to establish existence of martingale solutions. Then, path-wise uniqueness follows from a L1-contraction principle and we may apply the method of Gyöngy-Krylov to obtain stochastically strong solutions. These well-posedness results serve as a basis for the study of variational inequalities and Lewy-Stampacchia's inequalities for our problem.

math.PR↗

arXiv2025

Ergodicity for stochastic T-monotone parabolic obstacle problems

This work aims to investigate the existence of ergodic invariant measures and its uniqueness, associated with obstacle problems governed by a T-monotone operator defined on Sobolev spaces and driven by a multiplicative noise in a bounded domain of $\mathbb{R}^d$ with homogeneous boundary conditions. We show that the solution defines a Markov-Feller semigroup defined on the space of real bounded continuous functions of a convex subset related to the obstacle and we prove the existence of ergodic invariant measures and its uniqueness, under suitable assumptions. Our method relies on a combination of "Krylov Bogoliubov theorem", "Krein Milman theorem" and Lewy-Stampacchia inequalities to control the reflection measure.

math.PR↗

arXiv2024

A nonlinear stochastic diffusion-convection equation with reflection

We study a nonlinear, pseudomonotone, stochastic diffusion-convection evolution problem on a bounded spatial domain, in any space dimension, with homogeneous boundary conditions and reflection. The additive noise term is given by a stochastic Itô integral with respect to a Hilbert space valued $Q$-Wiener process. We show existence of a solution to the pseudomonotone stochastic diffusion-convection equation with non-negative initial value as well as the existence of a reflection measure which prevents the solution from taking negative values. In order to show a minimality condition of the measure, we study the properties of quasi everywhere defined representatives of the solution with respect to parabolic capacity.

math.AP↗

arXiv2024

Stretching of Polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary law

The aim of this work is understanding the stretching mechanism of stochastic models of turbulence acting on a simple model of dilute polymers. We consider a turbulent model that is white noise in time and activates frequencies in a shell $N\leq |k|\leq 2N$ and investigate the scaling limit as $N\rightarrow \infty$, under suitable intensity assumption, such that the stretching term has a finite limit covariance. The polymer density equation, initially an SPDE, converges weakly to a limit deterministic equation with a new term. Stationary solutions can be computed and show power law decay in the polymer length.

math.PR↗

arXiv2024

Large deviations for an obstacle problem with T-monotone operator and multiplicative noise

In this paper, we study the large deviation principle (LDP) for obstacle problems governed by a T-monotone operator and small multiplicative stochastic reaction. Our approach relies on a combination of new sufficient condition to prove LDP by Matoussi, Sabbagh and Zhang [Appl. Math. Optim. 2021] and Lewy-Stampacchia inequalities to manage the Lagrange-multiplier associated with the obstacle.

math.PR↗

arXiv2024

Background Vlasov equations and Young measures for passive scalar and vector advection equations under special stochastic scaling limits

In the last few years it was proved that scalar passive quantities subject to suitable stochastic transport noise, and more recently that also vector passive quantities subject to suitable stochastic transport and stretching noise, weakly converge to the solutions of deterministic equations with a diffusion term. In the background of these stochastic models, we introduce stochastic Vlasov equations which gives additional information on the fluctuations and oscillations of solutions: we prove convergence to non-trivial Young measures satisfying limit PDEs with suitable diffusion terms. In the case of a passive vector field the background Vlasov equation adds completely new statistical information to the stochastic advection equation.

math.PR↗

arXiv2024

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space H^3. Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada-Watanabe theorem. This leads to the existence of a local strong pathwise solution.

math.PR↗

arXiv2024

Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains

This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stabilty result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using "Krylov-Bogoliubov Theorem" we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subset of $L^2$, which leads to the existence of an ergodic invariant measure.

math.PR↗

arXiv2023

On the Existence and long time behaviour of $H^{1}$-Weak Solutions for $2, 3d$-Stochastic $3^{rd}$-Grade Fluids Equations

In the present work, we investigate stochastic third grade fluids equations in a $d$-dimensional setting, for $d = 2, 3$. More precisely, on a bounded and simply connected domain $\mathcal{D}$ of $\mathbb{R}^d$, $d = 2,3$, with a sufficiently regular boundary $\partial \mathcal{D},$ we consider incompressible third grade fluid equations perturbed by a multiplicative Wiener noise. Supplementing our equations by Dirichlet boundary conditions and taking initial data in the Sobolev space $H^1(\mathcal{D})$, we establish the existence of global stochastic weak solutions by performing a strategy based on the conjugation of stochastic compactness criteria and monotonicity techniques. Furthermore, we study the asymptotic behaviour of these solutions, as $t \to \infty$.

math.AP↗

arXiv2023

Optimal control of third grade fluids with multiplicative noise

This work aims to control the dynamics of certain non-Newtonian fluids in a bounded domain of $\mathbb{R}^d$, $d=2,3$ perturbed by a multiplicative Wiener noise, the control acts as a predictable distributed random force, and the goal is to achieve a predefined velocity profile under a minimal cost. Due to the strong nonlinearity of the stochastic state equations, strong solutions are available just locally in time, and the cost functional includes an appropriate stopping time. First, we show the existence of an optimal pair. Then,we show that the solution of the stochastic forward linearized equation coincides with the Gâteaux derivative of the control-to-state mapping, after establishing some stability results. Next, we analyse the backward stochastic adjoint equation; where the uniqueness of solution holds only when $d=2$. Finally, we establish a duality relation and deduce the necessary optimality conditions.

math.OC↗

arXiv2022

Optimal control of two dimensional third grade fluids

The aim of this work is to study the optimal control problems of flows governed by the incompressible third grade fluid equations with Navier-slip boundary conditions. After recalling a result on the well-posedness of the state equations, we study the existence and the uniqueness of solution to the linearized state and adjoint equations. Furthermore, we present a stability result for the state, and show that the solution of the linearized equation coincides with the Gâteaux derivative of the control-to-state mapping. Next, we prove the existence of an optimal solution and establish the first order optimality conditions. Finally, an uniqueness result of the coupled system constituted by the state equation, the adjoint equation and the first order optimality condition is established, under sufficiently large intensity of the cost.

math.OC↗