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Benedetta Ferrario

Publications and source records attributed to Benedetta Ferrario.

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↗
arXiv2025

Long time behavior of the stochastic 2D Navier-Stokes equations

We review some basic results on existence and uniqueness of the invariant measure for the two-dimensional stochastic Navier-Stokes equations. A large part of the literature concerns the additive noise case; after revising these models, we consider our recent result, arXiv:2307.03483, with a multiplicative noise.

math.PR↗
arXiv2024

Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schr\"odinger equation with multiplicative noise and arbitrary power of the nonlinearity

We consider the nonlinear Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2\sigma}u$. For any $\sigma \in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in $H^s(\mathbb T^d)$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover we prove existence of invariant measures and their uniqueness under more restrictive assumptions on the noise term.

math.PR↗
arXiv2023

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension 2. We construct a martingale solution using a modified Faedo-Galerkin's method, following arXiv:1707.05610. Then by means of the Strichartz estimates deduced from arXiv:math/0609455 but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov-Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler. This is the first result of this type for stochastic NLS on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well.

math.PR↗
arXiv2023

Uniqueness of the invariant measure and asymptotic stability for the 2D Navier Stokes equations with multiplicative noise

We establish the uniqueness and the asymptotic stability of the invariant measure for the two dimensional Navier Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an effectively elliptic setting, that is we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of Glatt Holtz,Mattingly,Richards(2017) and Kulik,Scheutzow(2018), used by these authors for the stochastic Navier Stokes equations with additive noise. Here we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias Prodi estimate in expected valued, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.

math.PR↗
arXiv2023

Stationary solutions for the nonlinear Schr\"odinger equation

We construct stationary statistical solutions of a deterministic unforced nonlinear Schr\"odinger equation, by perturbing it by a linear damping $\gamma u$ and a stochastic force whose intensity is proportional to $\sqrt \gamma$, and then letting $\gamma\to 0^+$. We prove indeed that the family of stationary solutions $\{U_\gamma\}_{\gamma>0}$ of the perturbed equation possesses an accumulation point for any vanishing sequence $\gamma_j\to 0^+$ and this stationary limit solves the deterministic unforced nonlinear Schr\"odinger equation and is not the trivial zero solution. This technique has been introduced in [KS04], using a different dissipation. However considering a linear damping of zero order and weaker solutions we can deal with larger ranges of the nonlinearity and of the spatial dimension; moreover we consider the focusing equation and the defocusing equation as well.

math.AP↗
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↗
arXiv2018

2D Navier-Stokes equation with cylindrical fractional Brownian noise

We consider the Navier-Stokes equation on the 2D torus, with a stochastic forcing term which is a cylindrical fractional Wiener noise of Hurst parameter $H$. Following [3,8] which dealt with the case $1/2$, we prove a local existence and uniqueness result when $7/16< H< 1/ 2$ and a global existence and uniqueness result when $ 1/2<H<1$.

math.AP↗
arXiv2018

Stochastic vorticity equation in $\mathbb R^2$ with not regular noise

We consider the Navier-Stokes equations in vorticity form in $\mathbb{R}^2$ with a white noise forcing term of multiplicative type, whose spatial covariance is not regular enough to apply the Itô calculus in $L^q$ spaces, $1<q<\infty$. We prove the existence of a unique strong (in the probability sense) solution.

math.PR↗
arXiv2017

Absolute continuity of the law for the two dimensional stochastic Navier-Stokes equations

We consider the two dimensional Navier-Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and coloured in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $ξ$ and we show that, if the initial vorticity $ξ_0$ is continuous in space, then there exists a space-time continuous version of the solution. In addition we show that the solution $ξ(t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.

math.PR↗
arXiv2017

Stationary solutions for stochastic damped Navier-Stokes equations in $\mathbb R^d$

We consider the stochastic damped Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$), assuming as in our previous work [4] that the covariance of the noise is not too regular, so Itô calculus cannot be applied in the space of finite energy vector fields. We prove the existence of an invariant measure when $d=2$ and of a stationary solution when $d=3$.

math.PR↗
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

Statistical properties of stochastic 2D Navier-Stokes equations from linear models

A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $λ$ which gives a system $(u^λ,w^λ)$; this system is studied for any $λ$ proving its well posedness and the uniqueness of its invariant measure $μ^λ$. The key point is that for any $λ\neq 0$ the fields $u^λ$ and $w^λ$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $λ\to 0$, proving that $μ^λ$ weakly converges to $μ^0$, where $μ^0$ is the only invariant measure for the joint system for $(u,w)$ when $λ=0$.

math.PR↗
arXiv2016

Characterization of the law for 3D stochastic hyperviscous fluids

We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term $-Δ\vec ξ$ of the Navier-Stokes equations is substituted by $(-Δ)^{1+c} \vec ξ$. We investigate how big the correction term $c$ has to be in order to prove, by means of Girsanov transform, that the vorticity equations are equivalent (in law) to easier reference equations obtained by neglecting the stretching term. This holds as soon as $c>\frac 12$, improving previous results obtained with $c>\frac 32$ in a different setting in [5,14].

math.PR↗
arXiv2015

A note on stochastic Navier-Stokes equations with not regular multiplicative noise

We consider the Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so Itô calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for $d=2,3$ and pathwise uniqueness for $d=2$.

math.PR↗
arXiv2012

On a Stochastic Leray-α model of Euler equations

We deal with the 3D inviscid Leray-α model. The well posedness for this problem is not known; by adding a random perturbation we prove that there exists a unique (in law) global solution. The random forcing term formally preserves conservation of energy. The result holds for initial velocity of finite energy and the solution has finite energy a.s.. These results are easily extended to the 2D case.

math.PR↗
arXiv2011

2D hydrodynamical systems: invariant measures of Gaussian type

Gaussian measures $μ^{β,ν}$ are associated to some stochastic 2D hydrodynamical systems. They are of Gibbsian type and are constructed by means of some invariant quantities of the system depending on some parameter $β$ (related to the 2D nature of the fluid) and the viscosity $ν$. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure $μ^{β,ν}$ is invariant for this flow and is unique. Finally, we prove that the deterministic inviscid equation has a $μ^{β,ν}$-stationary solution (for any $ν>0$).

math.PR↗