Theory of weak asymptotic autonomy of pullback stochastic weak attractors and its applications to 2D stochastic Euler equations driven by multiplicative noise
The two dimensional stochastic Euler equations (EE) perturbed by a linear multiplicative noise of Itô type on the bounded domain $\mathcal{O}$ have been considered in this work. Our first aim is to prove the existence of \textsl{global weak (analytic) solutions} for stochastic EE when the divergence free initial data $\boldsymbol{u}^*\in\mathbb{H}^1(\mathcal{O})$, and the external forcing $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H}^1(\mathcal{O}))$. In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if $\mathrm{curl}\ \boldsymbol{u}^*\in\mathrm{L}^{\infty}(\mathcal{O})$ and $\mathrm{curl}\ \boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$, we establish that the global weak (analytic) solution is unique. This work appears to be the first one to discuss the existence and uniqueness of global weak (analytic) solutions for stochastic EE driven by linear multiplicative noise. Secondly, we prove the existence of a \textsl{pullback stochastic weak attractor} for stochastic \textsl{non-autonomous} EE using the abstract theory available in the literature. Finally, we propose an abstract theory for \textsl{weak asymptotic autonomy} of pullback stochastic weak attractors. Then we consider the 2D stochastic EE perturbed by a linear multiplicative noise as an example to discuss that how to prove the weak asymptotic autonomy for concrete stochastic partial differential equations. As EE do not contain any dissipative term, the results on attractors (deterministic and stochastic) are available in the literature for dissipative (or damped) EE only. Since we are considering stochastic EE without dissipation, all the results of this work for 2D stochastic EE perturbed by a linear multiplicative noise are totally new.
This work is concerned about the asymptotic behavior of the solutions of the two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations driven by white noise with nonlinear diffusion terms. We prove the existence and uniqueness of weak pullback mean random attractors for the 2D SCBF equations (for $r\geq1$) as well as 3D SCBF equations (for $r>3$, any $μ,β>0$ and for $r=3$, $2μβ\geq1$) in Bochner spaces, when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we establish the existence of weak pullback mean random attractors for a class of locally mo…
We consider SDEs driven by two different sources of additive noise, which we refer to as intrinsic and common. We establish almost sure existence and uniqueness of pullback attractors with respect to realisations of the common noise only. These common noise pullback attractors are smooth probability densities that depend only on (the past of) a common noise realisation and to which the pullback evolution of a corresponding stochastic Fokker-Planck equation converges. Common noise pullback attractors have a natural motivation in the context of particle systems with intrinsic and common noise, d…
In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counter part with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$, $d=3$ with $r>3$ and…
The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors.
We provide a unified analytic approach to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part I deals with driving paths of finite $p$-variations with $1 \leq p <2$ so that the integrals are interpreted in the Young sense. Our method helps to generalize recent results \cite{GAKLBSch2010}, \cite{ducGANSch18}, \cite{duchongcong18} on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a sing…
This article concerns the long-term random dynamics in regular spaces for a non-autonomous Navier-Stokes equation defined on a bounded smooth domain $\mathcal{O}$ driven by multiplicative and additive noise. For the two kinds of noise driven equations, we demonstrate the existence of a unique pullback attractor which is backward compact and asymptotically autonomous in $\mathbb{L}^2(\mathcal{O})$ and $\mathbb{H}_0^1(\mathcal{O})$, respectively. The backward-uniform flattening property of the solution is used to prove the backward-uniform pullback asymptotic compactness of the non-autonomous ra…
We show that the stochastic flow generated by the Stochastic Navier-Stokes equations in a 2-dimensional Poincaré domain has a unique random attractor. This result complements a recent result by Brzeźniak and Li [10] who showed that the flow is asymptotically compact and generalizes a recent result by Caraballo et al. [12] who proved existence of a unique pullback attractor for the time-dependent deterministic Navier-Stokes equations in a 2-dimensional Poincaré domain.
The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with non-linear diffusion terms on bounded and unbounded ($\mathbb{R}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the litera…
In this work, we discuss the large time behavior of the solutions of the two dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations in bounded domains. Under the functional setting $\V\hookrightarrow\H\hookrightarrow\V'$, where $\H$ and $\V$ are appropriate separable Hilbert spaces and the embedding $\V\hookrightarrow\H$ is compact, we establish the existence of random attractors in $\H$ for the stochastic flow generated by the 2D SCBF equations perturbed by small additive noise. We prove the upper semicontinuity of the random attractors for the 2D SCBF equations in $\H$, when…
This article concerns the random dynamics and asymptotic analysis of the well known mathematical model, the Navier-Stokes equations. We consider the two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a \textsl{linear multiplicative white noise of Itô type} on the whole space $\mathbb{R}^2$. Firstly, we prove that the non-autonomous 2D SNSE generates a bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $\mathbb…
We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving paths of finite $ν$ - Hölder norms with $ν\in (\frac{1}{3},\frac{1}{2})$ so that the integrals are interpreted in the Gubinelli sense for controlled rough paths. We prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.
We prove the exponential stability of the zero solution of a stochastic differential equation with a Hölder noise, under the strong dissipativity assumption. As a result, we also prove that there exists a random pullback attractor for a stochastic system under a multiplicative fractional Brownian noise.
By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback random attractors if such attractors exist. We also prove pathwise periodicity and almost periodicity of inertial manifolds when non-autonomous deterministic forcing is periodic and almost periodic in time, respectively.
We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa \cite{AL0}. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE of Allen-Cahn type, we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of this random attractor and give a result on determ…
Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains
Hanoi Pedagogical University（河内师范大學）；Electric Power University（電力大學）；Tata Institute of Fundamental Research - Centre For Applicable Mathematics (TIFR-CAM)（塔塔基礎研究所應用數學中心）；NOVA School of Science and Technology (NOVA FCT)（新里斯本科技學院）；Indian Institute of Technology Roorkee-IIT Roorkee（魯爾凱印度理工學院）
Well-posedness, mean attractors and invariant measures of stochastic discrete long-wave-short-wave resonance equations driven by locally Lipschitz nonlinear noise
College of Science, National University of Defense Technology（国防科技大学理学院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
Asymptotically autonomous robustness in Probability of non-autonomous random attractors for stochastic convective Brinkman-Forchheimer equations on $\mathbb{R}^3$
The first goal of this paper is to prove multiple asymptotic results for a time-discrete and space-continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite volume limit for every fixed asymptotic slope; concentration inequalities for free energy implying a bound on its fluctuation exponent; straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one-sided infinite paths with given en…
