Armen Shirikyan

Coupling approach to white-forced nonlinear PDEs

We consider the 2D Navier–Stokes system, perturbed by a white in time random force, such that sufficiently many of its Fourier modes are excited (e.g., all of them are). It is proved that the system has a unique stationary measure and that all solutions exponentially fast converge in distribution to this measure. The proof is based on the same ideas as in our previous works on equations perturbed by random kicks. It applies to a large class of randomly forced PDEs with linear dissipation.

On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier–Stokes equations perturbed by various random forces of low dimension. AMS subject classifications: 35Q30, 60H15, 93C20

On Random Attractors for Mixing Type Systems

The paper deals with infinite-dimensional random dynamical systems. Under the condition that the system in question is of mixing type and possesses a random compact attracting set, we show that the support of the unique invariant measure is the minimal random point attractor. The results obtained apply to the randomly forced 2D Navier–Stokes system.

Euler equations are not exactly controllable by a finite-dimensional external force

We show that the Euler system is not exactly controllable by a finite-dimensional external force. The proof is based on the comparison of the Kolmogorov e-entropy for Holder spaces and for the class of functions that can be obtained by solving the two-dimensional Euler equations with various right-hand sides.

Some limiting properties of randomly forced two-dimensional Navier-Stokes equations

We consider random perturbations of two-dimensional Navier–Stokes equations. Under some natural conditions on random forces, we study asymptotic properties of solutions and stationary measures.

On dissipative systems perturbed by bounded random kick-forces

Abstract In this paper, we continue our investigation of dissipative PDE’s forced by random bounded kick-forces and of the corresponding random dynamical system (RDS) in function spaces. It was proved in [KS] that the domain of attainability from zero A (which is a compact subset of a function space) is invariant for the RDS associated with the original equation and carries a stationary measure μ, which is unique among all measures supported by A. Here we show that μ is the unique stationary measure for the RDS in the whole space and study its ergodic properties.

Entropic fluctuations in Gaussian dynamical systems

We study non-equilibrium statistical mechanics of a Gaussian dynamical system and compute in closed form the large deviation functionals describing the fluctuations of the entropy production observable with respect to the reference state and the non-equilibrium steady state. The entropy production observable of this model is an unbounded function on the phase space, and its large deviation functionals have a surprisingly rich structure. We explore this structure in some detail.

Rigorous results in space-periodic two-dimensional turbulence

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a unique stationary measure. We also discuss the Donsker-Varadhan and Freidlin-Wentzell type large deviations, as well as the inviscid limit and asymptotic results in 3d thin domains. We conclude with some open problems.

Approximate controllability of the viscous Burgers equation on the real line

The paper is devoted to studying the 1D viscous Burgers equation controlled by an external force. It is assumed that the initial state is essentially bounded, with no decay condition at infinity, and the control is a trigonometric polynomial of low degree with respect to the space variable. We construct explicitly a control space of dimension 11 that enables one to steer the system to any neighbourhood of a given final state in local topologies. The proof of this result is based on an adaptation of the Agrachev-Sarychev approach to the case of an unbounded domain.