arXiv subjects

Vahagn Nersesyan

Publications and source records attributed to Vahagn Nersesyan.

At least 19 recordsLinked to original sources

arXiv2026

On the chaotic behavior of the Lagrangian flow of the 2D Navier-Stokes system with bounded degenerate noise

We consider a fluid governed by the randomly forced 2D Navier-Stokes system. The force is bounded and highly degenerate, acting directly only on a small number of Fourier modes, and satisfies natural decomposability and observability properties. We prove that the Lagrangian flow associated with this system exhibits chaotic behavior, characterized by the strict positivity of the top Lyapunov exponent. To achieve this, we develop a new abstract criterion for random dynamical systems that enables one to derive the positivity of the top Lyapunov exponent and exponential mixing from controllability properties of the associated nonlinear and linearized deterministic systems. In particular, our mixing criterion removes the analyticity assumption required in previous works and establishes an exponential rate of convergence in the case where the unperturbed dynamics does not admit a unique globally stable equilibrium.

math.AP↗

arXiv2026

Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space

We consider the weakly damped stochastic nonlinear Schrödinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on $H^2$, and establish polynomial mixing in the dual-Lipschitz metric over $H^1$. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foiaş-Prodi-type estimate in the $H^1$-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincaré inequality and a space-time weight function quantifying the spatial decay of solutions.

math.PR↗

arXiv2026

Controllability and Exponential Mixing in Singular Interacting Particle Systems

This article concerns interacting particle systems with singular kernels, driven either by degenerate deterministic controls or by degenerate decomposable noise. In the deterministic setting, we establish global exact controllability and a topologically robust property called solid controllability. Moreover, we prove a result that guarantees global approximate controllability with prescribed trajectories. For stochastic dynamics, we obtain ergodicity and exponential mixing by utilizing coupling and recurrence mechanisms based on controllability. Our approach exploits the singularity and applies to a broad class of models, including Biot-Savart, Coulomb, Riesz, and Yukawa interactions, as well as heterogeneous multi-species systems.

math.OC↗

arXiv2025

Global controllability of Boussinesq flows by using only a temperature control

We show that buoyancy driven flows can be steered in an arbitrary time towards any state by applying as control only an external temperature profile in a subset of small measure. More specifically, we prove that the 2D incompressible Boussinesq system on the torus is globally approximately controllable via physically localized heating or cooling. In addition, our controls have an explicitly prescribed structure; even without such structural requirements, large data controllability results for Boussinesq flows driven merely by a physically localized temperature profile were so far unknown. The presented method exploits various connections between the model's underlying transport-, coupling-, and scaling mechanisms.

math.AP↗

arXiv2025

Relaxation enhancement by controlling incompressible fluid flows

We propose a PDE-controllability based approach to the enhancement of diffusive mixing for passive scalar fields. Unlike in the existing literature, our relaxation enhancing fields are not prescribed $\textit{ab initio}$ at every time and at every point of the spatial domain. Instead, we prove that time-dependent relaxation enhancing vector fields can be obtained as $\textit{state trajectories of control systems described by the incompressible Euler equations}$ either driven by finite-dimensional controls or by controls localized in space. The main ingredient of our proof is a new approximate controllability theorem for the incompressible Euler equations on $\mathbb{T}^2$, ensuring the approximate tracking of the full state all over the considered time interval. Combining this with a continuous dependence result yields enhanced relaxation for the passive scalar field. Another essential tool in our analysis is the exact controllability of the incompressible Euler system driven by spatially localized forces.

math.AP↗

arXiv2025

Localized and degenerate controls for the incompressible Navier-Stokes system

We consider the global approximate controllability of the two-dimensional incompressible Navier-Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier-Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.

math.AP↗

arXiv2024

Polynomial mixing for the white-forced Navier-Stokes system in the whole space

We study the mixing properties of the white-forced Navier-Stokes system in the whole space $\mathbb{R}^2$. Assuming that the noise is sufficiently non-degenerate, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The proof combines the coupling method with a Foiaş-Prodi type estimate, weighted growth estimates for trajectories, and an estimate for the Leray projector involving Muckenhoupt $A_2$-class weights.

math.AP↗

arXiv2024

Exponential mixing of constrained random dynamical systems via controllability conditions

We provide deterministic controllability conditions that imply exponential mixing properties for randomly forced constrained dynamical systems with possibly unbounded state space. As an application, new ergodicity results are obtained for non-smooth models in elasto-plasticity driven by various types of noise, including white noise. It is thereby illustrated how tools from control theory can be utilized to tackle regularity issues that commonly arise in the qualitative study of constrained systems.

math.OC↗

arXiv2023

Exponential mixing for the white-forced complex Ginzburg--Landau equation in the whole space

In the last two decades, there has been a significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In the present paper, we consider the damped complex Ginzburg--Landau equation on the real line driven by a white-in-time noise. Under the assumption that the noise is sufficiently non-degenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foiaş--Prodi estimate to the case of the real line and special space-time weighted estimates which help to handle the behavior of solutions at infinity.

math.AP↗

arXiv2022

Controllability and ergodicity of 3D primitive equations driven by a finite-dimensional force

We study the problems of controllability and ergodicity of the system of 3D primitive equations modeling large-scale oceanic and atmospheric motions. The system is driven by an additive force acting only on a finite number of Fourier modes in the temperature equation. We first show that the velocity and temperature components of the equations can be simultaneously approximately controlled to arbitrary position in the phase space. The proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the controllability of the linearisation of primitive equations around a non-stationary trajectory of randomly forced system. Assuming that the probability law of the forcing is decomposable and observable, we prove almost sure approximate controllability by using the same Fourier modes as in the nonlinear setting. Finally, combining the controllability of the linearised system with a criterion from arXiv:1802.03250, we establish exponential mixing for the nonlinear primitive equations with a random force.

math.AP↗

arXiv2022

Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions

We consider the 1D nonlinear Schrödinger equation with bilinear control. In the case of Neumann boundary conditions, local exact controllability of this equation near the ground state has been proved by Beauchard and Laurent in arXiv:1001.3288. In this paper, we study the case of Dirichlet boundary conditions. To establish the controllability of the linearised equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow to control approximately the linearised Schrödinger equation. We show that the reachable set for the linearised equation is closed. This is achieved by representing the resolving operator as a sum of two linear continuous mappings: one is surjective (here the control in generic direction is used) and the other is compact. A mapping with dense and closed image is surjective, so the linearised Schrödinger equation is exactly controllable. Then local exact controllability of the nonlinear equation is derived using the inverse mapping theorem.

math.AP↗

arXiv2022

Large deviations principle via Malliavin calculus for the Navier-Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker-Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier-Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in arXiv:1410.6188 in a discrete-time setting and extended in arXiv:1505.03686 to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman-Kac semigroup, which we verify by using Malliavin calculus.

math.PR↗

arXiv2021

Exponential mixing under controllability conditions for SDEs driven by a degenerate Poisson noise

We prove existence and uniqueness of the invariant measure and exponential mixing in the total-variation norm for a class of stochastic differential equations driven by degenerate compound Poisson processes. In addition to mild assumptions on the distribution of the jumps for the driving process, the hypotheses for our main result are that the corresponding control system is dissipative, approximately controllable and solidly controllable. The solid controllability assumption is weaker than the well-known parabolic Hörmander condition and is only required from a single point to which the system is approximately controllable. Our analysis applies to Galerkin projections of stochastically forced parabolic partial differential equations with asymptotically polynomial nonlinearities and to networks of quasi-harmonic oscillators connected to different Poissonian baths.

math.PR↗

arXiv2021

A proof of approximate controllability of the 3D Navier-Stokes system via a linear test

We consider the 3D Navier-Stokes system driven by an additive finite-dimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearised around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity.

math.AP↗

arXiv2021

The complex Ginzburg-Landau equation perturbed by a force localised both in physical and Fourier spaces

In the paper arXiv:1802.03250, a criterion for exponential mixing is established for a class of random dynamical systems. In that paper, the criterion is applied to PDEs perturbed by a noise localised in the Fourier space. In the present paper, we show that, in the case of the complex Ginzburg-Landau (CGL) equation, that criterion can be used to consider even more degenerate noise that is localised both in physical and Fourier spaces. This is achieved by checking that the linearised equation is almost surely approximately controllable. We also study the problem of controllability of the nonlinear CGL equation. Using Agrachev-Sarychev type arguments, we prove an approximate controllability property in the case of a control force which is again localised in physical and Fourier spaces.

math.AP↗

arXiv2021

Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of the NLS equation with random potential. More precisely, we assume that the amplitude of the potential is a random process whose law is $1$-periodic in time and non-degenerate. Combining the controllability with a stopping time argument and the Markov property, we show that the trajectories of the random equation are almost surely unbounded in regular Sobolev spaces.

math.AP↗

arXiv2020

Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension

In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the corresponding Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.

math.AP↗

arXiv2020

Multiplicative ergodic theorem for a non-irreducible random dynamical system

We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplica-tive ergodic theorem with an exponential rate of convergence. The assumptions are satisfied for a large class of parabolic PDEs, including the 2D Navier--Stokes and complex Ginzburg--Landau equations perturbed by a non-degenerate bounded random kick force. As a consequence of this er-godic theorem, we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system. In particular , we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible. The proof relies on a refined version of the uniform Feller property combined with some contraction and bootstrap arguments.

math.AP↗