My research interests and selected results

My past and current research interests concern the following three subjects:
1. Qualitative theory of nonlinear hyperbolic PDEs
2. Large-time asymptotics of stochastic PDEs
3. Control theory for nonlinear PDEs
I describe briefly the main results obtained in each of these directions and list five selected articles.

Qualitative theory of nonlinear hyperbolic PDEs

In a series of joint papers with L. Volevich, a qualitative theory is developed for linear and nonlinear hyperbolic PDEs with nearly constant coefficients. To be precise, let us consider a scalar equation of the form
\begin{equation}
P(\partial)u+\varepsilon Q(t,x,\partial^m u)=g(t,x),
\quad (t,x)\in{\mathbb R}^{1+n},
\end{equation}
where $P(\partial)$ is a strictly hyperbolic operator of order $m$ with constant coefficients, $\varepsilon\in{\mathbb R}$ is a small parameter, $\partial^m$ is the set of all partial derivatives up to order $m$, and $Q$ is a smooth function of its arguments. Under some natural conditions on the full symbol of $P(\partial)$ (which is regarded as a polynomial in the variable dual to time), we establish the following results:
• Existence of a time-bounded, time-periodic and almost periodic solution
• Asymptotic stability and exponential dichotomy for solutions of homogeneous problems and existence of Calderon projections
• Construction of stable, unstable, and centre manifolds for nonlinear equations
• Grobman-Hartman type theorems on the linearization of the phase portrait in the neighbourhood of a stationary point
Similar problems were thoroughly investigated for ODE's and different types of PDEs. However, there is a fundamental difference between hyperbolic equations and, say, ODE's. Namely, the solution of a hyperbolic equation of order $m$ possesses only $m-1$ derivatives. As a consequence, small (even linear) perturbation involving higher-order derivatives is not subordinate to the original unperturbed operator, and the above equation with $\varepsilon\ne0$ cannot be regarded as a small perturbation (in the "usual" sense) of the equation with constant coefficients corresponding to $\varepsilon=0$. To overcome this difficulty, we refined the technique of the Leray separating operator for a class of strictly hyperbolic operators whose full symbol does not vanish in some open strip. This enables one to obtain the results in the linear case. Investigation of nonlinear problems is based on the Leray-Schauder fixed point theorem and a general technique for reduction of a fully nonlinear equation to a quasi-linear system. A review of the results obtained in this direction (including the case of first-order systems) is given in this paper.

Large-time asymptotics of stochastic PDEs

A number of joint papers with S. Kuksin deal with a general theory for studying the problem of mixing for a class of dissipative stochastic PDEs with additive noise. To be precise, let us consider the case of the Navier-Stokes equations in a bounded domain $D\subset{\mathbb R}^2$:
\begin{equation}
\dot u+\langle u, \nabla\rangle u-\nu\Delta u+\nabla p=h(x)+\eta(t,x),
\quad {\mathop{\mathrm{div}}\nolimits}\,u=0.
\end{equation}
Here $\nu>0$ is the viscosity, $h$ is a deterministic function, and $\eta$ is a random process, which is assumed to be white in time and sufficiently regular in the space variables:
\begin{equation}
\eta(t,x)=\frac{\partial}{\partial t}\sum_{j=1}^\infty b_j\beta_j(t)e_j(x),
\end{equation}
where $b_j$ are positive decaying to zero sufficiently fast, $\{\beta_j\}$ is a family of independent standard Brownian motions, and $\{e_j\}$ is a complete set of normalised eigenfunctions for the Stokes operator. The equations are supplemented with the Dirichlet boundary condition for the velocity field $u$. It was proved that, for any $\nu>0$, the Markov process associated with the above problem has a unique stationary measure $\mu$, which is mixing in the Kantorovich-Wasserstein metric. Further study of the problem enabled one to establish the following results:
• Strong law of large numbers and central limit theorem
• Behaviour of stationary solutions at small values of the viscosity
• Description of the random point attractor in terms of a unique stationary measure
• Exponential mixing for PDEs perturbed by space-time localised noise
• Large deviations principle and entropy productions in PDEs perturbed by random kicks (in collaboration with V. Jaksic, V. Nersesyan, and C.-A. Pillet)
These investigations gave rise to a book written in collaboration with S. Kuksin and published by Cambridge University Press in 2012.

Control theory for nonlinear PDEs

My main contribution to the control theory concerns the 3D Navier-Stokes equations on a torus ${\mathbb T}^3\subset {\mathbb R}^3$. Writing these equations in the form
\begin{equation}
\dot u+\langle u, \nabla\rangle u-\nu\Delta u+\nabla p=h(t,x)+\zeta(t,x),
\quad {\mathop{\mathrm{div}}\nolimits}\,u=0,
\end{equation}
we now assume that $\zeta$ is a control taking values in a finite-dimensional subspace $E\subset L^2({\mathbb T}^3,{\mathbb R}^3)$. Developing the approach introduced by Agrachev and Sarychev in the two-dimensional case, I proved the following properties:
• Global approximate controllability
• Global exact controllability in finite-dimensional projections
These results, combined with some probabilistic techniques, enable one to investigate qualitative properties of stationary measures for the 3D Navier-Stokes system perturbed by a random force. Namely, I established that the support of stationary measures coincides with the whole phase space and that any finite-dimensional projection is minorised by a measure possessing a smooth positive density with respect to the Lebesgue measure.

Five selected articles

1. S. Kuksin, A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures, Comm. Math. Phys. 213 (2000), no. 2, 291–330.

This paper was the first in a series of articles devoted to studying the problem of uniqueness of a stationary measure and its stability for randomly forced PDEs. Using a Lyapunov–Schmidt-type reduction and a version of the Ruelle–Perron–Frobenius theorem, we proved the mixing for a large class of dissipative PDEs perturbed by a non-degenerate smooth kick force. Earlier results of the same type were dealing with the case of a rough noise. The ideas and techniques introduced in our paper were used later by many others to investigate mixing properties of various stochastic PDEs.

2. A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDEs, Prob. Theory Related Fields 134 (2006), no. 2, 215–247.

This paper is devoted to establishing limit theorems for trajectories of randomly forced PDEs. Building on some earlier results on the exponential mixing of the random flow, I proved that a large class of functionals calculated on trajectories satisfy the strong law of large numbers and central limit theorem and derived some optimal estimate for the corresponding rates of convergence. While it is a classical fact that sufficiently fast mixing in the total variation norm implies those results, we cannot make use of it in our context, because convergence to the stationary holds only in the weak topology. Our proof of the limit theorem is based on Gordin’s martingale approximation and limit theorems for martingales. This approach proved to be useful in other problems, such as the motion of a particle in a time-dependent random field defined by the 2D Navier–Stokes equation and forced shell model of turbulent flow.

3. A. Shirikyan, Approximate controllability for three-dimensional Navier–Stokes equations, Comm. Math. Phys. 266 (2006), no. 1, 123–151.

In this article, I developed an approach introduced by A. Agrachev and A. Sarychev to study the controllability of nonlinear PDEs in the situation when the linearised equation does not have good controllability properties. It was proved that the 3D Navier–Stokes equation considered on a torus is globally approximately controllable by a finite-dimensional force acting on low Fourier modes. The methods and results of this work were used later to establish the mixing character of the random dynamics corresponding to the 3D stochastic Navier–Stokes system, to prove the regularity of the law of finite-dimensional projections solutions, and to study controllability of more complicated PDEs, such as the 3D compressible Euler system.

4.  A. Shirikyan, Control and mixing for 2D Navier–Stokes equations with space-time localised noise, Ann. Sci. Éc. Norm. Supér. 48 (2015), no. 2, 253–280.

This paper is devoted to studying randomly forced 2D Navier–Stokes equations in a bounded domain with smooth boundary. It is assumed that the random perturbation is non-degenerate, and its law is periodic in time and has a support localised with respect to space and time. Concerning the unperturbed problem, we assume that it is approximately controllable in infinite time by an external force whose support is included in that of the random force. Under these hypotheses, we prove that the Markov process generated by the restriction of solutions to the instants of time proportional to the period possesses a unique stationary distribution, which is exponentially mixing. The proof is based on a coupling argument, a local controllability property of the Navier–Stokes system, an estimate for the total variation distance between a measure and its image under a smooth mapping, and some classical results from the theory of optimal transport.

While the problem of mixing for stochastic PDEs is now rather well understood when the random force acts on all determining modes, the situation is far more complicated when the random perturbation is highly degenerate. In such a context, the mixing was established for various models (such as the 2D Navier–Stokes and Boussinesq equations) when the noise is localised in the Fourier space. My paper provides a first result for random perturbations localised in the physical space and in time.

5. V. Jaksic, V. Nersesyan, C.-A. Pillet, A. Shirikyan, Large deviations from a stationary measure for a class of dissipative PDEs with random kicks, Comm. Pure Appl. Math. (2015), to appear.

This paper deals with a class of dissipative PDEs perturbed by a bounded random kick force. It is assumed that the random force is non-degenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviation principle for occupation measures of the Markov process in question. The proof is based on Kifer’s large deviation criterion, a coupling argument for Markov processes, and an abstract result on large-time asymptotic for generalised Markov semigroups. To the best of my knowledge, this paper provides a first result on Donsker–Varadhan-type LDP in a truly infinite-dimensional setting when the Markov process in question does not possess the strong Feller property.