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

Abstract

In the paper [KNS20a], 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..

Large deviations principle for 2D Navier-Stokes equations with space time localised noise

“…Besides [Shi15], Shirikyan [Shi17] proved the exponentially mixing for one-dimensional Burgers equation perturbed by a stochastic forcing which is white in time and localized in space. When the noise is localized in physical space and degenerate in Fourier space, Nersesyan [Ner21] proved that the complex Ginzburg-Landau equation is exponential mixing. For the 2D Navier-Stokes system driven by a random force acting through the boundary, Shirikyan [Shi21] established an exponential mixing property.…”