66 research outputs found
Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit
We consider the behaviour of the distribution for stationary solutions of the
complex Ginzburg-Landau equation perturbed by a random force. It was proved
earlier that if the random force is proportional to the square root of the
viscosity, then the family of stationary measures possesses an accumulation
point as the viscosity goes to zero. We show that if is such point, then
the distributions of the L^2 norm and of the energy possess a density with
respect to the Lebesgue measure. The proofs are based on It\^o's formula and
some properties of local time for semimartingales.Comment: 12 page
Global exponential stabilisation for the Burgers equation with localised control
We consider the 1D viscous Burgers equation with a control localised in a
finite interval. It is proved that, for any , one can find a
time of order such that any initial state can be
steered to the -neighbourhood of a given trajectory at time .
This property combined with an earlier result on local exact controllability
shows that the Burgers equation is globally exactly controllable to
trajectories in a finite time. We also prove that the approximate
controllability to arbitrary targets does not hold even if we allow infinite
time of control.Comment: 19 page
Exponential attractors for random dynamical systems and applications
The paper is devoted to constructing a random exponential attractor for some
classes of stochastic PDE's. We first prove the existence of an exponential
attractor for abstract random dynamical systems and study its dependence on a
parameter and then apply these results to a nonlinear reaction-diffusion system
with a random perturbation. We show, in particular, that the attractors can be
constructed in such a way that the symmetric distance between the attractors
for stochastic and deterministic problems goes to zero with the amplitude of
the random perturbation.Comment: 37 page
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
Exact controllability in projections for three-dimensional Navier-Stokes equations
The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Our sufficient condition is verified for any torus in R3. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a sim-ple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier–Stokes system has a unique strong solution fo
Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation
We study a damped semi-linear wave equation in a bounded domain with smooth
boundary. It is proved that any sufficiently smooth solution can be stabilised
locally by a finite-dimensional feedback control supported by a given open
subset satisfying a geometric condition. The proof is based on an investigation
of the linearised equation, for which we construct a stabilising control
satisfying the required properties. We next prove that the same control
stabilises locally the non-linear problem.Comment: 29 page
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