tailieunhanh - Đề tài " Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing "

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L2 (T2 ). Unlike previous 0 works, this class is independent of the viscosity and the strength of the noise. | Annals of Mathematics Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing By Martin Hairer and Jonathan C. Mattingly Annals of Mathematics 164 2006 993 1032 Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing By Martin Hairer and Jonathan C. MATTiNGLy Abstract The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in Lq T2 . Unlike previous works this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property introduced in this work and an approximate integration by parts formula. The first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a Hormander-type condition. This requires some interesting nonadapted stochastic analysis. 1. Introduction In this article we investigate the ergodic properties of the 2D Navier-Stokes equations. Recall that the Navier-Stokes equations describe the time evolution of an incompressible fluid and are given by dtu u -V u vầu Vp div u 0 where u x t E R2 denotes the value of the velocity field at time t and position x p x t denotes the pressure and x t is an external force field acting on the fluid. We will consider the case when x E T2 the two-dimensional torus. Our mathematical model for the driving force is a Gaussian field which is white in time and colored in space. We are particularly interested in the case when only a few Fourier modes of are nonzero so that there is a well-defined injection scale L at which energy is pumped into the system. Remember that both the energy u 2 J u x 2 dx and

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