Malliavin Calculus for the Stochastic 2d Navier Stokes Equation
نویسنده
چکیده
Abstract. We consider the incompressible, two dimensional Navier Stokes equation with periodic boundary conditions under the effect of an additive, white in time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite dimensional projection of the solution possesses a smooth density with respect to Lebesgue measure. We also show that under natural assumptions the density of such a projection is everywhere strictly positive. In particular, our conditions are viscosity independent. We are mainly interested in forcing which excites a very small number of modes. All of the results rely on the nondegeneracy of the infinite dimensional Malliavin matrix.
منابع مشابه
On Stochastic Navier-Stokes Equation Driven by Stationary White Noise
We consider an unbiased approximation of stochastic Navier-Stokes equation driven by spatial white noise. This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier-Stokes equation. The nonlinear term can be characterized as the highest stochastic order approximation of the original nonlinear term u∇u. We investigate the analyti...
متن کاملMalliavin Calculus for Infinite-dimensional Systems with Additive Noise
We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the system we develop a counterpart of Hörmander’s classical theory in this setting. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and ...
متن کاملAnticipating Stochastic 2D Navier-Stokes Equations
In this article, we consider the two-dimensional stochastic Navier-Stokes equation (SNSE) on a smooth bounded domain, driven by affine-linear multiplicative white noise and with random initial conditions and Dirichlet boundary conditions. The random initial condition is allowed to anticipate the forcing noise. Our main objective is to prove the existence and uniqueness of the solution to the SN...
متن کاملWeak Solutions of Non Coercive Stochastic Navier-stokes Equations in R
We prove existence of weak solutions of stochastic Navier-Stokes equations in R which do not satisfy the coercivity condition. The equations are formally derived from the critical point of some variational problem defined on the space of volume preserving diffeomorphisms in R. Since the domain of our equation is unbounded, it is more difficult to get tightness of approximating sequences of solu...
متن کاملErgodic properties of highly degenerate 2D stochastic Navier-Stokes equations
This note presents the results from “Ergodicity of the degenerate stochastic 2D Navier-Stokes equation” by M. Hairer and J. C. Mattingly. We study the Navier-Stokes equation on the two-dimensional torus when forced by a finite-dimensional Gaussian white noise and give conditions under which the system is ergodic. In particular, our results hold for specific choices of four-dimensional Gaussian ...
متن کامل