Existence of global stochastic flow and attractors for Navier-Stokes equations
The authors prove the existence of a random attractor for the flow given by the solutions of a 2D stochastic Navier-Stokes equation on a torus. A random attractor for a flow \(\varphi\) is a random compact set \(A\) such that \[ \varphi(t,A(\omega),\omega)=A(\theta_t\omega),\quad t\geq 0,\qquad \lim_{t\to\infty} \text{dist}(\varphi(t,x,\theta_{-t}\omega),A(\omega))=0 \] (\(\theta_t\) is a group of measure preserving maps which plays the role of adapting the events to the time instant), where the first property means the invariance of the attractor with respect to the flow, while the second means the asymptotic behaviour. The forcing term in the equation is a multiplicative noise. There are some rather technical hypotheses on this term, one of them being a sort of orthogonality. This property seems crucial in order to conclude some energy estimates and get rid of the random part, which is fully stochastic. This means that the problem of finding an attractor cannot be solved pathwise. The main tools used in the paper are from nonstandard analysis. The problem is put in a nonstandard framework and the random attractor is the standard part of the set of points reached by the flow at infinity. In particular, the compactness of the attractor is assured by a fundamental result of nonstandard analysis.
stochastic Navier-Stokes equation on a torus
scientific article; zbMATH DE number 1371489