Dissipativity and invariant measures for stochastic Navier-Stokes equations. (English) Zbl 0820.35108

The author considers the stochastic abstract equation \[ du(t) + Au(t) + B \bigl( u(t), u(t) \bigr) dt = fdt + dw(t). \tag{*} \] Let \(\nu\) be the space of infinitely differentiable two-dimensional vector fields \(u(x)\) on \(D\) (where \(D\) be a regular bounded open domain of \(\mathbb{R}^ 2)\) with compact support strictly contained in \(D\), satisfying \(\text{div} u(x) = 0\). We denote by \(V_ \alpha\) the closure of \(\nu\) in \([H^ \alpha (D)]^ 2\), for all real \(\alpha\), and we set in particular \(H = V_ 0\), \(V = V_ 1\). In equation \((*)\), we define the linear operator \[ A : D(A) \subset H \to H \] as \(Au = - \Delta u\), and \(D(A) = [H^ 2(D)]^ 2 \cap V\). Moreover in equation \((*)\), we define the bilinear operator \(B(u,v) : V \times V \to V_{-1}\) as \[ \bigl \langle B(u,v),z \bigr \rangle = \int_ Dz(x) \cdot \bigl( u(x) \cdot \nabla \bigr) v(x)dx, \] for all \(z \in V\). We assume that \(w(t)\) (in equation \((*)\)) is an infinite dimensional Brownian motion of the form \[ w(t) = \sum^ \infty_{j=1} \sigma_ j \beta_ j (t)e_ j, \] where \(\beta_ 1, \beta_ 2, \dots\) is a sequence of independent standard Brownian motions on a complete probability space \((\Omega, {\mathcal F},P)\) and the coefficients \(\sigma_ j\) satisfy the condition \[ \sum_{j=1}^ \infty {\sigma^ 2_ j \over \lambda_ j^{1/2} - 2 \beta_ 0} < \infty, \] for some \(\beta_ 0 > 0\), and we denote by \(0 < \lambda_ 1 \leq \lambda_ 2 \leq \dots\) the eigenvalues of \(A\), and by \(e_ 1,\) \(e_ 2, \dots\) a corresponding complete orthonormal system of eigenvectors.
The main result of this paper is the following theorem: Let \(\beta \in (0, \beta_ 0)\) and \(\theta \in (0,2 \beta_ 0)\), with \(\theta \leq 1/2\), be given. Denote by \(u_{t_ 0} (t, \omega)\) the solution of \((*)\) given in \([t_ 0, \infty)\), with the initial condition \(u(t_ 0) = 0\). Then there exists a real random variable \(r(\omega)\) (almost surely finite) such that \[ \sup_{- \infty < t_ 0 \leq 0} \biggl | A^{\min \{{1 \over 4} + \beta, \theta\}} u_{t_ 0} (0, \omega) \biggr | \leq r(\omega). \] In particular, the family of random variables \(\{u_{t_ 0} (0, \omega) : - \infty < t_ 0 \leq 0\}\) is bounded in probability in the space \(D(A^{\min \{{1 \over 4} + \beta, \theta\}})\); therefore the family of the laws of these random variables is tight in \(H\).


35Q30 Navier-Stokes equations
35R60 PDEs with randomness, stochastic partial differential equations
60J65 Brownian motion
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