## 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 $$H$$.

### MSC:

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

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