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Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force. (English) Zbl 0931.35124
The authors consider a viscous incompressible Newtonian fluid in a bounded open domain $D\subset\bbfR^3$ with smooth boundary $\partial D$, described by the classical Navier-Stokes equations $${\partial u\over\partial t}+(u\cdot\nabla)u+\nabla p=\nu\Delta u+f+{\partial g\over \partial t}\text{ in }[0,\infty)\times D,$$ \align \text{div} & u=0\text{ in }[0, \infty)\times D,\\ & u=0\text{ in }[0,\infty)\times\partial D,\\ & u(0,x)=u_0(x),\ x\in D,\endalign where $u$ is the velocity field, $p$ the pressure field, $\nu>0$ the kinematic viscosity, and $f+(\partial g/\partial t)$ the body force. Let $X=L^2_{\text{loc}}[0,\infty;H)$, and $W(g)\subset X$ be the space of generalized weak solutions. Let $G$ be the space of all $g\in C_0(\bbfR;V)$ that have polynomial growth at $-\infty$ and satisfy the property $$\liminf_{\alpha\to\infty}\limsup_{t\to\infty}{1\over t}\int^0_{-t} \bigl|z_\alpha(s,g)\bigr|^8_{L^4}ds=0,$$ where $z_\alpha(t,g)=g(t)-\int^t_{-\infty}(A+\alpha)e^{-(t-s)(A+\alpha)}g(s)ds$. Let $$B(M_0,g)= \left\{ u\in W(g): \int^1_0\bigl|u(s)\bigr|^2 ds\le M_0\right\}$$ be defined for all constants $M_0>0$ and all $g\in G$. Let $M(\cdot)$ be a function, defined on the backward orbit of $g$ (the set $\{\theta_{-\tau}g;\tau\ge 0\})$ with values in $(0,\infty)$, so that the subexponential growth condition is fulfilled, $$\lim_{\tau\to+\infty}{\log^+M(\theta-\tau g)\over\tau}=0.\tag 1$$ For $g\in G$ and a function $M(\cdot)$ satisfying the assumption (1) let us define the set $${\cal A}\bigl(g,M(\cdot)\bigr)=\bigcap_{\tau\ge 0}\overline {\bigcup_{t\ge\tau}\Phi(t,\theta_{-t}g)B\bigl(M(\theta_{-t}g), \theta_{-t}g \bigr)}$$ and let for $g\in G$ $${\cal A}(g)=\overline{\bigcup_{M(\cdot)}{\cal A}(g,M(\cdot))}.\tag 2$$ Let ${\cal D}$ be the set of all measurable multifunctions contained in a ball $B(M(g),g)$, so that (1) is fulfilled with $P_Q$-probability 1. There exists an $\alpha>0$ and a $\{\theta_t\}_{t \in\bbfR}$-invariant set $G_\alpha\subset C_0(\bbfR;V)$ of full $P_Q$-measure consisting of functions $g$ of polynomial growth so that $$2C^*\lim_{\tau \to\pm \infty} {1 \over\tau}\int^0_{-\tau} |z_\alpha(0, \theta_sg) |^8_{L^4} ds=\beta_\alpha.$$ The main theorem of this paper says that for sufficiently large $\alpha>0$ we have the following: (i) The attractor ${\cal A}(g)$ defined by (2) for $g\in G_\alpha$ (for $g\in C_0(\bbfR,V)\setminus G_\alpha$ we set ${\cal A}(g)= \{0\})$ defines a $\overline{\cal B}^{P_Q}_{C_0(-\infty,\infty;V)}$-measurable multifunction. (ii) ${\cal A}(g)$ is the unique attractor in ${\cal D}$. (iii) For any $D\in{\cal D}$, $\varepsilon>0$, $$P_Q\biggl\{d\bigl(\overline {\Phi (t,g)D(g)},{\cal A}(\theta_tg)\bigr)> \varepsilon\biggr\}\to 0\quad\text{for}\quad t\to\infty.$$

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35R60 PDEs with randomness, stochastic PDE 35B41 Attractors (PDE) 37L30 Attractors and their dimensions, Lyapunov exponents
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