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Trajectory attractors for the 2D Navier-Stokes system and some generalizations. (English) Zbl 0894.35011
The authors extend known results on global attractors of 2D-Navier-Stokes: $\partial_tu= P\Delta u- P(u\nabla)u+ g(x, t),\quad\text{div }u= 0,\quad u= 0\quad\text{on }\partial\Omega.\tag{1}$ Here $$\Omega$$ is a smooth bounded domain. Much is known about attractors of (1). The authors study global attractors of (1) in an extended setting, adapted to the nonautonomous case. Let $$U$$, $$V$$, $$H_2$$ be the closures in $$L^2(\Omega)^2$$, $$H^1(\Omega)^2$$, $$H^2(\Omega)^2$$ respectively of all $$v\in C^\infty_0(\Omega)^2$$ with $$\text{div }v= 0$$. The intention is to study (1) not in the usual phase space, say $$H$$, but rather in a space of trajectories. To this end spaces $$L^{\text{loc}}_2= L^{\text{loc}}_2(R_+, H)$$ and $$L^{\text{loc}}_{2,w}= L^{\text{loc}}_{2,w}(R_+, H)$$ are needed; $$L^{\text{loc}}_2$$ is a Fréchet space under the family of seminorms $$\int^{t_2}_{t_1} \int_\Omega| u(x, t)|^2dx dt$$, $$0< t_1<t_2$$, and $$L^{\text{loc}}_{2,w}$$ is endowed with a corresponding weak topology. A $$g\in L^{\text{loc}}_{2,w}$$ is translation compact (t-c for short) if the set of translates $$g(\cdot+h)$$, $$h\geq 0$$ is precompact in $$L^{\text{loc}}_{2,w}$$; $$H_+(g)$$ is the closure of the hull of translates $$g(\cdot+ h)$$, $$h\geq 0$$ in $$L^{\text{loc}}_{2,w}$$. With $$g_0$$ t-c in $$L^{\text{loc}}_{2,w}$$ and $$g\in H_+(g_0)$$, let $$K_g$$ be the set of solutions of (1) with $$g$$ as force, set $$K_+= \bigcup K_g$$, $$g\in H_+g_0$$. The action of the translation semigroup $$(T_hg)(t)=g(t+ h)$$ now extends in a natural way from $$H_+(g_0)$$ to $$K_+$$; moreover $$T_hK_+\subseteq K_+\subseteq L^{\text{loc}}_{2,w}$$ follows from the definitions. The authors now endow $$K_+$$ with a Fréchet topology which is adapted to the notion of strong solution of (1). Based on this notions, the concept of trajectory attractor in $$K_+$$ with respect to the translation semigroup $$T_h$$, $$h\geq 0$$ is defined. The main result states among others that if $$g_0\in L^{\text{loc}}_2$$ is t-c in $$L^{\text{loc}}_{2,w}$$ then $$K_+$$ has a trajectory attractor with respect to the translation semigroup $$T_h$$, $$h\geq 0$$, which is endowed with various properties. The proof is based on a number of lengthy estimates. The paper concludes with further material on an extension of these concepts to 3D-Navier-Stokes.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35Q30 Navier-Stokes equations
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