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Global attractors for small samples and germs of 3D Navier-Stokes equations. (English) Zbl 1075.35037
The author considers the Navier-Stokes equations $$u_t= \nu\Delta u- (u\nabla)u+ \nabla p+ f,\ \text{div}(u)= 0,\ u= 0\quad\text{on }\partial\Omega,\tag1$$ with $\Omega$ smooth and bounded, and $f$ a constant exterior force. One imposes on (1) a standard Hilbert space setting: $$|u|^2= (u,u)= \sum \int u^2_j\, dx.$$ $H$ is the $L^2$-closure of $V_0= \{u/u\in C_0(\Omega)^3,\text{div}(u)= 0\}$, while $V$ is the closure of $V_0$ with respect to the norm $$\Vert u\Vert^2= \sum\int(\partial_j u)^2\, dx,\quad \partial_j= \partial_{x_j}.$$ A weak solution of (1) is an element $u$ in $L^\infty(0,\infty; H)\cap L^2(0, T;V)$ (for all $T$) subject to three classical conditions, one requiring that $u$ satisfies a standard variational form of (1). The set $W$ of weak solutions is considered as a metric space under the norm $$[u]^2= \int^\infty_0 |u(t)|^2 e^{-t}\,dt.$$ It was proved by {\it G. R. Sell} [J. Dyn. Differ. Eq. 8, No. 1, 1--33 (1996; Zbl 0855.35100)] that the semiflow $S_t$ on $W$ given by $$(S_t u)(x)= u(t+ s),\quad s\ge 0$$ admits a global attractor $A$. The author now takes a generalization of this situation given by {\it J. M. Ball} [J. Nonlinear Sci. 7, No. 5, 475--502 (1997; Zbl 0903.58020)] as starting point who introduced the notion of generalized semiflow $G$ on a metric space $X$. Since this notion is not directly applicable to the space $W$, the author defines a notion of $\varepsilon$-samples as follows. With $u|\varepsilon$ the restriction of $u\in L^\infty(0,\infty; H)$ to $[0,\varepsilon]$ one sets: $$W_\varepsilon= \{u|\varepsilon\mid u\in W\}$$ and with $u\in W$ one associates $\varphi^\varepsilon_u: [0,\infty]\to W$ as follows: $$\varphi^\varepsilon_u(t)= u^t|\varepsilon,\quad\text{where }u^t(s)= u(t+ s),\quad s\ge 0.$$ One then sets: $$G_\varepsilon= \{\varphi^\varepsilon_u\mid u\in W\}.$$ The main results then are: (I) $G_\varepsilon$ is a generalized semiflow on $W_\varepsilon$, (II) with $A$ the attractor for $S_t$ and $A_\varepsilon= \{u\mid \varepsilon\mid u\in A\}$, $A_\varepsilon$ is a global attractor for $G_\varepsilon$. The author extends the above theory to a familiy of objects called “germs”, which are a kind of infinitesimal $\varepsilon$-samples $(\varepsilon= 0)$.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35B41 Attractors (PDE) 37L30 Attractors and their dimensions, Lyapunov exponents 76D05 Navier-Stokes equations (fluid dynamics)
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##### References:
 [1] Aubin, J. -P.: Un théorème de compacité. CR acad. Sci. Paris sér. I math. 256, 5042-5044 (1963) · Zbl 0195.13002 [2] J.M. Ball, Continuity properties and global attractors of generalised semiflows and the Navier -- Stokes equations, J. Nonlinear Sci. 7 (1997) 475 -- 502. Erratum ibid. 8 (1998) 233. Corrected version appears in Mechanics: from Theory to Computation, Springer, Berlin, 2000, pp. 447 -- 474. [3] Capiński, M.; Cutland, N. J.: A simple proof of existence of weak and statistical solutions of Navier -- Stokes equations. Proc. roy. Soc. London A 436, 1-11 (1992) · Zbl 0746.35059 [4] M. Capiński, N.J. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, Series on Advances in Mathematics for Applied Sciences, vol. 27, World Scientific, Singapore, 1995. · Zbl 0824.76003 [5] Capiński, M.; Cutland, N. J.: Attractors for three-dimensional Navier -- Stokes equations. Proc. roy. Soc. London ser. A 453, 2413-2426 (1997) · Zbl 0990.35118 [6] N.J. Cutland, H.J. Keisler, Attractors and neo-attractors for 3D stochastic Navier -- Stokes equations, Stoch. Dynam., to appear. · Zbl 1081.60048 [7] C. Foias, R. Temam, The connection between the Navier -- Stokes equations, dynamical systems, and turbulence theory, in: Directions in Partial Differential Equations (Madison, WI, 1985), Publications of the Mathematics Research Center University of Wisconsin, vol. 54, Academic Press, Boston, 1987, pp. 55 -- 73. [8] Málek, J.; Pražák, D.: Large time behavior via the method of l-trajectories. J. differential equations 181, 243-279 (2002) · Zbl 1187.37113 [9] Marín-Rubio, P.; Robinson, J. C.: Attractors for the stochastic 3D Navier -- Stokes equations. Stochastics dynamics 3, 279-297 (2003) · Zbl 1059.35100 [10] Sell, G. R.: Global attractors for the three-dimensional Navier -- Stokes equations. J. dynamics differential equations 8, 1-33 (1996) · Zbl 0855.35100 [11] Temam, R.: Navier -- Stokes equations and nonlinear functional analysis. (1983) · Zbl 0522.35002