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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 \[ u_t= \nu\Delta u- (u\nabla)u+ \nabla p+ f,\;\text{div}(u)= 0,\;u= 0\quad\text{on }\partial\Omega,\tag{1} \] 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 \[ \| u\|^2= \sum\int(\partial_j u)^2\, dx,\quad \partial_j= \partial_{x_j}. \] 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 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 \[ (S_t u)(x)= u(t+ s),\quad s\geq 0 \] admits a global attractor \(A\). The author now takes a generalization of this situation given by 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: \[ \varphi^\varepsilon_u(t)= u^t|\varepsilon,\quad\text{where }u^t(s)= u(t+ s),\quad s\geq 0. \] 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 Navier-Stokes equations
35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
76D05 Navier-Stokes equations for incompressible viscous fluids
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