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Attractors for three-dimensional Navier-Stokes equations. (English) Zbl 0990.35118

Introduction: Our interest lies in attractors for solutions of the time-homogeneous Navier-Stokes equations (NSE): \(\partial u/\partial t-\nu \Delta u+\langle u,\nabla\rangle u=f-\nabla p\), \(\text{div} u=0\), in a bounded domain \(D\subseteq \mathbb{R}^d\), \(d\leq 4\), with boundary of class \(C^2\), where \(u=(u^1,\cdots,u^d)\) is the velocity and \(p: D\times [0,\infty)\to\mathbb{R}\) is the pressure of an incompressible fluid \((\langle\cdot,\cdot\rangle\) is the scalar product in \(\mathbb{R}^d)\). The function \(f: D\to \mathbb{R}^d\) is given (it represents the external force). We impose the homogeneous Dirichlet condition \(u|_{\partial D}=0\).
In the two-dimensional case \((d=2)\), for each initial value \(v\in \mathbb{H}\), where \(\mathbb{H}\) is a suitable Hilbert space, it is well known that there exists a unique solution for all times, which we may denote by \(S(t)v\), thus defining a semigroup on \(\mathbb{H}\). An attractor \(A\) for this semigroup has been shown to exist; that is, an invariant set \(A\) (i.e. \(S(t)A=A\) for all \(t\)), such that for all \(v\in\mathbb{H}\) and any open neighbourhood \(\mathcal O\) of \(A\), \(S(t)v\in\mathcal O\) eventually. The attractor here has finite Hausdorff dimension, and estimates on its dimension are known [see R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, Springer, New York (1988; Zbl 0662.35001)].
The situation is different for the most interesting three-dimensional case. The uniqueness problem here is still open; for initial data \(v\in\mathbb{H}\), so-called weak solutions of the NSE exist for all time, but it is not known whether there is a unique weak solution. On the other hand, for initial data from a smaller Hilbert space \(\mathbb{V}\subseteq \mathbb{H}\) we have existence and uniqueness of the so-called strong solutions, but only locally in time. Thus for \(v\in\mathbb{V}\), \(S(t)v\) is defined only on a small interval, whose length depends on \(v\). The known results about existence of attractors in this case concern the hypothetical situation where the strong solution exists globally [see P. Constantin, C. Foias and R. Temam, Mem. Am. Math. Soc. 314 (1985; Zbl 0567.35070)].
The remedy for this lack of a semigroup that has been suggested by G. R. Sell [Global attractors for the Navier-Stokes equations, Preprint (1997); see also J. Dyn. Differ. Equations 8, No. 1, 1-33 (1996; Zbl 0855.35100)] is this: redefine the phase space to ensure the uniqueness property. To this end, he defines the phase space \(X\) to be the set of all functions defined on the time interval \([0,\infty)\) that are trajectories of weak solutions to the NSE. Then \(S(t)\) is uniquely defined on \(X\) as the shift of a weak solution by \(t\). Thus the non-uniqueness problem is bypassed and a semigroup obtained, but in a rather artificial way.
In this paper we present an approach using the nonstandard framework [M. Capinski and N. Cutland, Proc. R. Soc. Lond., Ser. A 436, 1-11 (1992; Zbl 0746.35059)] that greatly simplifies the solution of the NSE in dimensions \(d\leq 4\). The crucial feature that we exploit is the presence of a well-defined nonstandard semigroup \(T(t)\) on a hyperfinite-dimensional space \(\mathbb{H}_N\), which gives an internal (i.e. nonstandard) attractor \(C\). The semigroup \(T(t)\) has natural standard counterparts that are multivalued semiflows, each of which has an attractor \(A\subset \mathbb{H}\) that is obtained from \(C\).
In our first approach the phase space is the Hilbert space \(\mathbb{H}\). The clear advantage, compared with that of Sell for example, is that it is feasible to seek estimates of the Hausdorff and fractal dimensions of the attractor \(A\) in \(\mathbb{H}\) (although we do not pursue that here). We introduce two constructions of multi-valued semiflows \(S(t)\), \(\widehat S(t)\) on \(\mathbb{H}\), each obtained by taking standard parts of certain sets of points \(T(t)U\) in the weak topology.
We also present another approach that has some similarities to Sell’s, but is less radical and in particular does not bypass the non-uniqueness problem. The phase space here is a space \(\mathcal X\) of functions defined on the unit interval \([0,1]\) (the length of the interval is not material – it can be \([0,a]\) for any fixed \(a\)). As before, we have a well-defined internal semigroup \(\mathcal T(t)\) due to the uniqueness property on the nonstandard phase space. This semigroup also generates a multi-valued standard semiflow \(\mathcal S(t)\) with more satisfactory properties than those of \(S(t)\) or \(\widehat S(t)\), and in this case also we have a standard attractor \(\mathcal A\subset\mathcal X\).

MSC:

35Q30 Navier-Stokes equations
35B41 Attractors
76D05 Navier-Stokes equations for incompressible viscous fluids
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