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On the two-dimensional Navier-Stokes equations with the free boundary condition. (English) Zbl 0912.35127

The two-dimensional Navier-Stokes equations with free boundary condition is studied. Three different results are established. First, an orthogonality property for the trilinear from is given.
Based on this result, an upper bound of the dimension of the corresponding attractor which is given by \(CG^{2/3}(\log G+1)^{1/3}\) where \(G\) is the non-dimensional Grashof number and \(C\) is constant depending only on the domain, is derived. Finally, a new form of the corresponding Lieb-Thirring inequality for elongaged domains is proved and the explicit dependence of the upper bound of the dimension of the attractor on the length ratio of the domain is inferred.

MSC:

35Q30 Navier-Stokes equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
76D05 Navier-Stokes equations for incompressible viscous fluids
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