Ziane, M. On the two-dimensional Navier-Stokes equations with the free boundary condition. (English) Zbl 0912.35127 Appl. Math. Optimization 38, No. 1, 1-19 (1998). 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. Reviewer: A.Cichocka (Katowice) Cited in 7 Documents 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 Keywords:Navier-Stokes equations; trilinear form; global attractors; Hausdorff and fractal dimensions; Grashof number; elongated domains; upper bound of the dimension; Lieb-Thirring inequality PDFBibTeX XMLCite \textit{M. Ziane}, Appl. Math. Optim. 38, No. 1, 1--19 (1998; Zbl 0912.35127) Full Text: DOI