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A global attractor for the Navier-Stokes equations in an unbounded two-dimensional strip domain and an estimate of its dimension. (Chinese) Zbl 0909.35106

The authors discuss the Navier-Stokes equation in an unbounded two-dimensional strip domain: \[ u_t-\Delta u+ u_i{\partial u\over\partial x_i}= -\nabla p+ f,\quad (x,t)\in\Omega\times \mathbb{R},\quad \text{div }u= 0,\tag{1} \]
\[ u(x, t)\in H^1_0(\Omega),\quad t> 0,\quad u(x,0)= u_0(x)\in H, \] where \(\Omega= (0,d)\times \mathbb{R}\), \(d>0\) is a constant, \(u\) and \(p\) are unknown quantity, \(u= (u_1,u_2)\) is a velocity field, \(p\) expresses pressure. Using the compactness of the weighted space, the Galerkin method and the Gronwall uniform inequality, under \(u_0\in H\), \(f\in V\) and \(f[\log(e+| x|^2)]^{1/2}\in L^2(\Omega)\), the authors prove the existence of the global attractor \(A\) for (1) in \(H\), and \(A\) is a subset of \(H^2_0(\Omega)\). Also, the authors give estimates of the Hausdorff dimension and the fractal dimension of the attractor \(A\).

MSC:

35Q30 Navier-Stokes equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35B40 Asymptotic behavior of solutions to PDEs
47H20 Semigroups of nonlinear operators
76D05 Navier-Stokes equations for incompressible viscous fluids
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