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Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces. (English) Zbl 1112.35029

The purpose of this paper is to the investigate uniform attractor for the nonautonomous two-dimensional Navier-Stokes equations of viscous incompressible fluid on a bounded domain:

u t-νΔu+(u·)u+p=f(x,t),divu=0inΩ,u=0onΩ,u| t=τ =u τ ,τ,(1)

where ν>0 is the kinematic viscosity of the flow and f(x,t) is the external force. As usually H and V denote the Hilbert spaces

H=u L 2 (Ω) 2 : div u = 0 , u · n | Ω = 0,V=u H 0 1 (Ω) 2 : div u = 0

which are endowed with the scalar products and norms of L 2 (Ω) 2 and H 2 1 (Ω) 2 respectively. First, the existence and structure of uniform attractors in H is proved for equations with normal external forces in L loc 2 (;V ' ). Then, the properties of kernel sections are studied. Finally, the fractal dimension of the kernel sections of the uniform attractor is estimated.

MSC:
35B41Attractors (PDE)
35Q30Stokes and Navier-Stokes equations
76D05Navier-Stokes equations (fluid dynamics)