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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: $$\frac{\partial u}{\partial t}-\nu\Delta u+(u\cdot \nabla)u+\nabla p=f(x,t),\text{ div}u=0\text{ in }\Omega,\ u=0\text{ on }\partial\Omega, \ u|_{t=\tau}=u_\tau,\ \tau\in\bbfR,\tag 1$$ where $\nu>0$ is the kinematic viscosity of the flow and $f(x,t)$ is the external force. As usually $H$ and $V$ denote the Hilbert spaces $$\align H&=\bigl\{u\in L^2 (\Omega)^2:\operatorname{div}\,u=0,\ u\cdot\vec n|_{\partial\Omega}=0 \bigr\},\\ V&= \bigl\{u\in H^1_0(\Omega)^2:\operatorname{div}\,u=0\bigr\} \endalign$$ which are endowed with the scalar products and norms of $L^2 (\Omega)^2$ and $H^1_2(\Omega)^2$ respectively. First, the existence and structure of uniform attractors in $H$ is proved for equations with normal external forces in $L^2_{\text{loc}}(\bbfR;V')$. Then, the properties of kernel sections are studied. Finally, the fractal dimension of the kernel sections of the uniform attractor is estimated.

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