Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid
The first goal of this article is to prove the regularity of trajectory attractor and to reveal the trajectory smoothing effect for the following 2D incompressible non-Newtonian fluid: \[ \frac{\partial{u}}{\partial{t}}+(u\cdot\nabla)u+\nabla{p}=\nabla\cdot\tau(e(u))+g(x),\;x=(x_{1},x_{2})\in{\mathbb{D}},\qquad \nabla\cdot{u}=0, \] where \(\mathbb{D}\) is a smooth bounded domain of \(\mathbb{R}^{2}\), \(g(x)=(g^{(1)},g^{(2)})\) is the time-independent external force function, and \(\tau(e(u))=(\tau_{ij}(e(u)))_{2\times2}\), which is usually called the extra stress tensor of the fluid, is a matrix of order \(2\times2\) and \[ \tau_{ij}(e(u))=2\mu_{0}(\varepsilon+|e|^{2})^{-\alpha/2}e_{ij}-2\mu_{1}\Delta{e}_{ij},\quad i,j=1,2, \] \[ e_{ij}=e_{ij}(u)=\frac{1}{2}\left(\frac{\partial{u}_{i}}{\partial{x}_{j}}+\frac{\partial{u}_{j}}{\partial{x}_{i}}\right),\quad |e|^{2}=\sum_{i,j=1}^{2}|e_{ij}|^{2}, \] and \(\mu_{0}\), \(\mu_{1}\), \(\alpha\), \(\varepsilon\) are parameters which generally depend on the temperature and pressure. The second goal is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from \(\Omega_m\) to \(\Omega=\mathbb{R}\times(-L,L)\), where \(\{\Omega_m\}_{m=1}^\infty\) is an expanding sequence of simply connected, bounded and smooth subdomains of \(\Omega\) such that \(\Omega_m\rightarrow\Omega\) as \(m\rightarrow+\infty\).
scientific article; zbMATH DE number 5625444
scientific article