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Pullback attractors for a non-autonomous incompressible non-Newtonian fluid. (English) Zbl 1152.35012

In the present study the authors discuss the existence and regularity of pullback attractors for the following non-autonomous incompressible non-Newtonian fluid in 2D bounded domains: \[ {\partial u\over\partial t}+ (u\cdot\nabla) u+\nabla p= \nabla\cdot\tau(e(u))+ g(x, t),\quad x= (x_1,x_2)\in \Omega,\tag{1} \]
\[ \nabla\cdot u= 0,\tag{2} \] where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^2\), \(u= u(x,t)= (u^{(1)}(x,t), u^{(2)}(x, t))\), \(g(x,t)= g(t)= (g^{(1)}(x,t), g^{(2)}(x, t))\), the scalar function \(p\) represents the pressure. Equations (1)–(2) describe the motion of an isothermal incompressible viscous fluid, where \(\tau(e(u))= (\tau_{ij}(e(u)))_{2\times 2}\) which is usually called the extra stress tensor of the fluid.

MSC:

35B41 Attractors
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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