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Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. (English) Zbl 1170.35336
The paper deals with asymptotic behavior of solutions to the 2D viscous incompressible micropolar fluid flows in the whole space \(\mathbb R^2\). These flows described by the equations
\[ \begin{aligned} &\frac{\partial \bar{v}}{\partial t}-(\nu+\frac{k}{2})\Delta \bar{v}-k\nabla\times w+\nabla p+ (\bar{v}\cdot\nabla)\bar{v}=\bar{f},\\ &j\frac{\partial w}{\partial t}-\gamma\Delta w+2kw-k\nabla\times\bar{v}+ j\bar{v}\cdot \nabla w=g,\quad \text{div}\,\bar{v}=0,\\ &\bar{v}(x,0)=\bar{v}_0(x),\quad w(x,0)=w_0(x), \end{aligned} \] where \(\bar{v}=(v_1,v_2)\) is the velocity vector field, \(p\) is the pressure, \(w\) is the scalar gyration field, \(\bar{f}\) is the given body force, \(g\) is the given scalar body moment, \(\nu>0\) is the Newtonian kinetic viscosity, \(j>0\) is a gyration parameter, \(k\geq 0\) and \(\gamma>0\) are gyration viscosity coefficients. Here
\[ \nabla\times\bar{v}=\frac{\partial v_2}{\partial x_1}-\frac{\partial v_1}{\partial x_2},\qquad \nabla\times w=\left(\frac{\partial w}{\partial x_2},-\frac{\partial w}{\partial x_1}\right). \]
It is proved that the problem has a unique solution. The time decay estimates of this solution in \(L_2\) and \(L_\infty\) norms are obtained.

35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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