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Zbl 1170.35336
Dong, Bo-Qing; Chen, Zhi-Min
Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. (English)
[J] Discrete Contin. Dyn. Syst. 23, No. 3, 765-784 (2009). ISSN 1078-0947; ISSN 1553-5231/e

The paper deals with asymptotic behavior of solutions to the 2D viscous incompressible micropolar fluid flows in the whole space $\Bbb R^2$. These flows described by the equations $$\aligned &\frac{\partial \bar{v}}{\partial t}-(\nu+\frac{k}{2})\Delta \bar{v}-k\nabla\times w+\nabla p+ (\bar{v}\cdot\nabla)\bar{v}=\bar{f},\\ &j\frac{\partial w}{\partial t}-\gamma\Delta w+2kw-k\nabla\times\bar{v}+ j\bar{v}\cdot \nabla w=g,\quad \text{div}\,\bar{v}=0,\\ &\bar{v}(x,0)=\bar{v}_0(x),\quad w(x,0)=w_0(x), \endaligned$$ 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.
[Il'ya Sh. Mogilevskij (Tver')]

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
35Q35 Other equations arising in fluid mechanics
76A05 Non-Newtonian fluids
35A05 General existence and uniqueness theorems (PDE)

Keywords: spectral decomposition; time decay estimates

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Zentralblatt MATH Berlin [Germany]