Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. (English) Zbl 1170.35336

Dong, Bo-Qing; Chen, Zhi-Min

Discrete Contin. Dyn. Syst. 23, No. 3, 765-784 (2009).

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.

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

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)

Keywords:

spectral decomposition; time decay estimates