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Global well-posedness for the micropolar fluid system in critical Besov spaces. (English) Zbl 1234.35193

The authors consider an incompressible micropolar fluid system. This is a kind of non Newtonian fluid, and is a model of the suspensions, animal blood, liquid crystals which cannot be characterized appropriately by the Navier-Stokes system. It is described by the fluid velocity $u\left(x,t\right)=\left({u}_{1},{u}_{2},{u}_{3}\right)$, the velocity of rotation of particles $\omega \left(x,t\right)=\left({\omega }_{1},{\omega }_{2},{\omega }_{3}\right)$, and the pressure $\pi \left(x,t\right)$ in the following form:

$\left\{\begin{array}{cc}& {\partial }_{t}u-{\Delta }u+u·\nabla u+\nabla \pi -\nabla ×\omega =0,\hfill \\ & {\partial }_{t}\omega -{\Delta }\omega +u·\nabla \omega +2\omega -\nabla \text{div}\phantom{\rule{0.166667em}{0ex}}\omega -\nabla ×u=0,\hfill \\ & \text{div}\phantom{\rule{0.166667em}{0ex}}u=0,\hfill \\ & u\left(x,0\right)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}\omega \left(x,0\right)={\omega }_{0}\left(x\right)·\hfill \end{array}\right\$

They assume that the initial values ${v}_{0},{w}_{0}$ belong to the Besov space ${\stackrel{˙}{B}}_{p·\infty }^{\frac{p}{3}-1}$ for some $1\le p<6$ with small norms (this type of Besov space is called critical). They prove the existence of the solution in $C\left(0,\infty ;{\stackrel{˙}{B}}_{p·\infty }^{\frac{p}{3}-1}\right)$. They also prove the uniqueness under an additional assumption. For this purpose they consider an associated linear system

$\left\{\begin{array}{cc}& {\partial }_{t}u-{\Delta }u-\nabla ×\omega =0,\hfill \\ & {\partial }_{t}\omega -{\Delta }\omega +2\omega -\nabla ×u=0,\hfill \end{array}\right\$

and study the action of its Green matrix.

One can apply thier result directly to an incompressible Navier-Stokes equation, by setting $\omega =0$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76A05 Non-Newtonian fluids 76B03 Existence, uniqueness, and regularity theory (fluid mechanics) 35Q30 Stokes and Navier-Stokes equations
##### Keywords:
micropolar fluid; Besov space