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Zbl 1234.35193
Chen, Qionglei; Miao, Changxing
Global well-posedness for the micropolar fluid system in critical Besov spaces. (English)
[J] J. Differ. Equations 252, No. 3, 2698-2724 (2012). ISSN 0022-0396

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 $u(x,t)=(u_1 ,u_2 ,u_3)$, the velocity of rotation of particles $\omega (x,t)=(\omega_1 ,\omega_2 ,\omega_3 )$, and the pressure $\pi (x,t)$ in the following form: $$ \left\{\aligned & \partial_t u -\Delta u +u\cdot\nabla u +\nabla \pi -\nabla\times \omega =0,\\ & \partial_t \omega -\Delta \omega +u\cdot\nabla \omega +2\omega -\nabla\text{div}\,\omega -\nabla\times u=0,\\ & \text{div}\,u=0,\\ & u(x,0)=u_0 (x),\quad \omega (x,0)=\omega_0 (x). \endaligned\right. $$ They assume that the initial values $v_0, w_0$ belong to the Besov space $\Dot{B}^{\frac{p}{3}-1}_{p.\infty}$ for some $1\leq p<6$ with small norms (this type of Besov space is called critical). They prove the existence of the solution in $C(0,\infty ;\Dot{B}^{\frac{p}{3}-1}_{p.\infty})$. They also prove the uniqueness under an additional assumption. For this purpose they consider an associated linear system $$ \left\{\aligned & \partial_t u -\Delta u -\nabla\times \omega =0,\\ & \partial_t \omega -\Delta \omega +2\omega -\nabla\times u=0,\\ \endaligned\right. $$ and study the action of its Green matrix. \par One can apply thier result directly to an incompressible Navier-Stokes equation, by setting $\omega =0$.
[Keisuke Uchikoshi (Yokosuka)]

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
76A05 Non-Newtonian fluids
76B03 Existence, uniqueness, and regularity theory
35Q30 Stokes and Navier-Stokes equations

Keywords: micropolar fluid; Besov space

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