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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 \(u(x,t)=(u_1 ,u_2 ,u_3)\), 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: \[ \left\{\begin{aligned} & \partial_t u -\Delta u +u\cdot\nabla u +\nabla \pi -\nabla\times \omega =0,\\ & \partial_t \omega -\Delta \omega +u\cdot\nabla \omega +2\omega -\nabla\text{div}\,\omega -\nabla\times u=0,\\ & \text{div}\,u=0,\\ & u(x,0)=u_0 (x),\quad \omega (x,0)=\omega_0 (x). \end{aligned}\right. \] 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 \[ \left\{\begin{aligned} & \partial_t u -\Delta u -\nabla\times \omega =0,\\ & \partial_t \omega -\Delta \omega +2\omega -\nabla\times u=0,\\ \end{aligned}\right. \] and study the action of its Green matrix.
One can apply thier result directly to an incompressible Navier-Stokes equation, by setting \(\omega =0\).

35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q30 Navier-Stokes equations
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