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 $$\omega =0$$.

MSC:

 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

Keywords:

micropolar fluid; Besov space
Full Text:

References:

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