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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 ω(x,t)=(ω 1 ,ω 2 ,ω 3 ), and the pressure π(x,t) in the following form:

t u-Δu+u·u+π-×ω=0, t ω-Δω+u·ω+2ω-divω-×u=0,divu=0,u(x,0)=u 0 (x),ω(x,0)=ω 0 (x)·

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

t u-Δu-×ω=0, t ω-Δω+2ω-×u=0,

and study the action of its Green matrix.

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

MSC:
35Q35PDEs in connection with fluid mechanics
76A05Non-Newtonian fluids
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
35Q30Stokes and Navier-Stokes equations
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