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The generalized incompressible Navier-Stokes equations in Besov spaces. (English) Zbl 1075.35043
The following Cauchy problem is considered: \begin{aligned} &\frac{\partial v }{\partial t }+(v\cdot\nabla)v+\nu(-\Delta)^{\alpha} v+\nabla p=0,\quad \text{div\,}v=0, \quad x\in \mathbb R^n,\;t>0\\ &v(x,0)=v_0(x),\quad x\in \mathbb R^n. \end{aligned} Here $$\nu$$ and $$\alpha$$ are given positive constants. The case $$0<\alpha<\frac 12+\frac n2\;$$ is studied in the paper. It is proved that if $$\| v_0\|_B\leq C_0\,\nu$$ for some suitable constant $$C_0$$ then the Cauchy problem has a unique global solution in Besov spaces. $$\| \cdot\|_B$$ denotes the norm in a certain Besov space.

MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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