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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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