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Navier-Stokes equations in the Besov space near \(L^\infty\) and BMO. (English) Zbl 1067.35064

The following Cauchy problem for the Navier-Stokes equation is considered \[ \begin{aligned} &\frac{\partial v}{\partial t}-\Delta v+ v\cdot\nabla v+\nabla p=f, \quad \nabla\cdot v=0 \quad \text{in \;} x\in \mathbb R^n,\;t\in(0,T)\\ &v(x,0)=a(x). \end{aligned} \tag{1} \] The initial data \(a\) belongs to the Besov space \(B^0_{\infty,\infty}(\mathbb R^n)\). This space contains functions which do not decay at infinity. It is proved that for every \(a\in B^0_{\infty,\infty}\) with \(\text{div}\;a=0\) there exist \(T=T(\| a\|_{B^0_{\infty,\infty}})\) and a solution \(v\) to the problem (1) such that \(v\in C_w([0,T];B^0_{\infty,\infty})\) where \(C_w\) denotes weakly-* continuous functions. This solution can be extended on the interval \((0,T^\prime)\) for some \(T^\prime>T\) if \[ \int\limits^T_0\|\text{rot}v(t)\|_{\dot B^0_{\infty,\infty}}\,dt<\infty. \] This extension criterion is new for a solution which does not decay at infinity.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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