## 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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