The Space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 , Volumetric Sparseness, and 3D NSE

Abstract

In the context of the \({L^\infty}\)-theory of the 3D NSE, it is shown that smallness of a solution in Besov space \({B^{-1}_{\infty, \infty}}\) suffices to prevent a possible blow-up. In particular, it is revealed that the aforementioned condition implies a particular local spatial structure of the regions of high velocity magnitude, namely, the structure of local volumetric sparseness on the scale comparable to the radius of spatial analyticity measured in \({L^\infty}\).