Geodesic
Regularity of solutions to the Navier–Stokes equations in $\dot{B}_{\infty,\infty}^{-1}$
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 47, Tome 477 (2018), pp. 119-128
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We prove that if $u$ is a suitable weak solution to the three dimensional Navier–Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is shown that any axially symmetric suitable weak solution $u$, belonging to $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, is smooth. 
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G. Seregin; D. Zhou. Regularity of solutions to the Navier–Stokes equations in $\dot{B}_{\infty,\infty}^{-1}$. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 47, Tome 477 (2018), pp. 119-128. http://geodesic.mathdoc.fr/item/ZNSL_2018_477_a6/

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