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In the classic work of Beale-Kato-Majda ({[}2{]}) for the Euler equations in$\mathbb{R^{\mathrm{3}}}$, regularity of a solution throughout a given interval$[0,T_{*}]$ is obtained provided that the curl $\omega$ satisfies $\omega\inL^{1}((0,T);L^{\infty}(\mathbb{R^{\textrm{\ensuremath{3}}}})$ for all$T<T_{*}$, and the arguments apply equally well to the Navier-Stokes equations(NSE) in $\mathbb{R^{\mathrm{3}}}$. The spatial $L^{\infty}$-criterion imposedon the curl was generalized to other function spaces by various authors({[}9{]}, {[}10{]}, {[}11{]}). In {[}8{]} regularity results of this type areobtained on localized balls.  In this paper for the NSE case and on general bounded domains $\Omega$ in$\mathbb{R^{\mathrm{3}}}$, we obtain a regularity result of BKM type thatallows $\omega$ to be a distribution. Specifically, we show that if $u$ is aLeray solution of the 3-D NSE on the interval $(0,T)$ and if $\omega\inL^{s}((0,T);H^{-1,p}(\Omega))$ where $\frac{2}{s}+\frac{3}{p}=1$ for some$p\in(3,\infty]$, then $u$ is a regular solution on $\left(0,T]\right)$; inparticular for $p=\infty$ we have a regular solution when $\omega\inL^{2}((0,T);H^{-1,\infty}(\Omega))$, which directly strengthens the results in{[}2{]} by one order of (negative) derivative in terms of the spatial criteriafor regularity. Our results thus impose more stringent conditions on time thanthe BKM results and their generalizations described above, but as far as we areaware the results here represent the first of BKM type for the NSE that allow$\omega$ to spatially be a distribution.

Regularity Criteria of BKM type in Distributional Spaces for the 3-D Navier-Stokes Equations on Bounded Domains