Regularity Criteria of BKM type in Distributional Spaces for the 3-D Navier-Stokes Equations on Bounded Domains (1405.3709v1)

Abstract: 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 $\omega\in L^{1}((0,T);L^{\infty}(\mathbb{R^{\textrm{\ensuremath{3}}}})$ 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 imposed on the curl was generalized to other function spaces by various authors ({[}9{]}, {[}10{]}, {[}11{]}). In {[}8{]} regularity results of this type are obtained 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 that allows $\omega$ to be a distribution. Specifically, we show that if $u$ is a Leray solution of the 3-D NSE on the interval $(0,T)$ and if $\omega\in L^{s}((0,T);H^{-1,p}(\Omega))$ where $\frac{2}{s}+\frac{3}{p}=1$ for some $p\in(3,\infty]$, then $u$ is a regular solution on $\left(0,T]\right)$; in particular for $p=\infty$ we have a regular solution when $\omega\in L^{2}((0,T);H^{-1,\infty}(\Omega))$, which directly strengthens the results in {[}2{]} by one order of (negative) derivative in terms of the spatial criteria for regularity. Our results thus impose more stringent conditions on time than the BKM results and their generalizations described above, but as far as we are aware the results here represent the first of BKM type for the NSE that allow $\omega$ to spatially be a distribution.