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Regularity Criteria of BKM type in Distributional Spaces for the 3-D Navier-Stokes Equations on Bounded Domains
In the classic work of Beale-Kato-Majda ({[}2{]}) for the Euler equations in
, regularity of a solution throughout a given interval
is obtained provided that the curl satisfies \omega\in
L^{1}((0,T);L^{\infty}(\mathbb{R^{\textrm{\ensuremath{3}}}}) for all
, and the arguments apply equally well to the Navier-Stokes equations
(NSE) in . The spatial -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 in
, we obtain a regularity result of BKM type that
allows to be a distribution. Specifically, we show that if is a
Leray solution of the 3-D NSE on the interval and if where for some
, then is a regular solution on ; in
particular for we have a regular solution when , 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
to spatially be a distribution