Abstract: In
this paper we establish a new regularity criterion of Leray weak solutions to
the Navier-Stokes equation in Here we call u
to be a Leray weak solution if u is a
weak solution of finite energy, i.e.,
It
is known that if a Leray weak solution u
belongs to
for some(0.1)
then
u is regular (see [12]). We succeed in proving the regularity of
Leray weak solution u under the
condition
(0.2)
Here
E is a shift-invariant Banach space of
local measures (see Definition in the text section) and E is continuously embedded in the homogeneous Besov space for Since this space E
is wider than the above regularity criterion
(0.2) is an improvement of Serrin’s result (0.1). Moreover, this space E
is wider than the Lorentz space and hence the regularity criterion
(0.2) covers the recent results given by Sohr [13].
Keywords and phrases: Navier-Stokes equations, Banach space of local measure, regularity.
