We characterize a possible blowup for the 3D Navier-Stokes on the basis of dynamical equations
for vector potentials
A
.
This is motivated by a known interpolation
k
A
k
BMO
≤ k
u
k
L
3
,
together
with recent mathematical results. First, by working out an inversion formula for singular integrals
that appear in the governing equations, we derive a criterion using the nonlinear term of
A
as
R
t
∗
0
k
∂
A
∂t
−
ν
4
A
k
L
∞
dt
=
∞
for a blowup at
t
∗
.
Second, for a particular form of a scale-invariant
singularity of the nonlinear term we show that the vector potential becomes unbounded in its
L
∞
and BMO norms. Using the stream function, we also consider the 2D Navier-Stokes equations to
seek an alternative proof of their known global regularity. It is not yet proven that the BMO norm
of vector potentials in 3D (or, the stream function in 2D) serve as a blow up criterion in more
general cases.
CORE (COnnecting REpositories)
CORE (COnnecting REpositories)