Smooth solutions to the axisymmetric Navier–Stokes equations obey the following
maximum principle:
We prove that all solutions with initial data in
are smooth
globally in time if
satisfies a kind of form boundedness condition (FBC) which is invariant
under the natural scaling of the Navier–Stokes equations. In particular, if
satisfies
then our FBC is satisfied. Here
and
are
independent of neither the profile nor the norm of the initial data. So the gap from
regularity is logarithmic in nature. We also prove the global regularity of solutions if
or
is
small but the smallness depends on a certain dimensionless quantity of the initial
data.