530 research outputs found
Regularity Criterion to the axially symmetric Navier-Stokes Equations
Smooth solutions to the axially symmetric Navier-Stokes equations obey the
following maximum
principle:
We first prove the global regularity of solutions if
or is small compared with certain dimensionless quantity of the initial
data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a
corollary, we also prove the global regularity under the assumption that
$|ru_\theta(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\
0<r\leq\delta_0\in(0,1/2).$Comment: 13 pages, 0 figure
Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations
Consider axisymmetric strong solutions of the incompressible Navier-Stokes
equations in with non-trivial swirl. Such solutions are not known to be
globally defined, but it is shown in \cite{MR673830} that they could only blow
up on the axis of symmetry.
Let denote the axis of symmetry and measure the distance to the
z-axis. Suppose the solution satisfies the pointwise scale invariant bound for and
allowed to be large, we then prove that is regular at time zero.Comment: 25 page
Some Remarks on Regularity Criteria of Axially Symmetric Navier-Stokes Equations
Two main results will be presented in our paper. First, we will prove the
regularity of solutions to axially symmetric Navier-Stokes equations under a
supercritical assumption on the horizontally radial component and
vertical component , accompanied by a subcritical assumption on the
horizontally angular component of the velocity. Second, the precise
Green function for the operator under the axially
symmetric situation, where is the distance to the symmetric axis, and some
weighted estimates of it will be given. This will serve as a tool for the
study of axially symmetric Navier-Stokes equations. As an application, we will
prove the regularity of solutions to axially symmetric Navier-Stokes equations
under a critical (or a subcritical) assumption on the angular component
of the vorticity.Comment: Final version, to appear in Comm. Pure Appl. Ana
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