Criticality of the Axially Symmetric Navier-Stokes Equations

Abstract

Smooth solutions to the axi-symmetric Navier-Stokes equations obey the following maximum principle: $$\underset{t\ge 0}{\sup }\Vert r{v}^{\theta }(t,\cdot ){\Vert }_{L^{\infty }}\le \Vert r{v}^{\theta }(0,\cdot ){\Vert }_{L^{\infty }}.$$ We prove that all solutions with initial data in $H^{\frac{1}{2}}$ is smooth globally in time if $r{v}^{\theta }$ satisfies a kind of Form Boundedness Condition (FBC) which is invariant under the natural scaling of the Navier-Stokes equations. In particular, if $r{v}^{\theta }$ satisfies \begin{equation}\nonumber \sup_{t \geq 0}|rv^θ(t, r, z)| \leq C_\ast|\ln r|^{- 2},\ \ r \leq δ_0 \in (0, \frac{1}{2}),\ C_\ast < \infty, \end{equation} then our FBC is satisfied. Here ${\delta }_0$ and $C_{\ast }$ 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 $\Vert r{v}^{\theta }(0,\cdot ){\Vert }_{L^{\infty }}$ or ${\sup }_{t\ge 0}\Vert r{v}^{\theta }(t,\cdot ){\Vert }_{L^{\infty }(r\le r_0)}$ is small but the smallness depends on certain dimensionless quantity of the initial data.

arXiv commentThis is a merged article of "arXiv:1505.02628, version 1, Zhen Lei , Almost Criticality of the Axi-Symmetric Navier-Stokes Equations" and "arXiv:1505.00528, Qi S. Zhang, A critical regularity condition on the angular velocity of axially symmetric Navier-Stokes equations" We decided not to publish 1 and 2, but publish the merged one as a joint paper

References in corpus (4)

• Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl
• Regularity of 3D axisymmetric Navier-Stokes equations
• A priori bound on the velocity in axially symmetric Navier-Stokes equations
• A critical regularity condition on the angular velocity of axially symmetric Navier-Stokes equations

Cited by in corpus (4)

• Regularity Criterion to the axially symmetric Navier-Stokes Equations
• Global regularity for a family of 3D models of the axisymmetric Navier-Stokes equations
• On the global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang