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Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations

Chiun-Chuan Chen·Robert M. Strain·Tai-Peng Tsai·Horng-Tzer Yau
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Abstract

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ 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 $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large, we then prove that $v$ is regular at time zero.

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Chiun-Chuan Chen, Robert M. Strain, Tai-Peng Tsai, Horng-Tzer Yau. 2007-01-27. Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. https://doi.org/10.1093/imrn%2Frnn016

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