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

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}}$ or, for some \e > 0, $|v (x,t)| \le C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero.Comment: More explanations and a new appendi

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

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.Comment: 25 page