Chen, Chiun-Chuan; Strain, Robert M.; Yau, Horng-Tzer; Tsai, Tai-Peng Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. (English) Zbl 1154.35068 Int. Math. Res. Not. 2008, Article ID rnn016, 31 p. (2008). Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in \(\mathbb{R}^3\) with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown by L. Caffarelli, R. Kohn and L. Nirenberg [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)] 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 \(|v(x,t)|\leq C_*(r^2-t)^{-1/2}\) for \(-T_0\leq t<0\) and \(0<C_*<\infty\) allowed to be large, we then prove that \(v\) is regular at time zero. Reviewer: Pavel Burda (Praha) Cited in 1 ReviewCited in 60 Documents MSC: 35Q30 Navier-Stokes equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B65 Smoothness and regularity of solutions to PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000) Keywords:blow-up; axisymmetric Navier-Stokes equations; interior regularity; suitable weak solution Citations:Zbl 0509.35067 PDFBibTeX XMLCite \textit{C.-C. Chen} et al., Int. Math. Res. Not. 2008, Article ID rnn016, 31 p. (2008; Zbl 1154.35068) Full Text: DOI arXiv