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Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. (English) Zbl 1154.35068

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.

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)

Citations:

Zbl 0509.35067
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