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Letv be a solution of the axially symmetric Navier‐Stokes equation. We determine the structure of a certain (possible) maximal singularity ofv in the following sense. Let.x0; t0/ be a point where the flow speed Q0 D jv.x0; t0/j is comparable with the maximum flow speed at and before time t0. We show, after a space-time scaling with the factor Q0 and the center.x0; t0/, that the solution is arbitrarily close in C 2;1; local norm to a nonzero constant vector in a fixed parabolic cube, provided that r0 Q0 is sufficiently large. Here r0 is the distance from x0 to the z axis. Similar results are also shown to be valid if jr0v.x0; t0/j is comparable with the maximum of jrv.x; t/j at and before time t0. This mirrors a numerical result of Hou for the Euler equation: there exists a certain “calm spot” or depletion of vortex stretching in a region of high flow speed.

Structure of solutions of 3D axisymmetric Navier–Stokes equations near maximal points