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

Zhen Lei and Qi S. Zhang

Vol. 254 (2011), No. 2, 335–344

DOI: 10.2140/pjm.2011.254.335

Abstract

Let v be a solution of the axially symmetric Navier–Stokes equation. We determine the structure of a certain (possible) maximal singularity of v in the following sense. Let (x₀,t₀) be a point where the flow speed Q₀ = |v(x₀,t₀)| 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_local^2,1,α norm to a nonzero constant vector in a fixed parabolic cube, provided that r₀Q₀ is sufficiently large. Here r₀ is the distance from x₀ to the z axis. Similar results are also shown to be valid if |r₀v(x₀,t₀)| is comparable with the maximum of |rv(x,t)| 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.

Keywords

Axisymmetric Navier–Stokes equations, structure of singularities

Mathematical Subject Classification 2010

Primary: 35Q30, 76D05

Milestones

Received: 16 February 2011

Revised: 16 September 2011

Accepted: 31 October 2011

Published: 27 February 2012