Dimension reduction of axially symmetric Euler equations near maximal points off the axis

Abstract:

Let $v$ be a solution of the axially symmetric Euler equations (ASE) in a finite cylinder in $\mathbb {R}^3$. We show that suitable blow-up limits of possible velocity singularity and most self-similar vorticity singularity near maximal points off the vertical axis are two-dimensional ancient solutions of the Euler equation in either $\mathbb {R}^2 \times (-\infty , 0]$ or $\mathbb {R}^2_+ \times (-\infty , 0]$. This reduces the search of off-axis self-similar or other velocity blow-up solutions to a problem involving purely 2-dimensional Euler equations. Also, some asymptotic self-similar velocity blow-up and expected asymptotic self-similar vorticity blow up scenario at the boundary appear to be ruled out. On the other hand, this method may provide a path to velocity blow up if one can construct certain stable ancient solutions to the 2-d Euler equation in the half plane.