Structure of Singularities of 3D Axi-symmetric Navier-Stokes Equations

Abstract

Let $v$ be a solution of the axially symmetric Navier-Stokes equation. We determine the structure of certain (possible) maximal singularity of $v$ in the following sense. Let $(x_0,t_0)$ be a point where the flow speed $Q_0=|v(x_0,t_0)|$ is comparable with the maximum flow speed at and before time $t_0$. We show after a space-time scaling with the factor $Q_0$ and the center $(x_0,t_0)$, the solution is arbitrarily close in $C_{\mathrm{local}}^{2,1,\alpha }$ norm to a nonzero constant vector in a fixed parabolic cube, provided that $r_0{Q}_0$ is sufficiently large. Here $r_0$ is the distance from $x_0$ to the $z$ axis. Similar results are also shown to be valid if $|r_{0}v(x_0,t_0)|$ is comparable with the maximum of $|rv(x,t)|$ at and before time $t_0$.

References in corpus (1)

• Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl