On partial type I solutions to the Axially symmetric Navier-Stokes equations

Qi S Zhang

paper

On partial type I solutions to the Axially symmetric Navier-Stokes equations

arXiv:2604.07785

Abstract

Let $v = v_{r} e_{r} + v_{þ} e_{þ} + v_{3} e_{3}$ be a Leray-Hopf solution to the axially symmetric Navier-Stokes equations (ASNS). We call it a partial type I solution if $v_{r} (x, t) \geq - C / T - t$ for some constant $C > 0$ and $(x, t) \in R^{3} \times [0, T)$ . In this paper, it is proven that such solution does not blow up at time $T$ under the extra mild assumption that $∣ v_{θ} (x, 0) ∣∣ x^{'} ∣$ is bounded. This extends a well known result by two groups of people who proved the no blowup conclusion under the full type I condition: $∣ v (x, t) ∣ \leq C / T - t$ . The result also confirms the physical intuition that potential blow ups for ASNS are caused by super-critical inward radial velocity.