Vorticity-Response Blowup in Axisymmetric Models

Abstract

We study how the orientation and elliptic depth of swirl-generated vorticity response affect singularity formation in axisymmetric model equations. For a sign-reversed Hou--Li model on the whole line, the dynamics factorize into two real Riccati--diffusion channels. For every viscosity $\nu>0$, every nontrivial smooth datum in an infinite-dimensional cone of even, nonnegative, monotone channel data develops finite-time amplitude blowup; in the inviscid case, characteristics give the exact blowup time. For the same smooth swirling inviscid datum, the standard response sign is global whereas the reversed sign blows up. We then introduce a three-parameter 5D response family separating transport geometry, response orientation, and elliptic depth. Exact divergence, energy, and scaling identities show that physical three-dimensional incompressibility selects one geometry, weighted 5D incompressibility selects another, and Navier--Stokes scaling makes two elliptic inversions critical. The frozen physical response is critical and dissipative, while a natural one-inversion 5D-solenoidal model is supercritical and energy-pumping. These results identify response orientation and causal depth as load-bearing structure in axisymmetric model dynamics.