Euler Singularities II: Interior Quadrupole Blow-Up for Smooth Axisymmetric Euler with Swirl in \texorpdfstring{$\mathbb R^3$}

Abstract

We present a self-contained interior quadrupole mechanism for finite-time singularity formation in the axisymmetric three-dimensional incompressible Euler equations with swirl in the whole space. The construction is localized away from the axis. In local variables \[ x=r-r_*(t),\qquad y=z, \] centered at a tracked radial point, the active vorticity and swirl profiles are \[ G(x,y,t)\approx a(t)xy, \qquad \Gamma(x,y,t)\approx \Gamma_*(t)+\frac12 b(t)xy^2, \qquad \Gamma_*(t)>0. \] The first profile produces a positive interior Biot--Savart hyperbolic strain; the second profile makes the Euler source term in the equation for \(G=\omega^\theta/r\) regenerate the same quadrupole shape. The active quantity is the full four-quadrant quadrupole score, while a narrow diagonal sector is used only as a coercive subscore. We give the notation and the 5D recovery formula connecting the 3D axisymmetric variables to the lifted elliptic problem, construct explicit smooth decaying divergence-free data, verify their initial entry into the quadrupole bootstrap, prove the master propagation estimates, and derive the comparison system \[ Q'(t)\ge cC(t),\qquad C'(t)\ge cQ(t)C(t),\qquad C(t)\ge \kappa Q(t)^2. \] Consequently the tracked quadrupole score blows up in finite comparison time, and the strain lower bound gives blow-up of \(\norm{\nabla u(t)}_{L^\infty}\). All geometric and analytic constraints used by the construction are stated as named estimates: the interior quadrupole kernel sign expansion, source compatibility, swirl-jet amplification, full-score/coercive-subscore comparison, angular-profile defect persistence, radial-center tracking, neutral-jet hierarchy, and two-sided Dini bounds. This is Part II of a two-paper Euler series; Part I treats boundary blow-up in a periodic cylinder.