Hardy Criticality in Axisymmetric Swirl

For smooth axisymmetric Navier--Stokes flow with swirl, set \(F=u^θ/r\), \(Γ=ru^θ\), \(G=ω^θ/r\), and \(U=u^r/r\). Weighted swirl energies are classical; we determine the sharp two-parameter coercivity spectrum of the exact power-weight family \(\int |F|^p r^α\,dr\,dz\), \(p>1\), \(α>1\). The full d