Knowlpedia

Theorem

Integrability of the subtracted swirl heat tail

The Gamma-integral heat profile differs from its terminal power by two extra powers of radial decay.

Statement

For the with A>1/2A>1/2, put Kref(r)=c(r2/2)−AK_{\rm ref}(r)=c(r^2/2)^{-A}. For τ=T−t≥0\tau=T-t\ge0,

∣K−Kref∣≤Cτr−2A−2,∣∂tK∣≤Cr−2A−2.|K-K_{\rm ref}|\le C\tau r^{-2A-2},\qquad |\partial_tK|\le Cr^{-2A-2}.

The constants depend on the fixed parameters A,c,νA,c,\nu.

Weighted tail integral

The bounds ∣F(z)−1∣≤abz|F(z)-1|\le ab z and ∣F′(z)∣≤ab|F'(z)|\le ab give the stated estimates. Therefore for every R>0R>0,

∫R∞r2∣K−Kref∣ dr≤CτR1−2A2A−1.\int_R^\infty r^2|K-K_{\rm ref}|\,dr \le C\tau\frac{R^{1-2A}}{2A-1}.

The same bound without τ\tau controls the weighted time derivative. Thus the angular moment after reference subtraction converges at infinity and may be differentiated on compact time intervals. Behavior at the axis remains a separate condition; this tail estimate alone does not prove convergence of an integral starting at zero.