= Uniform phase-mixing bound from velocity derivatives
{title2=$\|\rho_t-\rho_\infty\|_\infty\leq 2\zeta(m)\|\partial_v^mf_0\|_1/(2\pi|t|)^m$}
For $m\geq2$ integrable weak velocity derivatives on a unit-period spatial circle, the <Fourier transform of a derivative> gives $|\widehat f_0(k,kt)|\leq\|\partial_v^mf_0\|_1/(2\pi|kt|)^m$ for $k\ne0$. Summing these modes uses $\sum_{k\ne0}|k|^{-m}=2\zeta(m)$. The resulting absolute Fourier summability gives the displayed uniform bound for the continuous representative of the density. For $m=1$ the same summation diverges, so this argument needs additional input.
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