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Under the uniform-in-time constraint $\\|\\nabla u(\\cdot,t)\\|_p\\leq 1$ we show that any function can be mixed to scale $\\epsilon$ in time $O(|\\log\\epsilon|^{1+\\nu_p})$, with $\\nu_p=0$ for $p<\\tfrac{3+\\sqrt 5}2$ and $\\nu_p\\leq \\tfrac 13$ for $p\\geq \\tfrac{3+\\sqrt 5}2$. Known lower bounds show that this rate is optimal for $p\\in(1,\\tfrac{3+\\sqrt 5}2)$. 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