= Square-root coordinates for primal congestion control
{title2=$x_r=\kappa_rz_r^2/4,\quad\dot z=\nabla V(z)$}
For positive rates in <congestion control>, put $x_r=\kappa_rz_r^2/4$ and $V(z)=U(x(z))$, where $U$ is the <primal congestion potential>. The equation $\dot x_r=\kappa_rx_r\partial_rU$ becomes $\dot z_r=\partial_rV$. Positive logarithmic utility weights make $\sum_rw_r\log(\kappa_rz_r^2/4)$ strictly concave on the positive orthant. Each integrated nonnegative nondecreasing resource price is a convex nondecreasing function of its load; composing it with the convex quadratic load in $z$ preserves convexity. Hence $V$ is a <strictly concave function>, even for prices that are merely continuous. For two solutions, $\frac12\frac d{dt}\|z-\widetilde z\|^2=(z-\widetilde z)\cdot[\nabla V(z)-\nabla V(\widetilde z)]\leq0$. This proves uniqueness of the flow without assuming local Lipschitz continuity of the price functions.
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