We prove that a certain (centered unimodal) rearrangement of coefficients in the moving average initial input process maximizes the variance (energy density) of the limit distribution of the spatiotemporal random field solution of a nonlinear partial differential equation called Burgers' equation. Our proof is in the spirit of domination principles developed in the book by Kwapien and Woyczynski.
"An Extremal Rearrangement Property of Statistical Solutions of Burgers' Equation." Ann. Appl. Probab. 4 (3) 838 - 858, August, 1994. https://doi.org/10.1214/aoap/1177004974