  We give a substantially simplified proof of near-optimal estimate on the
Kuramoto-Sivashinsky equation from [F. Otto, "Optimal bounds on the
Kuramoto-Sivashinsky equation", JFA 2009], at the same time slightly improving
the result. The result in the above cited paper relied on two ingredients: a
regularity estimate for capillary Burgers and an a novel priori estimate for
the inhomogeneous inviscid Burgers equation, which works out that in many ways
the conservative transport nonlinearity acts as a coercive term. It is the
proof of the second ingredient that we substantially simplify by proving a
modified K\'arm\'an-Howarth-Monin identity for solutions of the inhomogeneous
inviscid Burgers equation. This gives a new interpretation of the results
obtained in [F. Golse, B. Perthame "Optimal regularizing effect for scalar
conservation laws", Rev. Mat. Iber., 2013].
