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New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations
Michael Goldman, Marc Josien and Felix Otto
We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from , at the same time slightly improving the result. The result in  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 inmany 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 Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We showthat this provides a new interpretation of the results obtained in .