New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations

Michael Goldman, Marc Josien, and Felix Otto

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Submission date: 25. Mar. 2015
Pages: 34
published in: Communications in partial differential equations, 40 (2015) 12, p. 2237-2265 
DOI number (of the published article): 10.1080/03605302.2015.1076003
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Abstract:
We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from [14], at the same time slightly improving the result. The result in [14] 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 [7].