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arXiv · 1503.06059

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

Michael Goldman·Marc Josien·Felix Otto
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Abstract

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 Kármán-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].

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Michael Goldman, Marc Josien, Felix Otto. 2015-03-20. New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations. https://arxiv.org/abs/1503.06059

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