Calculus of Variations and Geometric Measure Theory

M. Goldman - M. Josien - F. Otto

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

created by goldman on 20 Mar 2015
modified on 26 Dec 2015


Accepted Paper

Inserted: 20 mar 2015
Last Updated: 26 dec 2015

Journal: CPDE
Year: 2015


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 Karman-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.