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

TL;DR

This paper simplifies and slightly improves the proof of near-optimal bounds for the Kuramoto-Sivashinsky equation by introducing a modified Kármán-Howarth-Monin identity for inhomogeneous inviscid Burgers solutions.

Contribution

It provides a simplified proof of existing bounds on the Kuramoto-Sivashinsky equation and introduces a new identity for inhomogeneous inviscid Burgers solutions, enhancing understanding of their regularity.

Findings

Simplified proof of near-optimal bounds for Kuramoto-Sivashinsky equation

Introduction of a modified Kármán-Howarth-Monin identity

Improved regularity estimates for inhomogeneous Burgers equations

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 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].