Global Dissipativity and Inertial Manifolds for Diffusive Burgers Equations with Low-Wavenumber Instability

TL;DR

This paper investigates the global behavior of certain diffusive Burgers equations, establishing dissipativity and inertial manifolds by using the Cole-Hopf transform to overcome spectral-gap limitations.

Contribution

It demonstrates the existence of inertial manifolds for a broad class of Burgers-type equations by circumventing spectral-gap restrictions with the Cole-Hopf transform.

Findings

Proves global dissipativity in 2D for periodic boundary conditions.

Establishes existence of inertial manifolds for the class of equations.

Valid in both 1D and 2D settings.

Abstract

Global well-posedness, existence of globally absorbing sets and existence of inertial manifolds is investigated for a class of diffusive Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, the Burgers-Sivashinsky equation and the Quasi-Stedy equation of cellular flames. The global dissipativity is proven in 2D for periodic boundary conditions. For the proof of the existence of inertial manifolds, the spectral-gap condition, which Burgers-type equations do not satisfy in its original form is circumvented by the Cole-Hopf transform. The procedure is valid in both one and two space dimensions.