Smoothness of Inertial Manifold for the Burgers Equation

TL;DR

This paper proves the existence of a smooth inertial manifold for the 1D Burgers equation, enabling its long-term dynamics to be described by explicit smooth ODEs, through a novel framework handling different nonlinearities.

Contribution

It introduces a new framework for constructing smooth inertial manifolds for equations with mixed nonlinearities, specifically applied to the Burgers equation.

Findings

Established a ${C^{n, ext{ε}}}$-smooth extension of the inertial manifold.

Demonstrated that long-time behavior can be captured by explicit smooth first-order ODEs.

Developed a framework for equations with two nonlinear terms, one preserving and one reducing regularity.

Abstract

This paper establishes a ${C^{n,\varepsilon }}$-smooth extension of the inertial manifold for the one-dimensional Burgers equation, which demonstrates that its long-time behavior can be completely determined by explicit smooth first-order ODEs. We first devise a new framework for an abstract equation with two nonlinear terms, where one preserves regularity and the other reduces regularity, and derive sufficient conditions for constructing the ${C^{n,\varepsilon}}$-smooth extension of the IM by treating these two nonlinear terms separately.