Smoothness of Inertial Manifold for the Burgers Equation

Abstract

This paper establishes a $C^{n,\epsilon }$-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 $C^{n,\epsilon }$-smooth extension of the IM by treating these two nonlinear terms separately.