Global Attractor For Weakly Damped, Forced Mkdv Equation Below Energy Space

TL;DR

This paper establishes the existence of a global attractor for the weakly damped and forced mKdV equation in a Sobolev space below the energy space, using the I-method and detailed frequency analysis.

Contribution

It proves the existence of a global attractor below the energy space for the damped and forced mKdV, overcoming challenges posed by damping and forcing terms.

Findings

Global attractor exists in rac{11}{12} for the mKdV.

Application of the I-method to damped and forced mKdV.

Detailed analysis of resonant frequency estimates.

Abstract

We prove the existence of the global attractor in $ \dot H^s$, $s > 11/12$ for the weakly damped and forced mKdV on the one dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space is established. We directly apply the I-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.