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

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.

34 pages. Submitted on December 15, 2017