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 has been 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.