Global attractor for weakly damped, forced mKdV equation in low regularity spaces

Abstract

The modified Korteweg-de Vries equation (for short, mKdV) models the propagation of nonlinear water waves in the shallow water approximation. We consider the weakly damped and forced mKdV under the periodic boundary condition. We often study mKdV equation by using the Miura transformation, which converts solutions of mKdV to solutions of the Korteweg-de Vries equation (KdV). However, if mKdV has damping and external forcing terms, the Miura transformation does not work well. To see the asymptotic behaviour of the solutions of mKdV equation, the study of global attractor is important. We prove the existence of the global attractor for $$ {s > 9/10} $$ in $$\dot{H}^{s}$$ for mKdV equation by applying the modified energies and almost conserved quantities.