Asymptotic Smoothing and the Global Attractor of a Weakly Damped KdV Equation on the Real Line

The existence of the global attractor of a weakly damped, forced Korteweg–de Vries equation in the phase space L2( ) is proved. An optimal asymptotic smoothing effect of the equation is also shown, namely, that for forces in L2( ), the global attractor in the phase space L2( ) is actually a compact set in H3( ). The energy equation method is used in conjunction with a suitable splitting of the solutions; the dispersive regularization properties of the equation in the context of Bourgain spaces are extensively exploited