Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line

Abstract

The existence of the global attractor of a weakly damped, forced Korteweg-de Vries equation in the phase space L2(ℝ) is proved. An optimal asymptotic smoothing effect of the equation is also shown, namely, that for forces in L2(ℝ), the global attractor in the phase space L2(ℝ) is actually a compact set in H3(ℝ). 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. © 2002 Elsevier Science (USA).