The global attractor for the weakly damped KdV equation on R RR has a finite fractal dimension

摘要

> In this work, we are interested in the finite fractal dimensionality of the global attractor for a weakly damped Kordeweg‐de Vries (KdV) equation. For the KdV equation with periodic boundary condition, this is done by Ghidaglia in 1988. But since then, it seems little is known on this topic for the KdV equation on unbounded domains. The main difficulties are (a) the dissipative effect of the KdV equation is weak, and (b) the Sobolev embeddings on R are not compact. To overcome these difficulties, we present some new idea to prove the Chueshov‐Lasicka quasi‐stable estimates for the KdV equation on R . In this way, we show that for the KdV equation on the real line, the global attractor has a finite fractal dimension in the sharp space H 3 ( R ) whenever the force belongs to L 2 ( R ) .