Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index

Pages: 301 - 318, Volume 3, Issue 2, June 2004

doi:10.3934/cpaa.2004.3.301       Abstract        Full Text (248.2K)       Related Articles

Kotaro Tsugawa - Institut des Hautes Etudes Scientifiques, Le Bois-Marie, 35, route de Chartres, 91440 Bures-sur-Yvette, France (email)

Abstract: In this paper, we treat the weakly damped, forced KdV equation on $\dot{H}^s$. We are interested in the lower bound of $s$ to assure the existence of the global attractor. The KdV equation has infinite conservation laws, each of which is defined in $H^j(j\in\mathbb Z, j\ge 0)$. The existence of the global attractor is usually proved by using those conservation laws. Because the KdV equation on $\dot{H}^s$ has no conservation law for $s<0$, it seems a natural question whether we can show the existence of the global attractor for $s<0$. Moreover, because the conservation laws restrict the behavior of solutions, the time global behavior of solutions for $s<0$ may be different from that for $s\ge 0$. By using a modified energy, we prove the existence of the global attractor for $s > -3/8$, which is identical to the global attractor for $s \ge 0$.

Keywords:  KdV equation, Cauchy problem, global attractor, Fourier restriction norm, low regularity, I-method.
Mathematics Subject Classification:  35Q53, 35B41, 37L30.

Received: January 2003;      Revised: January 2004;      Published: March 2004.