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

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June 2004, 3(2): 301-318. doi: 10.3934/cpaa.2004.3.301

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

1.	Institut des Hautes Etudes Scientifiques, Le Bois-Marie, 35, route de Chartres, 91440 Bures-sur-Yvette, France

Received January 2003 Revised January 2004 Published March 2004

Abstract
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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: Cauchy problem, low regularity, KdV equation, I-method., Fourier restriction norm, global attractor.

Mathematics Subject Classification: 35Q53, 35B41, 37L3.

Citation: Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301

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