Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation

• Published in: Journal of functional analysis Vol. 100; no. 1; pp. 87 - 109
• Main Authors: Christ, F.M, Weinstein, M.I
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.08.1991
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/0022-1236(91)90103-C

We study the long-time behavior of small solutions of the initial-value problem for the generalized Korteweg-de Vries equation ∂ tu + ∂ x 3u + ∂ xF(u) = 0 (gKdV) u(x, 0) = g(x) . For the case where F(w)=¦w¦ s , with s > ( 1 4 )(23 − √57) ≈ 3.8625 , our results imply that if ∥ g∥ L 1 1 + ∥ g∥ L 2 2 is sufficiently small then sup r(1 + ¦t¦) 1 3 ∥u(t)∥ L ∞ < ∞ . In particular, the solution tends to zero in the supremum norm. The proofs make use of Duhamel's formula and dispersion estimates for the linear propagator, as well as chain and Leibniz rules for fractional derivatives of compositions ∥ D α F( u)∥ L p and products ∥ D α ( fg)∥ L p , 0 < α < 1 and 1 < p < ∞.