Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation
Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation
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DOI：
10.1016/0022-1236(91)90103-c
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发表时间：
1991-08
影响因子：
1.7
通讯作者：
F. M. Christ;M. Weinstein
中科院分区：
文献类型：
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作者：
F. M. Christ;M. Weinstein
研究了广义Korteweg-de Vries方程∂tu +∂x 3u +∂x F (u)= 0 (gKdV) u (x, 0)= g (x)初值问题小解的长时间行为。对于F (w)= μ s，当s>(1 4)(23−√57)≈3.8625时，我们的结果表明，如果∥g∥l11 1+∥g∥l22 2足够小，则r (1+ μ t) 1 3∥u (t)∥L∞<∞。特别地，解在最高范数中趋向于零。该证明利用了线性传播子的Duhamel公式和频散估计，以及组合物∥D α F (u)∥L p和乘积∥D α (fg)∥L p, 0< α< 1和1< p<∞的分数阶导数的链规则和莱布尼兹规则。
We study the long-time behavior of small solutions of the initial-value problem for the generalized Korteweg-de Vries equation∂ t u+∂ x 3 u+∂ x F (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<∞.