WELL-POSEDNESS AND SCATTERING RESULTS FOR THE GENERALIZED KORTEWEG-DEVRIES EQUATION VIA THE CONTRACTION PRINCIPLE
WELL-POSEDNESS AND SCATTERING RESULTS FOR THE GENERALIZED KORTEWEG-DEVRIES EQUATION VIA THE CONTRACTION PRINCIPLE
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DOI：
10.1002/cpa.3160460405
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发表时间：
1993-04-01
影响因子：
3
通讯作者：
VEGA, L
中科院分区：
文献类型：
--
作者：
KENIG, CE;PONCE, G;VEGA, L
当k= 1时，该方程由Korteweg-de弗里斯在[45]中导出，作为在信道中传播的长波模型;我们将其称为KdV方程。随后，人们发现KdV及其修正形式（式1.1中k= 2）在许多不同的物理系统中都是相关的。事实上，一大类双曲模型已被简化为这些方程。此外，由于它们与逆散射理论和代数几何的关系，它们已经被研究;参见[22]，[52]和[63]以及其中的参考文献。关于IVP（1.1）的存在性问题已经有了大量的工作。对于k= 1，2的情况，其中应用逆散射方法（参见[26]和[27]），并且在数据的适当衰减假设下，已经建立了几个存在性结果;参见[131]。[141[37]，[60]，[73]。另一种继承自双曲问题的方法依赖于能量方法（L2-理论）;参见[5]，[6]，[33]，[34]，[61]，[62]和[64]。特别地，证明了IVP（1.1）在经典Sobolev空间HS（R）=（1-A）-S/2L 2（R）中是局部适定的，其中s> 3/2.利用这些结果
For k= 1 the equation was derived by Korteweg-de Vries in [45] as a model for long waves propagating in a channel; we shall refer to it as the KdV equation. Subsequently the KdV and its modified form (k= 2 in (1.1)) were found to be relevant in a number of different physical systems. In fact, a large class of hyperbolic models has been reduced to these equations. Also, they have been studied because of their relation to inverse scattering theory and to algebraic geometry; see [22],[52], and [63] and the references therein. A large amount of work has been devoted to the existence problem of the IVP (1.1). For the cases k= 1, 2 where the inverse scattering method (see [26] and [27]) applies and under appropriate decay assumptions on the data several existence results have been established; see [131.[141,[37],[60], and [73]. Another approach inherited from hyperbolic problems relies on the energy method (L2-theory); see [5],[6],[33],[34],[61],[62], and [64]. In particular, it shows that the IVP (1.1) is locally well-posed in the classical Sobolev spaces HS (R)=(1-A)-S/2L2 (R) with s> 3/2. Using these results