Well-posedness of the initial value problem for the Korteweg-de Vries equation
Well-posedness of the initial value problem for the Korteweg-de Vries equation
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DOI：
10.1090/s0894-0347-1991-1086966-0
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发表时间：
1991-05
影响因子：
3.9
通讯作者：
C. Kenig;G. Ponce;L. Vega
中科院分区：
文献类型：
--
作者：
C. Kenig;G. Ponce;L. Vega
(1.1)&ItU + axu + U1xU = O，x，t ∈ R { u（x，0）= uo（x）. KdV方程最初是作为非线性色散长波单向传播的模型导出的[21]，已经在不同的背景下被考虑，即在其与逆散射方法的关系中，在等离子体物理学中，以及在代数几何学中（参见[24]，以及其中的参考文献）。我们的目的是研究IVP（1.1）在经典Sobolev空间Hs（R）中的局部和整体适定性。我们可以说IVP（1.1）是局部的（分别为全局）适定的函数空间X中，如果它通过生成连续局部（分别）全局）流。[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][ 1)是本地的（resp.整体上）适定于Hs且s > 3/2（resp. s > 2）。粗略地说，Hs的整体适定性依赖于可用的局部理论和（1.1）的解所满足的守恒律，即：
(1.1) &ItU + axu + U1xU = O, x, t E R { u(x, 0) = uo(x). The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global well-posedness of the IVP (1.1) in classical Sobolev spaces Hs(R) . We shall say that the IVP (1.1) is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow. It was established in the works of Bona and Smith [3], Bona and Scott [2], Saut and Temam [30], and Kato [ 1 5] that the IVP (1. 1) is locally (resp. globally) well-posed in Hs with s > 3/2 (resp. s > 2). Roughly speaking, global well-posedness in Hs depends on the available local theory and on the conservation laws satisfied by solutions of (1.1), namely: