Remarks on $L^2$-wellposed Cauchy problem for some dispersive equations
Remarks on $L^2$-wellposed Cauchy problem for some dispersive equations
复制标题
关于某些色散方程的$L^2$-适定柯西问题的评论
DOI：
10.1215/kjm/1250518213
复制
发表时间：
1997
影响因子：
--
通讯作者：
S. Tarama
中科院分区：
文献类型：
--
作者：
S. Tarama
其中复值系数a（x）和B（x）属于空间Er，该空间Er由所有有界光滑函数组成，其任意阶导数也在真实的直线R上有界.如果系数a（x）和B（x）是一个常数，则通过B的Fourier变换证明，当系数a（x）的虚部不为零时，A的Cauchy问题不是P-适定的.这说明A的Cauchy问题不一定是L2良定的.通过构造渐近解，我们确实看到系数a（x）的无穷大部分满足以下条件，记为a1（x）：存在一个常数K，使得对任意x和y ∈ R
w ith the complex-valued coeffcients a (x ) and b (x) belonging to the space Er consisting o f all bounded smooth functions whose derivative o f any order is also bounded on real line R. If th e c o e ff ic ie n ts a ( x ) a n d b (x) a re c o n s ta n t, w e s e e b y Fourier transformation that, when the imaginary part of the coefficent a (x) is not zero, the Cauchy problem for A is not P-wellposed. T his im plies that the Cauchy problem fo r A is not always L 2 -wellposed. Indeed w e see by the construction of asymptotic solutions th a t the following condition on the im aginary part of the coefficent a (x), w hich is denoted by a1 (x) : there exists a constant K such that we have for any x and y E R