Translator Disclaimer
1995 A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain
Bing Yu Zhang
Differential Integral Equations 8(5): 1191-1204 (1995).

Abstract

The Cauchy problem of the Korteweg-de Vries equation on a periodic domain $S$, a unit circle in the complex plane establishes a nonlinear map $K$ from the initial data $\phi \in H^s(S) $ to the solution $u(x,t) \in C([-T,T];H^s(S) )$ for $s\geq 0$ (cf. [4] and [10]). Based on Bourgain's new approach [4] to periodic solutions of the KdV equation, it is shown that the nonlinear map $K$ is analytic from $H^s(S) $ to $C([-T,T];H^s(S) ) $ in the following sense. For any $\phi \in H^s(S) $, there exists a $\delta > 0 $ such that if $h\in H^s(S) $ with $\|h\| _{H^s(S) } \leq \delta $ and $\int _S h(x)dx = 0$, then $K (\phi + h) $ has the following Taylor series expansion: $$ K(\phi +h ) =\sum ^{\infty }_{n=0} \frac{K^{(n)} (\phi ) [h^n ] }{n!}, $$ where the series converges in the space $C([-T,T];H^s(S) ) $ uniformly for $\|h\| _{H^s (S)} \leq \delta $ and $K^{(n)} (\phi ) $ is the n-th order Fr\'{e}chet derivatives of $K$ at $\phi $. As a consequence, the periodic solution $u(x,t)$ of the KdV equation can be obtained by solving a series of linear problems since each term in the above Taylor series is a solution of a linearized KdV equation.

Citation

Download Citation

Bing Yu Zhang. "A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain." Differential Integral Equations 8 (5) 1191 - 1204, 1995.

Information

Published: 1995
First available in Project Euclid: 20 May 2013

zbMATH: 0826.35114
MathSciNet: MR1325553

Subjects:
Primary: 35Q53
Secondary: 35B65, 35C10

Rights: Copyright © 1995 Khayyam Publishing, Inc.

JOURNAL ARTICLE
14 PAGES


SHARE
Vol.8 • No. 5 • 1995
Back to Top