Differential and Integral Equations

Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity

Nobu Kishimoto

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Abstract

The Cauchy problem for the nonperiodic KdV equation is shown by the iteration method to be locally well-posed in $H^{-3/4}(\mathbb R )$. In particular, solutions are unique in the whole Banach space for the iteration. This extends the previous well-posedness result in $H^s$, $s>-3/4$ obtained by Kenig, Ponce and Vega (1996) to the limiting case, and improves the existence result in $H^{-3/4}$ given by Christ, Colliander and Tao (2003). Our result immediately yields global well-posedness for the KdV equation in $H^{-3/4}(\mathbb R )$ and for the modified KdV equation in $H^{1/4}(\mathbb R )$, combined with the argument of Colliander, Keel, Staffilani, Takaoka and Tao (2003).

Article information

Source
Differential Integral Equations Volume 22, Number 5/6 (2009), 447-464.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019601

Mathematical Reviews number (MathSciNet)
MR2501679

Zentralblatt MATH identifier
1240.35461

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]

Citation

Kishimoto, Nobu. Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Differential Integral Equations 22 (2009), no. 5/6, 447--464. https://projecteuclid.org/euclid.die/1356019601.


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