## Differential and Integral Equations

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

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

#### 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