## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 12 (2005), 1333-1339.

### A bilinear Airy-estimate with application to gKdV-3

#### Abstract

The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem \[u_t + u_{xxx} + (u^4)_x=0,\hspace{1cm}u(0)=u_0 \in H^s_x({\bf R}), \,\,\,s>-\tfrac{1}{6}\] for the generalized Korteweg-deVries equation of order three, for short gKdV-3. For real-valued data $u_0 \in L^2_x({\bf R})$ global wellposedness follows by the conservation of the $L^2$ norm. The main new tool is a bilinear estimate for solutions of the Airy-equation.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 12 (2005), 1333-1339.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356059713

**Mathematical Reviews number (MathSciNet)**

MR2174975

**Zentralblatt MATH identifier**

1212.35412

**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

Grünrock, Axel. A bilinear Airy-estimate with application to gKdV-3. Differential Integral Equations 18 (2005), no. 12, 1333--1339. https://projecteuclid.org/euclid.die/1356059713