Global well-posedness for KdV in Sobolev spaces of negative index. (English) Zbl 0967.35119

Colliander, James E.; Keel, Markus; Staffilani, Gigliola; Takaoka, Hideo; Tao, Terence C.

Electron. J. Differ. Equ. 2001, Paper No. 26, 7 p. (2001).

Summary: The initial value problem for the Korteweg-de Vries equation on the line \[ \partial_t u+\partial^3_x u+\textstyle{{1\over 2}} \partial_x(u^2)= 0,\quad x\in\mathbb{R}, \]
\[ u(0)= \phi \] is shown to be globally well-posed for rough data \(\phi\). In particular, we show global well-posedness for initial data in \(H^s(\mathbb{R})\) for \(-3/10< s\).

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
42B35	Function spaces arising in harmonic analysis
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

rough initial data; initial value problem; Korteweg-de Vries equation; global well-posedness