The purpose of this paper is to study local and global well-posedness of the initial value problem for the generalized Korteweg-de Vries (gKdV) equation in Lr = (f ε S' (ℝ): ||f||Lr = ||f||Lr' ≤ ∞). We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical Lr-space. A key ingredient is a Stein-Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for Lr-framework.
- Generalized korteweg-de vries equation
- Scattering problem
ASJC Scopus subject areas
- Numerical Analysis
- Applied Mathematics