Well-posedness of the Korteweg-de Vries Equation: An Application
of Fixed Point Theory and Harmonic Analysis
By Mamikon Gulian
November 6, 2013
The KdV equation is a time-dependent equation describing the evolution of water waves in a shallow channel. We will start by discussing basic one-dimensional stationary phase analysis of oscillatory integrals. Then, we will apply these results to prove a Kato smoothing inequality for the linear part of the KdV equation. This inequality states that solutions to the linear part of the KdV equation obtain higher differentiability than the initial data from which they propogate. If time permits, we will discuss how this result can be used to prove well-posedness for the (nonlinear) KdV equation.