A survey on the Cauchy problem for the Korteweg-de Vries equation

Official URL: http://192.168.1.20/record=b1534393 (Table of Contents)

In this thesis, we study the Cauchy problem for the classic Korteweg-de Vries equation ut + ux + uux + uxxx = 0 for x ∈ R, t > 0 u(x, 0) = u0(x) for x ∈ R describing the propagation of long waves in shallow waters. We first use Bona and colleagues' approach of adding a regularizing term to the equation and show that the equation is well-posed for initial data u0 2 Hs, s ≥ 3, with solution lying in this space for each t globally. We then use Kato's methods of semigroup theory in nonlinear study to lower the bound on s to s > 3=2 for local solutions and to s ≥2 for global solutions.