A SURVEY ON THE CAUCHY PROBLEM FOR THE KORTEWEG-DE VRIES EQUATION
Ali Kutlu Durşen
Mathematics, MSc Thesis, 2013
Thesis Jury
Prof. Dr. Albert Erkip (Thesis Supervisor), Assoc. Prof. Cem Güneri, Assoc. Prof. Hans Frenk, Asst. Prof. Yasemin Şengül, Prof. Dr. Saadet Erbay
Date &Time: August,06th, 2013 – 10:30
Place: FENS L055
Keywords: Korteweg – de Vries equation, Global existence, Cauchy problem
Abstract
In this thesis, we study the classic Korteweg – de Vries equation
ut + ux + uux + uxxx = 0
u(x,0)=u0(x)
in one dimensional spatial variable, expressing the behaviour of waves of long amplitude and shallow water depth. We first use Bona and colleagues' approach of adding a regularizing term to the equation and show that equation is well-posed for initial data u0 in Hs, , with solution lying in this space for each t, globally. We then use semigroup theory, introduced to nonlinear study mostly by Kato, to lower the bound on s to s > 3/2 for local solutions and to s = 2 for global solutions.