Master Thesis Defense: Ali Kutlu Durşen
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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.