IDEALS Home University of Illinois at Urbana-Champaign logo The Alma Mater The Main Quad

Discrete Fourier restriction phenomenon associated with some periodic dispersive equations

Show full item record

Bookmark or cite this item: http://hdl.handle.net/2142/34289

Files in this item

File Description Format
PDF Hu_Yi.pdf (406KB) (no description provided) PDF
Title: Discrete Fourier restriction phenomenon associated with some periodic dispersive equations
Author(s): Hu, Yi
Director of Research: Li, Xiaochun
Doctoral Committee Chair(s): Erdogan, M. Burak
Doctoral Committee Member(s): Li, Xiaochun; Tzirakis, Nikolaos; Berndt, Bruce C.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): discrete Fourier restriction periodic dispersive equations Schrodinger equations Korteweg-de Vries (KdV) equations fifth order Korteweg-de Vries (KdV) equations
Abstract: The thesis consists of six chapters. In Chapter 1, we will briefly introduce the background of the topic, as well as some results we already know. The next five chapters can be divided into two parts. The first part is about the discrete Fourier restriction phenomenon. In Chapter 2, we consider the discrete Fourier restriction phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain's level set result on Strichartz estimates associated with periodic Schrodinger equations. Some sharp estimates on $L^\frac{2(d+2)}{d}$ norms of certain exponential sums in higher dimensional cases are established. In Chapter 3 we further discuss the discrete Fourier restriction problem associated with higher order dispersive equations, with the method developed in Chapter 2. We obtain some sharp bound on the size of the Fourier transform of a function for large indices. Some new Strichartz estimates of this type are obtained. Also, we can use the method to prove some exponential sum estimates, which are classic in number theory. The second part of the thesis is about the local well-posedness of some dispersive equations. In Chapter 4, we prove the local well-posedness of the periodic gKdV equations. The method we apply here is a generalization of Bourgain's "denominator manipulation". With this idea, we further discuss a more general type of KdV equation in Chapter 5. We establish the local well-posedness of the periodic KdV equations with nonlinear terms $F(u)u_x$, provided $F\in C^5$ and the initial data $u_0\in H^s$ with $s>1/2$ (1/2 is sharp). In Chapter 6 we focus on the local well-posedness of the periodic fifth order KdV type dispersive equations with nonlinear terms $P_1(u)u_x + P_2(u)u_x^2$, provided the initial data $u_0\in H^s$ with $s>1$. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. Some Strichartz estimates derived in Chapter 3 are used in the proof. Also, a couple of counterexamples are given to exhibit the sharpness of the indices.
Issue Date: 2012-09-18
URI: http://hdl.handle.net/2142/34289
Rights Information: Copyright 2012 Yi Hu
Date Available in IDEALS: 2012-09-18
Date Deposited: 2012-08
 

This item appears in the following Collection(s)

Show full item record

Item Statistics

  • Total Downloads: 129
  • Downloads this Month: 2
  • Downloads Today: 0

Browse

My Account

Information

Access Key