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Title:  A study on certain periodic Schrödinger equations 
Author(s):  Demirbas, Seckin 
Director of Research:  Tzirakis, Nikolaos; Erdogan, Burak 
Doctoral Committee Chair(s):  Junge, Marius 
Doctoral Committee Member(s):  Bronski, Jared C. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Periodic Schrodinger equation
Fractional Schrodinger equation 
Abstract:  In the first part of this thesis we consider the cubic Schrödinger equation
iu_t+Delta u =+/u^2u, x in T_theta^2, t\in [T,T],
u(x,0)=u_0(x) in H^s(T_theta^2).
T is the time of existence of the solutions and T_theta^2 is the irrational torus given by R^2/theta_1 Z * \theta_2 Z for theta_1, theta_2 > 0 and theta_1/theta_2 irrational. Our main result is an improvement of the Strichartz estimates on irrational tori using a counting argument by Huxley [43], which estimates the number of lattice points on ellipsoids. With this Strichartz estimate, we obtain a local wellposedness result in H^s for s>131/416. We also use energy type estimates to control the H^s norm of the solution and obtain improved growth bounds for higher order Sobolev norms.
In the second and the third parts of this thesis, we study the Cauchy problem for the 1d periodic fractional Schrödinger equation:
iu_t+(Delta)^alpha u =+/ u^2u, x in T, t in R,
u(x,0)=u_0(x) in H^s(T),
where alpha in (1/2,1). First, we prove a Strichartz type estimate for this equation. Using the arguments from Chapter 3, this estimate implies local wellposedness in H^s for s>(1alpha)/2. However, we prove local wellposedness using direct X^(s,b) estimates. In addition, we show the existence of globalintime infinite energy solutions. We also show that the nonlinear evolution of the equation is smoother than the initial data. As an important consequence of this smoothing estimate, we prove that there is global wellposedness in H^s for s>(10*alpha+1)/(12).
Finally, for the fractional Schrödinger equation, we define an invariant probability measure mu on H^s for s 
Issue Date:  20150629 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/87978 
Rights Information:  Capyright 2015 Seckin Demirbas 
Date Available in IDEALS:  20150929 
Date Deposited:  August 201 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois