Spherical Averaged Endpoint Strichartz Estimates for the Two-dimensional Schrödinger Equations with Inverse Square Potential

Title of dissertation: Spherical Averaged Endpoint Strichartz Estimates for the Two-dimensional Schrödinger Equations with Inverse Square Potential I-Kun Chen, Doctor of Philosophy, 2009 Dissertation directed by: Professor Manoussos G. Grillakis Department of Mathematics In this dissertation, I investigate the two-dimensional Schrödinger equation with repulsive inverse square potential, i.e., { i∂tu+4u− a 2 |x|2u = 0 u : R 2 × R → C, u(x, 0) = u0(x). (1) I prove the following version of the homogeneous endpoint Strichartz estimate: ‖u‖L2t (Lr Lθ) ≤ C‖u0‖L2 , (2) where the Lr Lθ is a norm that takes L 2 average in angular variable first and then supremum norm on radial variable, i.e., ‖f(x, y)‖L∞r Lθ = sup r>0 ( 1 2π ∫ 2π 0 |f(r cos θ, r sin θ)|dθ ) 1 2 . (3) The main result is presented in chapter 4. In chapter 2, I give a brief introduction on the equations that inspired my research, namely the Landau-Lifshitz equation and the Schrödinger map equation. In chapter 3, I introduce a geometric concept in order to obtain a gauge system suitable for analysis. Spherical Averaged Endpoint Strichartz Estimates for The Two-dimensional Schrödinger Equations with Inverse Square Potential by I-Kun Chen Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2009 Advisory Committee: Professor Manoussos G. Grillakis, Chair/Advisor Professor Matei Machedon Professor Dionisios Margetis Professor Kasso A. Okoudjou Distinguished University Professor John D. Weeks c © Copyright by I-Kun Chen 2009 Dedication For my parents, Chao-Nan Chen and Fang-Yu Chang