A Dissertation entitled Sobolev Gradient Semi-flows & Applications to Nonlinear Problems by Ramesh Karki Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics Dr. Alessandro Arsie, Committee Chair Dr. Denis A. White, Committee Member Dr. Franco Cardin, Committee Member Dr. Gregory S. Spradlin, Committee Member Dr. Mao-Pei Tsui, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo May 2015 Copyright 2015, Ramesh Karki This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Sobolev Gradient Semi-flows & Applications to Nonlinear Problems by Ramesh Karki Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics The University of Toledo May 2015 We are interested in solving nonlinear pseudo-differential equations (in particular, partial differential equations as well) involving fractional powers of uniformly ellip- tic self-adjoint operators of order two with suitable smoothness conditions on the coefficients subject to given (Dirichlet or periodic) boundary conditions. Under the stronger assumptions, we are interested in studying solutions in a special class whose elements satisfy non-selfintersecting property and have bounded distance from a given hyperplane, since such solutions are the analogue for Aubrey-Mather sets for ODEs and leaves of minimal foliations or laminations for PDEs . To solve such a ΨDE, we will start by introducing an energy type functional whose Euler-Lagrange equation is the pseudo-differential equation itself. As we seek to minimize this functional, we will introduce the Sobolev gradient of the functional as an element of a suitable Sobolev space and then we consider the gradient descent equation subject to appropriate ini- tial and boundary conditions. The equilibrium solutions of this Sobolev gradient descent equation are the critical points we are looking for. Now the first step of our work will be to construct a semi-flow corresponding to the aforementioned initial-boundary value problem. So we will prove the existence, uniqueness, regularity, and comparison properties related to the semi-flow. Then iii the next step will be to analyze the convergence of this semi-flow to an equilibrium solution to this initial-boundary value problem. In our work, we will adapt two methods: analytical method and numerical method. We apply various analytical tools to establish the general results and numerical tools to study concrete solutions of particular pseudo-differential or partial differential equations. iv Dedicated to \My Professors, Teachers, Family, and Friends" who have directly or indirectly supported me in my academic life Acknowledgments I would like to sincerely thank my Ph. D. adviser Prof. Alessandro Arsie for his continuous advice and tireless support to develop foundation in my research area and successfully establish main results and many other technical results included in this dissertation. I would like to thank my dissertation committee members Dr. Denis A. White, Dr. Mao-Pei Tsui, Dr. Franco Cardin and Dr. Gregory Spradlin for their careful reading of this manuscript, feedback and suggestions. I would also like to thank Dr. Timothy Blass and Prof. Rafael de lLave for their support in many ways. Their kind responses of my questions in emails and in person during several meetings, their help on providing notes, papers and books, and giving comments were instrumental to proceed my research work ahead. I am particularly indebted to my good friends and professors Dr. Akaki Tikaradze, Dr. Vani Cheruvu, Dr. Santosh Kandel, Dr. Harihar Khanal, who have been help- ing me discuss on several topics and problems, giving me valuable suggestions and encouragements to obtain my academic achievements by several means. Finally, I would like to express my sincere thanks to my family, many other friends, colleagues, and professors who have influenced directly and indirectly in my academic life so that their support has immensely helped me obtain my academic achievements. Ramesh Karki vi Contents Abstract iii Acknowledgments vi Contents vii List of Abbreviations x List of Symbols xi Preface xii 1 Introduction 1 2 Some Preliminaries 5 2.1 Euler-Lagrange Equation . .5 2.2 Sobolev Spaces . .7 2.2.1 When Ω = Td ...........................7 2.2.2 When Ω is an open bounded domain . 10 2.3 Sobolev Gradients . 11 2.4 Sobolev Gradient Descent Equations . 14 2.5 Semigroups And Fractional Powers of Linear Operators . 16 2.6 The Spectral Property For An Unbounded Self-adjoint Operator On A Hilbert Space . 22 2.7 Main Results . 23 vii 2.7.1 Existence, Uniqueness & Regularity . 23 2.7.2 The Comparison Principles . 24 2.7.3 Applications To Semilinear ΨDEs . 25 2.7.4 Numerical Solutions To Some ΨDEs . 25 3 Characterization of Sobolev Spaces and Fractional Powers of Oper- ators 27 3.1 Characterization of L2(Ω)........................ 27 3.2 Characterizations of Hα(Ω); Hαβ(Ω) and Fractional Powers of Oper- ators on L2(Ω).............................. 28 3.3 Fractional Powers of Operators on Hαβ(Ω)............... 30 4 Constructions of Semigroups, Comparison, Regularity & Bounded- ness 32 4.1 Constructions of Semigroups and Comparison . 33 4.2 Regularity and Boundedness of e−tAα .................. 37 4.3 Constructions of Semigroups and Comparison, revisited . 41 4.4 Regularity and Boundeness Properties of etL .............. 46 5 The Proofs of Main Results 50 5.1 The Proofs of Main Theorems Under L2-gradient . 50 5.1.1 Some Preliminaries . 50 5.1.2 The Proof of Theorem 2.7.1 . 54 5.1.3 The Proof of Theorem 2.7.3 . 58 5.2 The Proofs of Main Theorems Under Hαβ-gradient . 62 5.2.1 Some Preliminaries (Revisited) . 62 5.2.2 The Proof of Theorem 2.7.2 . 65 5.2.3 The Proof of Theorem 2.7.4 . 70 viii 6 Applications to Semilinear Pseudo Differential Equations 75 6.1 Nonself-intersecting Solutions . 76 6.2 A Generalization of Minimal Foliations (or Minimal Laminations) & Invariant Tori (or Aubry-Mather Sets) . 77 6.3 The Proof of Theorem 2.7.5 . 82 6.4 Regularity in Some Special Cases . 91 6.5 Numerical Solutions to Some Semilinear PDEs and ΨDEs . 92 References 100 A Pseudo-differential Operators 103 B MATLAB Programmings 107 B.1 Code for Example 6.5.1 . 107 ix List of Abbreviations E-L . Euler-Lagrange ACP ...................... abstract Cauchy aroblem IVP . initial value problem IBVP . initial-boundary value problem ΨDE . pseudo-differential equation PDE . partial differential equation ODE . ordinary differetial equation FFT . fast Fourier transform DFT . discrete Fourier transform x List of Symbols R ........... The set of real numbers N ........... The set of positive integers NTd ........ The torus or N copies of [0;N]d or Rd=NZd C1(Ω)..... The space of smooth functions in Ω 1 Cc (Ω)..... The space of smooth functions with compact supports in Ω S(R)........ The Schwarz space of rapidly decreasing smooth functions 0 1 D (Ω)....... The space of distributions or dual of Cc (Ω) S0(R)....... The space of tempered distributions or dual of S(R) L1(Ω)...... The space of functions whose essentialsup norm is finite L2(Ω)....... The space of functions (more generally distributions) which are square integrable over Ω and satisfy either periodic or Dirichlet boundary conditions Hk(Ω)...... Sobolev space of order k consisting of distributions in L2(Ω) whose weak partial derivatives of orders ≤ k, k 2 Z+ are in L2(Ω) Hs(Ω)...... Fractional order Sobolev space of order s Hαs(Ω)..... Fractional order Sobolev space of order αs Inner Products( :; :)L2 = (:; :)L2(Ω) ; (:; :)αβ = (:; :)Hαβ (Ω) ; (:; :)αs = (:; :)Hαs(Ω) ; (:; :)s = (:; :)Hs(Ω) ; (:; :)k = (:; :)Hk(Ω) ; (:; :)0 = (:; :)H0(Ω) Norms . k:k1 = k:kL1(Ω), k:kL2 = k:kL2(Ω), k:kαβ = k:kHαβ (Ω), k:kαs = k:kHαs(Ω), k:ks = k:kHs(Ω), k:kk = k:kHk(Ω), k:k0 = k:kH0(Ω) L(H)....... The Banach space of all bounded linear operators on Hilbert space H under supnorm of operators. α @jαj d D ..........= for multi-index α = (α1; : : : ; αd) 2 N with jαj = α1 + @xα1 :::@xαd @αi th ··· + αd, where each is αi weak partial derivative. @xαi α α1 αd d jxj .........= x1 ::::::::xn for multi-index α = (α1; : : : ; αd) 2 N with jαj = α1 + ··· + αd. xi Preface The goal of this dissertation work is to give a brief presentation of Sobolev gradi- ents and to emphasize their applications on the theory of partial differential equations and pseudo-differential equations. We explore in depth the solutions to such equa- tions and some interesting behavior of the solutions using powerful tools from some of the rich theories of analysis such as the abstract semigroup theory, the spectral theory of linear unbounded self-adjoint operator on a Hilbert space, the theory of gen- eralized functions, functional analysis, and theory of partial differential equations and pseudo-differential equations. It, further, highlights on some numerical applications of these equations and opens up some problems in the future directions. xii Chapter 1 Introduction In this chapter we are going to discuss briefly what our main goal is, how we set up problem, and what we do to solve the problem we set up. We also discuss about what we prove so far, work in progress and the possible extensions of our problem or some future projects. Here our main interest is to solve certain types of nonlinear partial differential equations (PDEs) and pseudo-differential equations (ΨDEs), look for some special classes of solutions and investigate various properties. We will investigate these equations and their solutions via analytical method as well as numerical method. To deal with each of these nonlinear PDEs and ΨDEs we consider in section 2.1 a certain energy type functional Eα defined on an infinite dimensional Hilbert space H, namely a Sobolev space Hs(Ω); s ≥ 0, so that the equation satisfied by its critical points is the given PDE or ΨDE.