Table of Contents

Chapter 0: Introduction

Chapter 1: Condensate growth term: existence, uniqueness, propagations of moments (Joint work with Ricardo Alonso and Irene M. Gamba)

Chapter 2: Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons (Joint work with Toan Nguyen)

Chapter 3: What else besides the condensate growth term? (Joint work with Avy Soffer)

Chapter 4: Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature (Joint work with Miguel Escobedo)

Chapter 5: A toric dynamical system approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases (Joint work with Gheorghe Craciun)

Chapter 6: Derivation of quantum kinetic equations (Joint work with Linda Reichl)

Chapter 7: Coupling kinetic and Schrodinger equations – A simplified model by Soffer and Tran (Joint work with Avy Soffer)

Chapter 8: Quantum hydrodynamic approximations to finite temperature trapped Bose gases  (Joint work with Shi Jin)

Chapter 9: Wave turbulence theory for capillary water waves (Joint work with Toan Nguyen)

Chapter 10: On the wave turbulence theory for stratified flows in the ocean (Joint work with Irene M. Gamba and Leslie M. Smith)

Chapter 11: Optimal local well-posedness theory for the kinetic wave equation. (Joint work with Pierre Germain and Alexandru D. Ionescu)

EXTRA CHAPTERS: SCIENTIFIC COMPUTING-COMPUTATIONAL SCIENCES and UNCERTAINTY QUANTIFICATION-SENSITIVITY ANALYSIS

Extra Chapter 1: iHDG – Iterative Hybridized Discontinuous Galerkin method (Joint work with Tan Bui-Thanh and Sriramkrishnan Muralikrishnan)

Extra Chapter 2: iHDG – Iterative Hybridized Discontinuous Galerkin method (Part 2) (Joint work with Tan Bui-Thanh and Sriramkrishnan Muralikrishnan)