Analysis for Nonlinear Partial Differential Equations

This book offers a rigorous and self-contained introduction to the analytical methods used in nonlinear partial differential equations, with a special emphasis on dispersive equations. The book addresses a central difficulty of the field: the prerequisite tools required for applying modern PDE theory are often scattered across different sources, either developed with incompatible notation or presented without considering dispersive analysis. In this book, they are organized into a coherent framework, offering rigorous, self-contained proofs and a unified point of view. Readers are guided from foundational topics such as Lebesgue and Lorentz spaces, convolution, weak convergence, and Fourier transforms to more advanced techniques including oscillatory integrals, maximal functions, singular integrals, dyadic methods, and spaces of regular functions. These foundational techniques can help readers in studying regularity, decay, dispersive estimates, and the behavior of solutions to evolution equations. The final part of the book shows how this analytical toolbox is applied to the Schrödinger and wave equations, highlighting the core ideas of Euclidean dispersive analysis in a concrete and mathematically rigorous way. By combining breadth, detail, and an explicit focus on applications, the book serves both as a reference work and as a pathway into a demanding area of research. It is intended for graduate students and early career researchers in analysis, partial differential equations, harmonic analysis, and mathematical physics.