The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.

This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems. This second edition has been thoroughly revised and in a new chapter the authors discuss several methods for proving the existence of solutions of primarily the Dirichlet problem for various types of elliptic equations.

The notes collected here originated in seminars on analysis in partial differential equations, given during the academic year of 2014 to 2015 at the Morningside Center of Mathematics, Chinese Academy of Sciences (CAS), and at the Center of Harmonic Analysis and Its Application, Academy of Mathematics and Systems Science, CAS. Included are: Hajer Bahouri, on critical Sobolev embeddings in Orlicz spaces and applications to PDEs with exponential nonlinearity; Hongjie Dong, on nonstandard Schauder estimates for parabolic equations; Frederic Herau, on hypocoercive methods and applications for simple linear inhomogeneous kinetic models; Nicolas Lerner, on Carleman inequalities; Jean-Pierre Puel, on controllability of Navier-Stokes equations; and Jiahong Wu, on 2D magnetohydrodynamic equations with partial or fractional dissipation.

The works presented in this volume originated in lectures on analysis in partial differential equations, given in at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Included are: Jean-Yves Chemin, on profile decomposition and its applications to the Navier-Stokes system; Hongjie Dong, on Lp estimates for parabolic equations; Xiaochun Li, on the Hardy-Littlewood circle method; Fanghua Lin, on elliptic free boundary problems; Alexis Vasseur, on the De Giorgi method for elliptic and parabolic equations; Jiahong Wu, on 2D Boussinesq equations with partial or fractional dissipation; and Xiaoyi Zhang, on analytic tools for critical dispersive PDEs.

This volume presents some of the most recent progress in the mathematical theory of fluid mechanics. The eight papers herein originated in a series of seminars held in 2011 at the Chinese Academy of Sciences in Beijing. Among them are Nicolas Burq on the wellposedness of the water wave problem with rough data, Jean-Yves Chemin on the wellposedness of the Navier-Stokes system, and Isabelle Gallagher on the semiclassical limit of a geostrophic system. This third volume of the series is a good reference for those working on nonlinear partial differential equations, especially as applied to fluid mechanics equations and micro-local analysis.