Lectures on the Analysis of Nonlinear Partial Differential 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; Frédéric Hérau, 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.