In this talk, we will give a brief introduction of paradifferential calculus and its application in the study of differential equations involving "loss of regularity".
The first part consists of a concise description of paradifferential calculus as a toolbox in Fourier analysis, including the definition of paraproduct, regularity of paraproduct operator, calculus of paradifferential operators, and Bony's paralinearization theorem. It is emphasized that paradifferential operators preserve the algebraic structure of ordinary (pseudo)differential operators while keep control of loss of regularity. This feature gives a rigorous, elegant justification of "formal computations".
The second part is devoted to a new proof of classical KAM type results avoiding the usual Nash-Moser/KAM iteration schemes. Following Hörmander's notion of "parainverse equation", it will be explained how the Newtonian step of solving homological equation may be replaced by its paradifferential counterpart, called the "para-homological equation". Being simpler than Newton type iteration, such a proof has the potential of extending to "realistic" threshold of perturbation. If time permits, various generalizations will also be briefly discussed, including applications to "KAM for PDEs".
Alain Albouy, Alain Chenciner, Jacques Laskar