Paracomposition Operators and Paradifferential Reducibility
Thomas Alazard, Chengyang Shao
TL;DR
This work introduces a paradifferential framework to reduce nonlinear quasilinear PDEs to constant-coefficient form modulo smoothing, enabling construction of quasiperiodic solutions without Nash-Moser iterations. It develops a robust paracomposition toolbox based on a paratransport flow, providing precise, Lipschitz dependence on diffeomorphisms and a refined paralinearization theory. The authors prove fixed-point reducibility results for both matrix differential operators and nearly parallel vector fields on the torus, deriving tame estimates and measure results for the admissible frequency sets. The paradigm is then extended to PDEs by paralinearizing the nonlinear equation and conjugating it to a constant-coefficient system via a paracomposition and a paradifferential change-of-variables, yielding a contractions-based existence theory with Cantor-type frequency sets. The approach delivers a conceptually simpler alternative to Nash-Moser/KAM schemes for KAM-type problems in hyperbolic PDEs and opens new avenues for paradifferential analysis of change-of-variables in nonlinear PDEs.
Abstract
Reducibility methods, aiming to simplify systems by conjugating them to those with constant coefficients, are crucial for studying the existence of quasiperiodic solutions. In KAM theory for PDEs, these methods help address the invertibility of linearized operators that arise in a Nash-Moser/KAM type scheme. The goal of this paper is to prove paradifferential reducibility results, enabling the reduction of nonlinear equations themselves, rather than just their linearizations, to constant coefficient form, modulo smoothing terms. As an initial application, we demonstrate the existence of quasiperiodic solutions for certain hyperbolic systems. Despite the small denominator problem, our proof does not rely on traditional Nash-Moser/KAM-type schemes, but instead on the Banach fixed point theorem. To achieve this, we develop two key toolsets. The first focuses on the calculus of paracomposition operators introduced by Alinhac, interpreted as the flow map of a paraproduct vector field. We refine this approach to establish new estimates that precisely capture the dependence on the diffeomorphism in question. The second toolset addresses two classical reducibility problems, one for matrix differential operators and the other for nearly parallel vector fields on torus. We resolve these problems by paralinearizing the conjugacy equation and exploiting, at the paradifferential level, the specific algebraic structure of conjugacy problems, akin to Zehnder's approximate Nash-Moser approach.