A Nash-Moser-H\"ormander implicit function theorem with applications to control and Cauchy problems for PDEs

TL;DR

This paper introduces a sharp Nash-Moser implicit function theorem tailored for control and Cauchy problems in PDEs, reducing regularity loss and simplifying assumptions compared to existing methods.

Contribution

It develops a novel Nash-Moser theorem combining Hörmander's iteration scheme with a new series splitting technique, enhancing applicability to PDE control and Cauchy problems.

Findings

Improves regularity requirements for solutions of PDE control problems.

Successfully applied to quasi-linear KdV perturbations, enhancing previous results.

Simplifies assumptions needed for PDE control and Cauchy problem analysis.

Abstract

We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The proof is a combination of: (i) the iteration scheme by H\"ormander (ARMA 1976), based on telescoping series, and very close to the original one by Nash; (ii) a suitable way of splitting series in scales of Banach spaces, inspired by a simple, clever trick used in paradifferential calculus (for example, by M\'etivier). As an example of application, we apply our theorem to a control and a Cauchy problem for quasi-linear perturbations of KdV equations, improving the regularity of a previous result. With respect to other approaches to control and Cauchy problems, the application of our theorem requires lighter assumptions to be verified.