Nash--Moser iteration and singular perturbations

TL;DR

This paper introduces a simplified Nash--Moser iteration theorem designed for singular perturbation problems, accommodating loss of powers of the small parameter and derivatives, with applications to quasilinear Schrödinger equations and relaxation systems.

Contribution

It provides a new Nash--Moser theorem tailored for singular perturbations with parameter-dependent approximate solutions, allowing for loss of powers of the small parameter.

Findings

Applicable to quasilinear Schrödinger systems

Proves existence of small-amplitude profiles in relaxation systems

Demonstrates utility through sample applications

Abstract

We present a simple and easy-to-use Nash--Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter $\eps\to 0.$ The novel feature is to allow loss of powers of $\eps$ as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schr\"odinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems.