Contents Online
Methods and Applications of Analysis
Volume 27 (2020)
Number 1
Inversion of a non-uniform difference operator
Pages: 65 – 86
DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n1.a3
Authors
Abstract
The problem of applying Nash–Moser Newton methods to obtain periodic solutions of the compressible Euler equations has led the authors to identify the main obstacle, namely, how to invert operators which impose periodicity when they are based on non-uniform shift operators. Here we begin a theory for finding the inverses of such operators by proving that a scalar non-uniform difference operator does in fact have a bounded inverse on its range. We argue that this is the simplest example which demonstrates the need to use direct rather than Fourier methods to analyze inverses of linear operators involving nonuniform shifts.
Keywords
periodic solutions, shock waves, Nash–Moser, nonlinear shift operator
2010 Mathematics Subject Classification
35L60
Received 19 June 2019
Accepted 11 February 2020
Published 12 August 2020