Inversion of a Non-uniform Difference Operator

TL;DR

This paper develops a theoretical framework for inverting non-uniform difference operators, crucial for applying Nash-Moser methods to periodic solutions of the Euler equations, highlighting the limitations of Fourier methods.

Contribution

It proves that scalar non-uniform difference operators have bounded inverses on their range, establishing a foundation for analyzing such operators beyond Fourier techniques.

Findings

Scalar non-uniform difference operators have bounded inverses.

Fourier methods are insufficient for analyzing non-uniform shift operators.

The work provides a basis for applying Nash-Moser methods to fluid dynamics problems.

Abstract

The problem of applying Nash-Moser Newton methods to obtain periodic solutions of the compressible Euler equations has led 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.