Strict condition for the $L^{2}$-wellposedness of fifth and sixth order dispersive equations

TL;DR

This paper establishes necessary and sufficient conditions for the $L^{2}$-wellposedness of fifth and sixth order variable-coefficient dispersive equations, simplifying the process of proving well-posedness using an inductive approach.

Contribution

It refines previous necessary conditions into a complete characterization, enabling easier construction of pseudodifferential operators for well-posedness proofs.

Findings

Necessary and sufficient conditions for $L^{2}$-wellposedness are identified.

An inductive method simplifies the proof process.

Conditions are split into parts for easier application.

Abstract

We provide a set of conditions that is necessary and sufficient for the $L^{2}$-wellposedness of the Cauchy problem for fifth and sixth order variable-coefficient linear dispersive equations. The necessity of these conditions had been presented by Tarama, and we scrutinized their proof to split the conditions into several parts so that an inductive argument is applicable. This inductive argument simplifies the engineering process of the appropriate pseudodifferential operator needed for the proof of $L^{2}$-wellposedness.