Generator functions and their applications

TL;DR

This paper presents generator functions as a novel tool for analyzing the regularity and existence of analytic solutions to various classical equations, offering an alternative to traditional methods.

Contribution

It introduces generator functions and demonstrates their application in proving existence of analytic solutions for hyperbolic, Euler, hydrostatic Euler, and Vlasov equations.

Findings

Generator functions effectively track regularity of solutions.

Existence results for classical equations using generator functions.

Alternative to classical Cauchy-Kovalevskaya theorem.

Abstract

In [Grenier-Nguyen], we introduced so called {\em generators} functions to precisely follow the regularity of analytic solutions of Navier Stokes equations. In this short note, we give a presentation of these generator functions and use them to give existence results of analytic solutions to some classical equations, namely to hyperbolic equations, to incompressible Euler equations, and to hydrostatic Euler and Vlasov models. The use of these generator functions appear to be an alternative way to the use of the classical abstract Cauchy-Kovalevskaya theorem [Asano,Caflisch,Nirenberg,Safonov].