Adaptation of the generic PDE's results to the notion of prevalence

TL;DR

This paper extends the concept of prevalence, a measure-theoretic notion of genericity, to the qualitative analysis of PDE solutions, adapting existing generic results to this new framework.

Contribution

It introduces the adaptation of generic PDE results to the prevalence framework, especially in contexts involving Sard-Smale theorems and analytic perturbations.

Findings

Prevalence sets align with full Lebesgue measure sets in finite dimensions.

The adaptation applies to Sard-Smale theorems in PDE analysis.

Analytic perturbation arguments are compatible with prevalence-based genericity.

Abstract

Many generic results have been proved, especially concerning the qualitative behaviour of solutions of partial differential equations. Recently, a new notion of "almost always", the prevalence, has been developped for vectorial spaces. This notion is interesting since, for example, prevalence sets are equivalent to the full Lebesgue measure sets in finite dimensional spaces. The purpose of this article is to adapt the generic PDE's results to the notion of prevalence. In particular, we consider the cases where Sard-Smale theorems or arguments of analytic perturbations of the parameters are used.