Stieltjes differential systems with non monotonic derivators

TL;DR

This paper investigates Stieltjes differential systems with non-monotonic derivators, introducing controlled variation functions, characterizing precompact sets, and applying the results to fluid stratification models.

Contribution

It defines functions of controlled variation, characterizes precompact sets of g-continuous functions, and provides explicit g-exponential maps, extending the theory of Stieltjes systems with sign-changing derivators.

Findings

Established the notion of functions of controlled variation.

Characterized precompact sets of g-continuous functions.

Derived explicit expressions for g-exponential maps.

Abstract

In this work we study Stieltjes differential systems of which the derivators are allowed to change sign. This leads to the definition of the notion of \emph{function of controlled variation}, a characterization of precompact sets of $g$-continuous functions, and an explicit expression of $g$-exponential maps. Finally, we prove a Peano-type existence result and apply it to a model of fluid stratification on buoyant miscible jets and plumes.