Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stieltjes differential systems with non monotonic derivators

Marlène Frigon, F. Adrián F. Tojo

TL;DR

This work extends Stieltjes differential calculus to derivators that can change sign by introducing the notion of a function of controlled variation, the $g$-derivative, a $g$-absolutely continuous framework, and an explicit $g$-exponential map. It proves a Peano-type existence result for systems with $g$-derivatives and develops a generalized Fundamental Theorem of Calculus and chain rule in this setting, supported by a Schauder fixed-point existence argument for $g$-Carathéodory systems. A local existence theory is established for multi-dimensional systems with $g_j\in CV_-(I,\mathbb{R})$, and the theory is applied to a buoyant miscible jet/plume model, recasting density variations in terms of a $g$-derivative to obtain local solutions without distributions. Overall, the paper broadens the Stieltjes framework beyond monotone derivators, enabling rigorous treatment of physical processes with sign-changing variation and non-smooth density effects.

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.

Stieltjes differential systems with non monotonic derivators

TL;DR

This work extends Stieltjes differential calculus to derivators that can change sign by introducing the notion of a function of controlled variation, the -derivative, a -absolutely continuous framework, and an explicit -exponential map. It proves a Peano-type existence result for systems with -derivatives and develops a generalized Fundamental Theorem of Calculus and chain rule in this setting, supported by a Schauder fixed-point existence argument for -Carathéodory systems. A local existence theory is established for multi-dimensional systems with , and the theory is applied to a buoyant miscible jet/plume model, recasting density variations in terms of a -derivative to obtain local solutions without distributions. Overall, the paper broadens the Stieltjes framework beyond monotone derivators, enabling rigorous treatment of physical processes with sign-changing variation and non-smooth density effects.

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 -continuous functions, and an explicit expression of -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.
Paper Structure (9 sections, 25 theorems, 101 equations)

This paper contains 9 sections, 25 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functions of controlled variation
  4. The $g$-continuity and the relative compactness in $BC_g(I,\mathbb R)$
  5. The $g$-derivative
  6. The Fundamental Theorem of Calculus
  7. The exponential map
  8. Existence results
  9. An application to fluid stratification on buoyant miscible jets and plumes

Key Result

Lemma 2.2

The measure $|\mu|_g$ satisfies $|\mu_g|=\mu_g^++\mu_g^-$.

Theorems & Definitions (67)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3: Jordan Decomposition
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 57 more