# A mode of convergence arising in diffusive relaxation

## TL;DR

This paper introduces a new mode of convergence for measurable functions, proves its completeness, and explores its preservation under composition, motivated by convergence results in diffusive relaxation systems.

## Contribution

It defines a novel convergence mode, establishes its mathematical properties, and connects it to the convergence of densities in Euler and diffusion systems.

## Key findings

- New mode of convergence for measurable functions
- Completeness of the convergence mode
- Application to convergence of densities in relaxation limits

## Abstract

In this work, a mode of convergence for measurable functions is introduced. A related notion of Cauchy sequence is given and it is proved that this notion of convergence is complete in the sense that Cauchy sequences converge. Moreover, the preservation of convergence under composition is investigated. The origin of this mode of convergence lies in the path of proving that the density of a Euler system converges almost everywhere (up to subsequences) towards the density of a non-linear diffusion system, as a consequence of the convergence in the relaxation limit.