Premi\`ere approche de la densit\'e d'un op\'erateur de Perron Frobenius. III - Applications : EDP, EDO, etc

TL;DR

This paper introduces a novel approach to invariant densities of Perron-Frobenius operators, focusing on their asymptotic behaviors in differential equations and applications to complex dynamical systems.

Contribution

It presents a new method for analyzing invariant densities and their asymptotic behaviors in PDEs and ODEs, with applications to fluid dynamics and chaos theory.

Findings

Asymptotic profiles of Lorenz, Navier-Stokes, and Hamilton's equations identified.

Partially linear operators allow for asymptotic solutions via random distributions.

The approach provides insights into the long-term behavior of complex dynamical systems.

Abstract

First approach of invariant densities of a Perron Frobenius operator. Asymptotic behaviours of ODE or PDE, as, are most interesting. The associed infinitesimal iteration is. If is partially linear, a random distribution can be asymptotic solution. Among applications, are asymptotic profiles of of Lorenz, Navier Stokes or Hamilton's \'equations.