A dyadic model on a tree

TL;DR

This paper analyzes a tree-structured infinite system of nonlinear differential equations modeling turbulence, proving existence, dissipation, and stationary solutions, thus providing insights into simplified turbulence models related to Euler and Navier-Stokes equations.

Contribution

It introduces rigorous mathematical results for a dyadic tree model of turbulence, including existence, dissipation, and stationary solutions, advancing understanding of simplified fluid dynamics models.

Findings

Existence of finite energy solutions

Anomalous dissipation in the inviscid case

Existence and uniqueness of stationary solutions

Abstract

We study an infinite system of non-linear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3d Euler and Navier-Stokes equations in a rough approximation of a wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.