Stationary solutions for dyadic mixed model of the Euler equation. A complete spectrum

TL;DR

This paper analyzes dyadic models of the Euler equations, providing a complete spectrum of existence and uniqueness results for stationary solutions, including constant and self-similar solutions, extending previous partial findings.

Contribution

It offers a comprehensive classification of stationary solutions for a generalized dyadic Euler model, advancing understanding of their structure and properties.

Findings

Complete spectrum of existence and uniqueness for stationary solutions.

Identification of conditions for constant and self-similar solutions.

Extension of previous partial results to a full classification.

Abstract

Dyadic models of the Euler equations were introduced as toy models to study the behaviour of an inviscid fluid in turbulence theory. In 1974 Novikov proposed a generalized mixed dyadic model that extends both Katz-Pavlovic and Obukhov models giving birth to a more complex structure: no results were found in literature until 2015 where blow up in finite time for smooth solutions and existence of self-similar solution for particular values of the model parameters were shown by Jeong I.J. We extend such partial results by giving a complete spectrum of existence and uniqueness results for two cardinal classes of finite energy stationary solutions, namely constant and self-similar solutions.