Energy dissipation and self-similar solutions for an unforced inviscid dyadic model

TL;DR

This paper studies an inviscid shell model of fluid dynamics without external forcing, proving energy dissipation, decay rates, and classifying self-similar solutions, revealing phenomena like coalescence and blow-up.

Contribution

It introduces and classifies self-similar solutions for an inviscid shell model, demonstrating energy decay and phenomena such as coalescence and blow-up.

Findings

Energy dissipation for positive solutions is proved.

Energy decays like t^{-2} over time.

Existence and classification of self-similar solutions are established.

Abstract

A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved and decay of energy like $t^{-2}$ is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.