# Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics

## TL;DR

This paper uses Littlewood-Paley theory to define determining wavenumbers for the Hall-MHD system, demonstrating that strong solutions exhibit almost finite dimensional long-term behavior based on bounded average wavenumbers.

## Contribution

It introduces a new method to identify determining wavenumbers for Hall-MHD using Littlewood-Paley theory, linking boundedness to long-term solution behavior.

## Key findings

- Long-term behavior of strong solutions is almost finite dimensional.
- Determining wavenumbers are bounded in certain average senses.
- Method provides a new perspective on Hall-MHD dynamics.

## Abstract

Using Littlewood-Paley theory, one formulates the determining wavenumbers for the Hall-MHD system, defined for each individual solution $(u,b)$. It is shown that the long time behaviour of strong solutions is almost finite dimensional as the wavenumbers are bounded in certain average senses.