On $p$-adic cascade equations of hydrodynamic type in modeling fully developed turbulence

TL;DR

This paper introduces a $p$-adic hydrodynamic cascade model for turbulence energy dissipation, showing stationary solutions that align with the Kolmogorov-Obukhov law, and discusses potential future research directions.

Contribution

It proposes a novel $p$-adic cascade equation with integrals of motion, modeling turbulence energy dissipation and connecting to classical turbulence laws.

Findings

Stationary solutions match the 2/3 Kolmogorov-Obukhov law.

Special cases of the equation are analyzed.

The model offers a new mathematical framework for turbulence modeling.

Abstract

A $p$-adic hydrodynamic type equation with two integrals of motion is proposed. It can be considered as a model cascade equation for energy dissipation in fully developed turbulence. Some of special cases of the proposed equation are detailed and it is shown that for a specific choice of parametrs they have stationary solutions that correspond to the 2/3 Kolmogorov-Obukhov law. Possible further studies of the proposed model are discussed.