Backwards semi-martingales into Burgers' turtulence

TL;DR

This paper models turbulence in the inviscid Burgers equation using Markov processes for turbulent intervals and clusters, revealing that the associated velocity processes are backward semi-martingales, thus linking turbulence dynamics with stochastic processes.

Contribution

It introduces a novel Markov process framework for turbulent intervals and clusters in Burgers' turbulence, showing their velocity processes are backward semi-martingales.

Findings

Markov processes describe turbulent interval and cluster dynamics.

Velocity processes are backward semi-martingales.

Provides stochastic process perspective on Burgers' turbulence.

Abstract

In fluid dynamics governed by the one dimensional inviscid Burgers equation $\partial_t u+u\partial_x(u)=0$, the stirring is explained by the sticky particles model. A Markov process $([Z^1_t,Z^2_t],\,t\geq0)$ describes the motion of random turbulent intervals which evolve inside an other Markov process $([Z^3_t,Z^4_t],\,t\geq0)$, describing the motion of random clusters concerned with the turbulence. Then, the four velocity processes $(u(Z^i_t,t),\,t\geq0)$ are backward semi-martingales.