Stochastic turbulence for Burgers equation driven by cylindrical L\'evy process

TL;DR

This paper studies the effects of cylindrical Lévy noise on the stochastic turbulence and regularity of solutions in the one-dimensional viscous Burgers equation, including the inviscid limit.

Contribution

It provides a rigorous analysis of solution regularity, statistical properties, and turbulence behavior under Lévy noise, especially as viscosity approaches zero.

Findings

Established regularity and statistical properties of solutions.

Analyzed the energy spectrum and structure functions of turbulence.

Described the inviscid limit as a regime of strong stochastic turbulence.

Abstract

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by L\'evy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities are such as the moment estimate, the structure function and the energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity {\nu} tends towards zero. The inviscid limit describes the strong stochastic turbulence.