Global existence of smooth solutions to three-dimensional turbulent flow equations

TL;DR

This paper proves the global existence of smooth solutions for three-dimensional turbulent flow equations in compressible fluids, using energy estimates and novel techniques, marking the first such result for the $k$-$psilon$ model.

Contribution

It establishes the first global existence result for the $k$-$psilon$ turbulence model equations in 3D compressible flows under near-equilibrium initial data.

Findings

Global well-posedness proved for initial data close to equilibrium

New techniques developed for handling product of functions and higher order norms

First result on $k$-$psilon$ model equations in 3D compressible turbulence

Abstract

In this paper we are concerned with the global existence of smooth solutions to the turbulent flow equations for compressible flows in $\mathbb{R}^3$. The global well-posedness is proved under the condition that the initial data are close to the standard equilibrium state in $H^3$-framework. The proof relies on energy estimates about velocity, temperature, turbulent kinetic energy and rate of viscous dissipation. We use several new techniques to overcome the difficulties from the product of two functions and higher order norms. This is the first result concerning $k$-$\varepsilon$ model equations.