Well-posedness and singularity formation for the Kolmogorov two-equation model of turbulence in 1-D

TL;DR

This paper proves local well-posedness of the 1-D Kolmogorov turbulence model even with zero initial energy and demonstrates that solutions generally develop finite-time singularities, highlighting fundamental behaviors of the model.

Contribution

It establishes the first well-posedness results for vanishing initial data and shows finite-time blow-up for the Kolmogorov two-equation turbulence model in one dimension.

Findings

Well-posedness in Sobolev spaces for zero initial energy

Finite-time singularity formation in solutions

First such results for this turbulence model

Abstract

We study the Kolomogorov two-equation model of turbulence in one space dimension. Two are the main results of the paper. First of all, we establish a local well-posedness theory in Sobolev spaces even in the case of vanishing mean turbulent kinetic energy. Then, we show that, in general, those solutions must blow up in finite time. To the best of our knowledge, these results are the first establishing the well-posedness of the system for vanishing initial data and the occurence of finite time singularities for the model under study.