Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces

TL;DR

This paper establishes local well-posedness of the Kolmogorov two-equation turbulence model in Sobolev spaces, addressing degeneracy when turbulent kinetic energy vanishes, using advanced harmonic analysis techniques.

Contribution

It proves the well-posedness of the model in optimal Sobolev spaces considering degeneracy, and provides lifespan estimates using Littlewood-Paley and paradifferential calculus.

Findings

Well-posedness in Sobolev spaces for the model with degeneracy

Optimal regularity threshold identified as s > 1 + d/2

Lower bounds for solution lifespan established

Abstract

In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain $\mathbb{T}^d$, for space dimensions $d=2,3$. We admit the average turbulent kinetic energy $k$ to vanish in part of the domain, \textsl{i.e.} we consider the case $k \geq 0$; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces $H^s$, for any $s>1+d/2$. We expect this regularity to be optimal, due to the degeneracy of the system when $k \approx 0$. We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the non-linear terms involved in the computations.