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arXiv · 2306.16014

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

Ophélie Cuvillier·Francesco Fanelli·Elena Salguero
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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.

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Ophélie Cuvillier, Francesco Fanelli, Elena Salguero. 2023-06-28. Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces. https://arxiv.org/abs/2306.16014

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