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

Ophélie Cuvillier (1, 2) , Francesco Fanelli (2, 1) , Elena Salguero (3)

1 ICJ - Institut Camille Jordan
2 EDPA - Équations aux dérivées partielles, analyse
3 Universidad de Sevilla = University of Seville

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.

Domaines: Mathématiques [math]

Ophélie Cuvillier, Francesco Fanelli, Elena Salguero. Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces. Journal of Evolution Equations, 2023, pp.68. ⟨10.1007/s00028-023-00914-x⟩. ⟨hal-04149696⟩