  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.
