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, i.e. we consider the case \(k \ge 0\); 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 nonlinear terms involved in the computations.

在本文中，我们研究了周期域\(\mathbb {T}^d\)中湍流的柯尔莫哥洛夫二方程模型的适定性，空间维度为\(d=2,3\)。我们承认平均湍流动能k在部分域中消失，即我们考虑\(k \ge 0\)的情况；在这种情况下，方程的抛物线结构变得简并。对于这个系统，我们证明了 Sobolev 空间\(H^s\)中的局部适定性结果，对于任何\(s>1+d/2\)。我们期望这种规律性是最优的，因为当\(k \approx 0\)时系统具有简并性。我们还证明了连续性标准，并为解决方案的寿命提供了下限。结果的证明基于 Littlewood-Paley 分析和圆环上的准微分计算，以及计算中涉及的非线性项的精确换向器分解。