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

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, 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.