We study a simplified equation governing turbulent kinetic energy in a bounded domain, arising from turbulence modeling where the eddy diffusion is given by ρ(x)+ε, with ρ representing the Prandtl mixing length of the order of the distance to the boundary, and a right-hand side in L 1. We establish the convergence toward the formal limit equation as ε approaches 0, within fractional Sobolev spaces W 1/2,q .