Optimal convergence rates for the three-dimensional turbulent flow equations

In this paper we are concerned with the convergence rate of solutions to the three-dimensional turbulent flow equations. By combining the $L^p$-$L^q$ estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space, when the initial perturbation of the equilibrium state is small in $H^3$-framework. More precisely, the optimal convergence rates of the solutions and its first order derivatives in $L^2$-norm are obtained when the $L^p$-norm of the perturbation is bounded for some $p\in[1, 6/5)$.