Abstract: In this paper, the convergence rates of solutions to the three-dimensional turbulent flow equations are considered. By combining the L^p-L^q estimate 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 the H³-framework. More precisely, the optimal convergence rates of the solutions and their first-order derivatives in the L²-norm are obtained when the L^p-norm of the perturbation is bounded for some p ∈ [1, 6/5 ).

Key words: neural network, centre manifold, pitchfork bifurcation, Hopf bifurcation, model, turbulent flow equation, optimal convergence rate, k-&epsilon, energy estimate