The optimal upper and lower bounds of convergence rates for the 3D Navier–Stokes equations under large initial perturbation

Abstract This paper is concerned with the optimal algebraic convergence rates for Leray weak solutions of the 3D Navier–Stokes equations in Morrey space. It is shown that if the global Leray weak solution u ( x , t ) of the 3D Navier–Stokes equations satisfies ∇ u ∈ L r ( 0 , ∞ ; M ˙ p , q ( R 3 ) ) , 2 r + 3 p = 2 , 3 2 p ∞ , p ≥ q > 2 , then even for the large initial perturbation, every weak solution v ( x , t ) of the perturbed Navier–Stokes equations converges algebraically to u ( x , t ) with the optimal upper and lower bounds C 1 ( 1 + t ) − γ 2 ≤ ‖ v ( t ) − u ( t ) ‖ L 2 ≤ C 2 ( 1 + t ) − γ 2 , for large  t > 1 , 2 γ 5 2 . The findings are mainly based on the developed Fourier splitting methods and iterative process.