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

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∈Lr(0,∞;M˙p,q(R3)),2r+3p=2,322, 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 boundsC1(1+t)−γ2≤‖v(t)−u(t)‖L2≤C2(1+t)−γ2,for large t>1,2<γ<52. The findings are mainly based on the developed Fourier splitting methods and iterative process.