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 del u is an element of L-r(0, infinity; (M) over dot(p,q)(R-3)), 2/r + 3/p = 2, 3/2 < p < infinity, 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)(-gamma/2) <= parallel to v(t) - u(t)parallel to(L2) <= C-2(1 + t)(-gamma/2), for large t > 1, 2 < gamma < 5/2. The findings are mainly based on the developed Fourier splitting methods and iterative process. (C) 2017 Elsevier Inc. All rights reserved.