  In this paper, we show that Landau solutions to the Navier-Stokes system are
asymptotically stable under $L^3$-perturbations. We give the local
well-posedness of solutions to the perturbed system with initial data in
$L_{\sigma}^3$ space and the global well-posedness with small initial data in
$L_{\sigma}^3$ space, together with a study of the $L^q$ decay for all $q>3.$
Moreover, we have also studied the local well-posedness, global well-posedness
and stability in $L^p$ spaces for $3<p<\infty$.
