Asymptotic stability of Landau solutions to Navier-Stokes system under $L^p$-perturbations

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 $3<p<\mathrm{\infty }$.