L²(R^n) estimate of the solution to the Navier-Stokes equations with linearly growth initial data

In this article, we consider the incompressible Navier-Stokes equations with linearly growing initial data U0 := u0(x) − Mx. Here M is an n ˣ n matrix, trM = 0, M² is symmetric and u0 ϵ L²(R^n) ∩ L^n(R^n). Under these conditions, we consider v(t) := u(t) − e−^tA u0, where u(x) := U(x) − Mx and U(x) is the mild solution of the incompressible Navier-Stokes equations with linearly growing initial data. We shall show that D ^β v(t) on the L²(R^n) norm like t- |β|−1- n/4 for all |β| ≥ 0.