L2(Rn) 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 × n matrix, trM = 0, M2 is symmetric and u0 ∈ L2(Rn) ∩ Ln(Rn). Under these conditions, we consider v(t) := u(t) − eu0, 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 Dβv(t) on the L2(Rn) norm like t− |β|−1 2 − n 4 for all |β| > 0. c ©2017 All rights reserved.