Upper bound of decay rate for solutions to the Navier-Stokes-Voigt equations in R3

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the Navier-Stokes-Voigt equations in R3. Then we combine the Fourier splitting method of Schonbek and the Gronwall inequality to prove that the solutions have the following decay rates‖∇mu(x,t)‖2+‖∇m+1u(x,t)‖2⩽c(1+t)-3/2-m,for largetwhen u0∈Hm(R3)∩L1(R3) and m=0,1.