Decay rate of solutions to 3D Navier–Stokes–Voigt equations in HmHm spaces

In this paper, we first prove the regularity in Hm(R3)Hm(R3) of weak solutions to the Navier–Stokes–Voigt equations with initial data in HK(R3)HK(R3) for all m≤Km≤K. Then we compute the upper bound of decay rate for these solutions, specifically, we prove that ‖∇mu(t)‖2+‖∇m+1u(t)‖2≤c(1+t)−3/2−m,for large  t, when u0∈Hσm+1(R3)∩L1(R3), m∈Nm∈N.