On the regularity and convergence of solutions to the 3D Navier–Stokes–Voigt equations

Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces

“…)-(44), we readily obtaind dt y(t) ≤ hy(t) + G,wherey(t) = w 2 + α w 2 1 , h(t) = C(1 + v 1 + u 0 1 ), G(t) = C v4 Since (w(τ ), η τ ) = (0, 0), the Gronwall lemma implies thaty(t) ≤ t τ G(s)e C tτ h(r)dr ds ≤ e C t τ hhand, from (37) and (39), we learn thatt τ h(s)ds ≤ t τ C(1 + v(s) 1 + u 0 (s) 1 )ds ≤ C(R 0 + )(t − τ + 1),…”

Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay

“…Introduction. Let Ω ⊂ R 3 be an open bounded domain with smooth boundary ∂Ω. The three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay is the following system:…”

“…The existence, long-time behavior and regularity of solutions to the 3D Navier-Stokes-Voigt equations without delays in bounded domains or unbounded domains satisfying the Poincaré inequality have attracted the attention of many mathematicians [2,3,6,15,16,20,21,30,31,32]. There are many results involving PDEs in fluid mechanics with delay (see e.g., [1,8,9,10,11,12,13,27,28,29]).…”

“…Denote also by H and V , the closures of V in the (L 2 (Ω)) 3 and (H 1 (Ω)) 3 , respectively. Then H and V are Hilbert spaces with inner products given by…”

“…We use • * for the norm in V and •, • V ,V for the dual pairing between V and V . Let P : (L 2 (Ω)) 3 → H be the Helmholtz-Leray projector, and denote by A = −P ∆ the Stokes operator such that Au, v V ,V = ((u, v)), with domain D(A) = (H 2 (Ω)) 3 ∩V . The Stokes operator A is a positive self-adjoint operator with compact inverse.…”