On the regularity of the Navier–Stokes equation in a thin periodic domain

We address the global regularity of solutions of the Navier–Stokes equations in a thin domain Ω=[0,L1]×[0,L2]×[0,ϵ]Ω=[0,L1]×[0,L2]×[0,ϵ] with periodic boundary conditions, where L1,L2>0L1,L2>0 and ϵ∈(0,1/2)ϵ∈(0,1/2). We prove that if‖∇u0‖L2(Ω)⩽νC(L1,L2)ϵ1/2|logϵ|3/2 where u0u0 is the initial datum, then there exists a unique global smooth solution with the initial datum u0u0. Also, if‖u0‖H˙1/2(Ω)⩽νC(L1,L2)|logϵ|1/4 the global regularity of the corresponding solution holds.