Using periodic boundary conditions to approximate the Navier–Stokes equations on R3 and the transfer of regularity

Abstract

Abstract This paper considers solutions u α of the three-dimensional Navier–Stokes equations on the periodic domains Q α ≔ (−α,α)3 as the domain size α → ∞, and compares them to solutions of the same equations on the whole space. For compactly-supported initial data u α 0 ∈ H 1 ( Q α ) , an appropriate extension of u α converges to a solution u of the equations on R 3 , strongly in L r ( 0 , T ; H 1 ( R 3 ) ) , r ∈ [1, ∞). The same also holds when u α 0 is the velocity corresponding to a fixed, compactly-supported vorticity. A consequence is that if an initial compactly-supported velocity u 0 ∈ H 1 ( R 3 ) or an initial compactly-supported vorticity ω 0 ∈ H 1 ( R 3 ) gives rise to a smooth solution on [0, T*] for the equations posed on R 3 , a smooth solution will also exist on [0, T*] for the same initial data for the periodic problem posed on Q α for α sufficiently large; this illustrates a ‘transfer of regularity’ from the whole space to the periodic case.