Quantitative Transfer of Regularity of the Incompressible Navier–Stokes Equations from R3 to the Case of a Bounded Domain

Let u 0 ∈ C 0 5 ( B R 0 ) be divergence-free and suppose that u is a strong solution of the three-dimensional incompressible Navier–Stokes equations on [0, T ] in the whole space R 3 such that ‖ u ‖ L ∞ ( ( 0 , T ) ; H 5 ( R 3 ) ) + ‖ u ‖ L ∞ ( ( 0 , T ) ; W 5 , ∞ ( R 3 ) ) ≤ M < ∞ . We show that then there exists a unique strong solution w to the problem posed on B R with the homogeneous Dirichlet boundary conditions, with the same initial data and on the same time interval for R ≥ max ( 1 + R 0 , C ( a ) C ( M ) 1 / a exp ( C M 4 T / a ) ) for any a ∈ [ 0 , 3 / 2 ) , and we give quantitative estimates on u - w and the corresponding pressure functions.