Quantitative transfer of regularity of the incompressible Navier-Stokes equations from $\mathbb{R}^3$ to the case of a bounded domain

Abstract

Let u 0 ∈ C 5 0 (B R0 ) be divergence-free and suppose that u is a strong solution of the threedimensional incompressible Navier-Stokes equations on [0, T ] in the whole spaceWe 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(CM 4 T /a))) for any a ∈ [0, 3/2), and we give quantitative estimates on u − w and the corresponding pressure functions.

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

“…While the results here demonstrate convergence, they give no error estimates; this appears to be a significantly harder problem, but a particularly interesting one if one is to view solving the equations on a periodic domain as a 'numerical approximation' to the solution of the equations on the whole space. Ożański (2021) has recently obtained such error estimates, comparing solutions of the equations on the whole space and on bounded domains with Dirichlet boundary conditions, by finding a way to treat the bounded domain problem as a perturbation of the problem posed on the whole space.…”