Spacetime decay of mild solutions and conditional quantitative transfer of regularity of the incompressible Navier–Stokes Equations from ℝⁿ to bounded domains

This paper is motivated by the “transfer of regularity” phenomenon for the incompressible Navier–Stokes equations (NSE) in dimension n ≥ 3 n ≥ 3 ; that is, the strong solutions of NSE on R n R^n can be nicely approximated by those on sufficiently large domains Ω ⊂ R n Ω ⊂ R^n under the no-slip boundary condition. Based on the spacetime decay estimates of mild solutions of NSE established by Miyakawa [On space-time decay properties of nonstationary incompressible Navier-Stokes flows in R n R^n , Funkcial. Ekvac. 43 (2000), no. 3, 541–557], Schonbek [ L 2 L^2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985), no. 3, 209–222], and others, we obtain quantitative estimates on higher-order derivatives of velocity and pressure for the incompressible Navier–Stokes flow on large domains under certain additional smallness assumptions of the Stokes system and/or the initial velocity, thus complementing the results obtained by Robinson [Using periodic boundary conditions to approximate the Navier-Stokes equations on R 3 R^3 and the transfer of regularity, Nonlinearity 34 (2021), no. 11, 7683–7704] and Ożánski [Quantitative transfer of regularity of the incompressible Navier-Stokes equations from R 3 R^3 to the case of a bounded domain, J. Math. Fluid Mech. 23 (2021), no. 4, Paper No. 98, 14].