Spacetime decay of mild solutions and conditional quantitative transfer of regularity of the incompressible Navier--Stokes Equations from $\mathbb{R}^n$ to bounded domains

Abstract

We are concerned with the "transfer of regularity" phenomenon for the incompressible Navier--Stokes Equations (NSE) in dimension $n \geq 3$; that is, the strong solutions of NSE on $\mathbb{R}^n$ can be nicely approximated by those on sufficiently large domains $\Omega \subset \mathbb{R}^n$ under the no-slip boundary condition. Based on the space-time decay estimates of mild solutions of NSE established by [On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac. 43 (2000);$L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985)] and others, we obtain quantitative estimates for the ``transfer of regularity'' on higher-order derivatives of velocity and pressure under the smallness assumptions of the Stokes' system and/or the initial velocity, thus complementing the results obtained by [Using periodic boundary conditions to approximate the Navier-Stokes equations on $\mathbb{R}^n$ and the transfer of regularity, Nonlinearity 34 (2021)] and [Quantitative transfer of regularity of the incompressible Navier-Stokes equations from $\Bbb R^3$ to the case of a bounded domain, J. Math. Fluid Mech. 23 (2021)].