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

TL;DR

This paper studies how regularity properties of solutions to the Navier-Stokes equations in unbounded space can be transferred to large bounded domains, using decay estimates and smallness conditions to quantify this transfer.

Contribution

It provides new quantitative estimates for the transfer of higher-order regularity from the whole space to bounded domains under smallness assumptions.

Findings

Established decay estimates for mild solutions of NSE.

Derived quantitative bounds for regularity transfer.

Complemented previous results with explicit estimates.

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)].