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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; that is, the strong solutions of NSE on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be nicely approximated by those on sufficiently large domains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subset \mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the no-slip boundary condition. Based on the spacetime decay estimates of mild solutions of NSE established by Miyakawa [<italic>On space-time decay properties of nonstationary incompressible Navier-Stokes flows in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula></italic>, Funkcial. Ekvac. 43 (2000), no. 3, 541–557], Schonbek [<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>decay for weak solutions of the Navier-Stokes equations</italic>, 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 [<italic>Using periodic boundary conditions to approximate the Navier-Stokes equations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the transfer of regularity</italic>, Nonlinearity 34 (2021), no. 11, 7683–7704] and Ożánski [<italic>Quantitative transfer of regularity of the incompressible Navier-Stokes equations from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the case of a bounded domain</italic>, J. Math. Fluid Mech. 23 (2021), no. 4, Paper No. 98, 14].