<p>Let us consider the spatial pointwise behavior of time-periodic solutions to the Navier-Stokes equation in the exterior of a rigid body, moving by time-periodic motion. For the translational and angular velocity of the body, assuming besides smallness and regularity, either of the following conditions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. If time average over a period of translational velocity, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> (say), is non-zero (resp. zero), we then show that gradient of the velocity of the fluid decays like the one of the gradient of the Oseen fundamental solution (resp. decays at the rate <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(|x|^{-2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>). As applications, in the case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we show the attainability of the time-periodic solution for small data. In the case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the stability/attainability of the time-periodic solution with sharp decay properties are also deduced.</p>

Spatial Pointwise Behavior of Gradient of Navier-Stokes Flow Around a Rigid Body Moving by Time-Periodic Motion, with Applications to Stability/attainability of time-periodic Flow

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发表日期：

2026年4月3日

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Tomoki Takahashi

摘要

Let us consider the spatial pointwise behavior of time-periodic solutions to the Navier-Stokes equation in the exterior of a rigid body, moving by time-periodic motion. For the translational and angular velocity of the body, assuming besides smallness and regularity, either of the following conditions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. If time average over a period of translational velocity, $\lambda $ $λ$ (say), is non-zero (resp. zero), we then show that gradient of the velocity of the fluid decays like the one of the gradient of the Oseen fundamental solution (resp. decays at the rate $O(|x|^{-2})$ $O (| x |^{- 2})$ ). As applications, in the case $\lambda =0$ $λ = 0$ , we show the attainability of the time-periodic solution for small data. In the case $\lambda \ne 0$ $λ \neq 0$ , the stability/attainability of the time-periodic solution with sharp decay properties are also deduced.