<p>Consider the motion of a viscous incompressible fluid filling a 3D exterior domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> subject to the Navier slip-with-friction boundary condition as well as outflow at infinity. For the Oseen system as the linearization, we discuss the resolvent set under a certain relationship among the geometry of the boundary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, friction coefficient <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the outflow <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>. We then study the regularity of the resolvent near the origin in the complex plane to develop <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> decay estimates of the Oseen semigroup provided that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha (x)+u_\infty \cdot \nu (x)/2\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mi>∞</mi> </msub> <mo>·</mo> <mi>ν</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mn>2</mn> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x\in \partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\nu (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> stands for the outward unit normal to the boundary <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Large Time Decay of the Oseen Flow in Exterior Domains Subject to the Navier Slip-with-Friction Boundary Condition

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2026年7月25日

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Toshiaki Hishida

摘要

Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $\Omega $ $Ω$ subject to the Navier slip-with-friction boundary condition as well as outflow at infinity. For the Oseen system as the linearization, we discuss the resolvent set under a certain relationship among the geometry of the boundary $\partial \Omega $ $\partial Ω$ , friction coefficient $\alpha (x)$ $α (x)$ and the outflow $u_\infty $ $u_{\infty}$ . We then study the regularity of the resolvent near the origin in the complex plane to develop $L^q$ $L^{q}$ - $L^r$ $L^{r}$ decay estimates of the Oseen semigroup provided that $\alpha (x)+u_\infty \cdot \nu (x)/2\ge 0$ $α (x) + u_{\infty} \cdot ν (x) / 2 \geq 0$ for every $x\in \partial \Omega $ $x \in \partial Ω$ , where $\nu (x)$ $ν (x)$ stands for the outward unit normal to the boundary $\partial \Omega $ $\partial Ω$ .