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<p>We investigate Runge-type approximation theorems for solutions to the 3D unsteady Stokes system. More precisely, we establish that on any compact set with connected complement, local smooth solutions to the 3D unsteady Stokes system can be approximated with an arbitrarily small positive error in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5364_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> norm by a global solution of the 3D unsteady Stokes system, where the velocity grows at most exponentially at spatial infinity and the pressure grows polynomially. Additionally, by considering a parasitic solution to the Stokes system, we establish that some growths at infinity are indeed necessary. These results markedly differ from the Runge-type theorem for the heat equation in Enciso–García-Ferrero–Peralta-Salas (Duke Math J 168(5):897–939, 2019), where the approximations with decay at infinity can be achieved.</p>

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A Runge-Type Approximation Theorem for the 3D Unsteady Stokes System

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  • 发表日期：

    2025年7月2日

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  • Mitsuo Higaki,
  • Franck Sueur

摘要

We investigate Runge-type approximation theorems for solutions to the 3D unsteady Stokes system. More precisely, we establish that on any compact set with connected complement, local smooth solutions to the 3D unsteady Stokes system can be approximated with an arbitrarily small positive error in \(L^\infty \) L ∞ norm by a global solution of the 3D unsteady Stokes system, where the velocity grows at most exponentially at spatial infinity and the pressure grows polynomially. Additionally, by considering a parasitic solution to the Stokes system, we establish that some growths at infinity are indeed necessary. These results markedly differ from the Runge-type theorem for the heat equation in Enciso–García-Ferrero–Peralta-Salas (Duke Math J 168(5):897–939, 2019), where the approximations with decay at infinity can be achieved.