An L²-quantitative global approximation for the Stokes initial-boundary value problem

2025/11/20 by Higaki, Mitsuo
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.16079

Abstract

We establish the first quantitative Runge approximation theorem, with explicit L²-estimates, for the 3d nonstationary Stokes system on a bounded spatial domain. This result addresses the two primary limitations of the qualitative result [H.-Sueur, 2025] obtained in collaboration with Franck Sueur: first, it bypasses the non-constructive Hahn-Banach theorem used in [H.-Sueur, 2025], precluding quantitative estimates; and second, it extends the scope of the theory from interior approximations to the physically important initial-boundary value problem. Our proof is founded on the modern quantitative framework of [Rüland-Salo, 2019], which we adapt to the Stokes system by combining semigroup theory with a quantitative approximation for the associated resolvent problem.

Citations

• A Runge-type approximation theorem for the 3D unsteady Stokes system
• An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows
• Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators
• Approximation theorems for the Schr "odinger equation and quantum vortex\n reconnection
• Quantitative Approximation Properties for the Fractional Heat Equation
• Quantitative Runge Approximation and Inverse Problems
• Local approximation of arbitrary functions by solutions of nonlocal equations
• Analyticity of the Stokes semigroup in spaces of bounded functions
• Vector spherical harmonics and their application to magnetostatics
• Zur Theorie der Eindeutigen Analytischen Functionen

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