Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An $L^2$-quantitative global approximation for the Stokes initial-boundary value problem

Mitsuo Higaki

TL;DR

This work delivers the first quantitative $L^2$ Runge-type approximation for the 3D nonstationary Stokes system on bounded domains, addressing two key gaps of prior work: a non-constructive Hahn–Banach-based approach and the restriction to interior approximations. By adapting the Rüland–Salo quantitative framework to the Stokes setting and employing a Dunford integral representation of the Stokes semigroup, the authors construct global approximations that decompose into a Stokes resolvent part and a heat resolvent part, yielding explicit $L^2$-estimates linked to the initial data. The results overcome the qualitative nature of previous interior-only theorems and provide constructive, time-dependent and finite-time approximations with explicit growth bounds, at the cost of exponential-in-time behavior in the global scheme. These advances have potential implications for inverse problems and control in fluid dynamics, where precise quantitative Runge-type approximations are essential.

Abstract

We establish the first quantitative Runge approximation theorem, with explicit $L^2$-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.

An $L^2$-quantitative global approximation for the Stokes initial-boundary value problem

TL;DR

This work delivers the first quantitative Runge-type approximation for the 3D nonstationary Stokes system on bounded domains, addressing two key gaps of prior work: a non-constructive Hahn–Banach-based approach and the restriction to interior approximations. By adapting the Rüland–Salo quantitative framework to the Stokes setting and employing a Dunford integral representation of the Stokes semigroup, the authors construct global approximations that decompose into a Stokes resolvent part and a heat resolvent part, yielding explicit -estimates linked to the initial data. The results overcome the qualitative nature of previous interior-only theorems and provide constructive, time-dependent and finite-time approximations with explicit growth bounds, at the cost of exponential-in-time behavior in the global scheme. These advances have potential implications for inverse problems and control in fluid dynamics, where precise quantitative Runge-type approximations are essential.

Abstract

We establish the first quantitative Runge approximation theorem, with explicit -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.

Paper Structure

This paper contains 4 sections, 9 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Global approximation for the Stokes resolvent problem
  4. Proof of Theorems

Key Result

Proposition 2.1

Let $D \subset \mathbb{R}^3$ be a bounded domain whose complement $\mathbb{R}^3\setminus D$ is connected. For $\lambda\in \Sigma_{\pi-\delta} \cap \{|\lambda| \ge 1\}$ with $\delta\in(0,\pi/2)$, let $v\in H^2(D)^3$ and $q\in H^1(D)$ satisfy Then, for any $\varepsilon>0$, there exists $(u,p)$ solving such that $u$ approximates $v$ in $D$ as Moreover, $u$ can be decomposed into $u = u_1 + u_2$, w

Theorems & Definitions (10)

  • Remark 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3: logarithmic stability
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9