Approximation of divergence-free vector fields vanishing on rough planar sets

2024/09/15 by Giacomo Del Nin, Bian Wu, Del Nin, Giacomo +1
Engineering · Mathematics · #31A99 #41A30 #46E35 #76D07 #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2409.09880

openalex publication_date 2024/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given any divergence-free vector field of Sobolev class W^m,p₀(Ω) in a bounded open subset Ω⊂ ℝ², we are interested in approximating it in the W^m,p norm with divergence-free smooth vector fields compactly supported in Ω. We show that this approximation property holds in the following cases: For p>2, this holds given that ∂ Ω has zero Lebesgue measure (a weaker but more technical condition is sufficient); For p ≤ 2, this holds if Ω^c can be decomposed into finitely many disjoint closed sets, each of which is connected or d-Ahlfors regular for some d∈[0,2). This has links to the uniqueness of weak solutions to the Stokes equation in Ω. For Hölder spaces, we prove this approximation property in general bounded domains.

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