Decomposition of vector-valued divergence free Sobolev functions and shape optimization for stationary Navier-Stokes equations

2005/10/31 by Gengsheng Wang, Wang, Gengsheng, Donghui Yang +1
Computer Science · Engineering · Mathematics · #35Q30: 49Q30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #math.OC #msc:49Q30

paper · pdf · doi:10.48550/arxiv.math/0510669

25 pages, 0 figures, 15 conference

arxiv created 2005/10/31 · openalex publication_date 2005/10/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a divergence free partition for vector-valued Sobolev functions with free divergence in \bf Rⁿ, n≥ 1. We prove that for any domain \om of class \cal C in \bf Rⁿ,n=2,3, the space D₀¹(\om)≡\v ∈ H¹₀(Ω)ⁿ ; divv=0\ and the space H_0,σ¹(\om)≡ \v∈ C^∞₀(Ω)ⁿ;div\mathbf v=0\^‖⋅‖_H¹(\om)ⁿ, which is the completion of \v ∈ C^∞₀(Ω)ⁿ; div\mathbf v=0\ in the H¹(Ω)ⁿ-norm, are identical. We will also prove that H_0,σ¹(D∖\om)=\\mathbf v∈ H_0,σ¹(D); \mathbf v=0 a.e. in \om\, where D is a bounded Lipschitz domain such that \om⊂⊂ D. These results, together with properties for domains of class \mathcal C, are used to solve an existence problem in the shape optimization theory of the stationary Navier-Stokes equations.

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