Vortex and the Gradient of Divergence in Sobolev Spaces

2021/12/29 by R. S. Saks, Saks, Romen Semenovich
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2201.08818

openalex publication_date 2021/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The properties of the vortex and the gradient of divergence operators ( rot and ∇ div ) are studied in the space \mathbf L₂ (G) in a bounded domain G ⊂ \textrm R³ with a smooth boundary Γ and in the Sobolev spaces: C(2k, m)(G)≡ A^2k(G) ⊕ W^m(G). S.L. Sobolev studied boundary value problems for the scalar polyharmonic equation Δ^m u=ρ in the spaces W₂^m(Ω) with a generalized right-hand side and laid the foundation for the theory of these spaces. Its constructions have matrix analogs, here are some of them. Analogues of the spaces W₂^(m)(G) in the classes \mathcal A and \mathcal B are the space A^2k(G) and W^m(G) of orders 2k> 0 and m> 0 , and \mathbf A^-2k (G) and their dual spaces W^{- m}(G) . Pairs of spaces form a net of Sobolev spaces, its elements are classes C(2k, m)(G)≡ A^2k(G) ⊕ W^m(G); the class C(2k, 2k)coincides with the Sobolev space H^2k(G). They belong to L₂(G), if k≥ 0 and m≥ 0. A wide field of problems has opened up: studying the operators (rot)^p, (∇ div)^p for p = 1,2, ..., and others in the network Sobolev spaces.

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• Fourier series of the ∇ div operator and Sobolev spaces II
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