The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to…

Abstract

We consider the Stokes operator A on smooth exterior domains Ω of ℝⁿ in homogeneous Sobolev spaces Ĥ^κ,q_w(Ω) with radially symmetric Muckenhoupt weights w∈ 𝒜_q. A fundamental property is the existence of a bounded ℋ^∞-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous L^q Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers A^it, t∈ℝ, and the characterization of domains of fractional powers A^θ equipped with nonhomogeneous (| u|_{L^q_w} + |A^θu|_{L^q_w}) as well as homogeneous norm (|A^θu|_{L^q_w}) as complex interpolation spaces. The final aim is the identification with homogeneous spaces 𝒟̂((−Δ_q,w)^θ) = [L^q_w,𝒟̂(−Δ_q,w)]_θ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted L^q-L^r decay estimates of the Stokes semigroup and L^p-maximal regularity on L^q_σ,w(Ω).

Weighted L^q-L^r estimates of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes flow past a rigid body
Yosuke Asami
arXiv1 Nov 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Characterization of the Boundedness of a Generalized Fractional Maximal Operator in Weighted Lorentz Spaces for Certain Parameter Values
Azhar Abek, Amiran Gogatishvili, Nurzhan Bokayev
Mathematics2 Nov 2026Peer-reviewedCC BY 4.0General mathematicsCrossref
Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces
Amiran Gogatishvili, Nurzhan A. Bokayev, Nurgul K. Kuzeubayeva
arXiv5 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation
Zhihui Zhang
arXiv3 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem
Mukul Dwivedi, Jesse Railo, Andreas Rupp
arXiv7 Oct 2026PreprintCC0Full textNumerical analysis & scientific computingarXiv

Cite

BibTeX RIS

BibTeX

@misc{farwig2026stokes,
  title = {{The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to $\mathscr H^\infty$-calculus to Fractional Domains}},
  author = {Reinhard Farwig and Kazuyuki Tsuda},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29002},
  eprint = {2608.29002},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29002},
}

RIS

TY  - UNPB
TI  - The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to ℋ^∞-calculus to Fractional Domains
AU  - Reinhard Farwig
AU  - Kazuyuki Tsuda
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29002
UR  - https://arxiv.org/abs/2608.29002
AB  - We consider the Stokes operator A on smooth exterior domains Ω of ℝ^n in homogeneous Sobolev spaces Ĥ^κ,q_w(Ω) with radially symmetric Muckenhoupt weights w∈ 𝒜_q. A fundamental property is the existence of a bounded ℋ^∞-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous L^q Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers A^it, t∈ℝ, and the characterization of domains of fractional powers A^θ equipped with nonhomogeneous (| u|_L^q_w + |A^θu|_L^q_w) as well as homogeneous norm (|A^θu|_L^q_w) as complex interpolation spaces. The final aim is the identification with homogeneous spaces 𝒟̂((−Δ_q,w)^θ) = [L^q_w,𝒟̂(−Δ_q,w)]_θ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted L^q-L^r decay estimates of the Stokes semigroup and L^p-maximal regularity on L^q_σ,w(Ω).
ER  -

Abstract

Related

Cite