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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1093–1121 | Cite as

\({\varvec{L}}^{\varvec{q}}\)-Helmholtz Decomposition on Periodic Domains and Applications to Navier–Stokes Equations

  • Jens Babutzka
  • Peer Christian Kunstmann
Article

Abstract

We prove the existence of the Helmholtz decomposition \( L^q(\Omega _{\mathrm {p}},\mathbb {C}^d)=L_\sigma ^q(\Omega _{\mathrm {p}})\oplus G^q(\Omega _{\mathrm {p}})\) for periodic domains \(\Omega _{\mathrm {p}}\subseteq \mathbb {R}^d\) with respect to a lattice \(L\subseteq \mathbb {R}^d\), i.e. \(\Omega _{\mathrm {p}}=\Omega _{\mathrm {p}}+z\) for all \(z\in L\), and for a suitable range of q depending on the regularity of the boundary. The proof of the Helmholtz decomposition builds upon recent Bloch multiplier theorems due to B. Barth. We give several applications to Stokes operators and Navier–Stokes equations on such domains.

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Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology (KIT) Institute for AnalysisKarlsruheGermany

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