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

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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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Authors and Affiliations

  1. Karlsruhe Institute of Technology (KIT) Institute for Analysis, 76128, Karlsruhe, Germany

    Jens Babutzka & Peer Christian Kunstmann

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  1. Jens Babutzka
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  2. Peer Christian Kunstmann
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Correspondence to Peer Christian Kunstmann.

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The authors declare that they have no conflict of interest.

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Communicated by M. Hieber

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via GRK 1294.

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Babutzka, J., Kunstmann, P.C. \({\varvec{L}}^{\varvec{q}}\)-Helmholtz Decomposition on Periodic Domains and Applications to Navier–Stokes Equations. J. Math. Fluid Mech. 20, 1093–1121 (2018). https://doi.org/10.1007/s00021-017-0356-z

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