3 years ago

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

Jens Babutzka, Peer Christian Kunstmann

### 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.

DOI: 10.1007/s00021-017-0356-z

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