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

We prove the existence of the Helmholtz decomposition Lq(Ωp,Cd)=Lσq(Ωp)⊕Gq(Ωp) for periodic domains Ωp⊆Rd with respect to a lattice L⊆Rd, i.e. Ωp=Ωp+z for all z∈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.