Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions

In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L ₂(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot u + λu = f and the Stokes equation −ν(Δv + λ ²v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276.