ABSTRACT
This paper begins by developing the weak formulation of a spectral problem for the Stokes operator. A classical description of the Stokes eigenproblem is given in chapter 4 of Constantin and Foias [4]. The development there is done in the setting of a closed, densely defined linear operator on the Hilbert space of L2-vector fields on Ω. Another version is described in section 2.6 of chapter 1 of Temam [8]. Here the Stokes eigenfields will be constructed using a direct variational characterization on a natural Sobolev-Hilbert space of vector fields. These eigenfields will be proven to be a basis of the space of H1-incompressible flow fields obeying no-slip boundary conditions. It will be called the Stokes basis. Various properties of these base fields will be proved and spectral formulae for the energy and enstrophy will be derived. In particular, the helicity of an eigenfield and of its vorticity are related by a simple formula, and various formulae for the coefficients in spectral expansions of the field are derived and used.
