The solution of a~spectral problem for the curl and the Stokes operators with periodic boundary

R. S. Saks

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 246-276

Résumé

In this paper, the relations between eigenvalues and eigenfunctions of the curl operator and the Stokes operator (with periodic boundary condition) are considered. These relations show that the curl operator is a square root of the Stokes operator with $\nu=1$. The multiplicity of zero eigenvalue of the curl operator is infinite. The space $\mathbf{L}_2(Q,2\pi)$ is decomposed into a directe sum of the eigensubspaces of the operator curl. For any complex number $\lambda$, the equation $\operatorname{rot}\mathbf{u}+\lambda\mathbf{u}=\mathbf{f}$ and the Stokes equation $-\nu(\Delta v+\lambda^2v)+\nabla p=\mathbf{f}$, $\operatorname{div}v=0$, are solved.