Conjugate points along Kolmogorov flows on the torus
Résumé
The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold M , for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of conjugate points along such geodesics reveal that these cease to be infinitesimally length-minimizing between their endpoints. In this work, we focus on the case of the torus M = T 2 and on geodesics corresponding to steady solutions of the Euler equation generated by stream functions ψ = − cos(mx) cos(ny) for positive integers m and n, called Kolmogorov flows. We show the existence of conjugate points along these geodesics for all (m, n), with the sole exception of m = n = 1. We also discuss the unusual features of this special case and conjecture that there are no conjugate points in this case.
Origine : Fichiers produits par l'(les) auteur(s)