On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows

Abstract

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to \Gamma(T M)$ is the velocity field and $p: [0,T] \to C^\infty(M)$ is the pressure field. In this paper we show that if one is permitted to extend the base manifold $M$ by taking an arbitrary warped product with a torus, then the space of solutions to this equation becomes "non-rigid'"in the sense that a non-empty open set of smooth incompressible flows $u: [0,T] \to \Gamma(T M)$ can be approximated in the smooth topology by (the horizontal component of) a solution to these equations. We view this as further evidence towards the "universal" nature of Euler flows.