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 $u:[0,T]\to \Gamma (TM)$ 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 $u:[0,T]\to \Gamma (TM)$ 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.

arXiv comment21 pages, no figures. To appear, Pure Appl. Func. Anal. Referee suggestions implemented (and a clearer distinction between abstract index notation and local coordinate notation is now in place)

Cited by in corpus (6)

• Constructing Turing complete Euler flows in dimension $3$
• Universality of Euler flows and flexibility of Reeb embeddings
• Computability and Beltrami fields in Euclidean space
• Steady Euler flows and Beltrami fields in high dimensions
• Undecidability in Physics: a Review
• A note on the Turing universality of homogeneous potential wells and geodesible flows