%0 Journal Article %1 tao2019universality %A Tao, Terence %D 2019 %K Euler flows manifolds %T On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows %U http://arxiv.org/abs/1902.06313 %X The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form align* \partial_t u + \nabla_u u &= - grad_g p \\ div_g u &= 0, align* where $u: 0,T \Gamma(T M)$ is the velocity field and $p: 0,T C^ınfty(M)$ 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 $u: 0,T \Gamma(T M)$ 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 üniversal" nature of Euler flows.

@article{tao2019universality, 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.}, added-at = {2019-02-27T03:08:16.000+0100}, author = {Tao, Terence}, biburl = {https://www.bibsonomy.org/bibtex/2738fda25b614723d43db8bb9ecdbfc51/kirk86}, description = {[1902.06313] On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows}, interhash = {336d767e234d86f57ea04ff45c01257e}, intrahash = {738fda25b614723d43db8bb9ecdbfc51}, keywords = {Euler flows manifolds}, note = {cite arxiv:1902.06313Comment: 17 pages, no figures. Submitted, Pure Appl. Func. Anal}, timestamp = {2019-02-27T03:08:16.000+0100}, title = {On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows}, url = {http://arxiv.org/abs/1902.06313}, year = 2019 }