Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the universality of the incompressible Euler equation on compact
  manifolds   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*} We show that any quadratic ODE $\partial_t
y = B(y,y)$, where $B : {\bf R}^n \times {\bf R}^n \to {\bf R}^n$ is a
symmetric bilinear map, can be linearly embedded into the incompressible Euler
equations for some manifold $M$ if and only if $B$ obeys the cancellation
condition $\langle B(y,y), y \rangle = 0$ for some positive definite inner
product $\langle,\rangle$ on $ {\bf R}^n$. This allows one to construct
explicit solutions to the Euler equations with various dynamical features, such
as quasiperiodic solutions, or solutions that transition from one steady state
to another, and provides evidence for the "Turing universality" of such Euler
flows.

29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph