[Paper Review] On the universality of the incompressible Euler equation on compact manifolds

Terence Tao|arXiv (Cornell University)|Jul 25, 2017

Navier-Stokes equation solutions7 references3 citations

TL;DR

This paper establishes a complete characterization for when a quadratic ordinary differential equation (ODE) of the form $\partial_t y = B(y,y)$, with symmetric bilinear map $B$, can be linearly embedded into the incompressible Euler equations on a compact Riemannian manifold. The key result is that such embedding is possible if and only if $B$ satisfies the cancellation condition $\langle B(y,y), y \rangle = 0$ for some positive definite inner product, which ensures energy conservation and enables the construction of complex fluid dynamics such as quasiperiodic or transition solutions.

ABSTRACT

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + abla_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 $B : {\bf R}^n imes {\bf R}^n o {\bf R}^n$ 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 $\langle B(y,y), y angle = 0$ for some positive definite inner product $\langle, angle$ 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.

Motivation & Objective

• To determine the necessary and sufficient conditions under which a finite-dimensional quadratic ODE can be embedded into the incompressible Euler equations on a compact Riemannian manifold.
• To establish a universality result for the incompressible Euler equation by showing that any such ODE satisfying energy conservation can be realized as a finite-dimensional invariant subspace of the Euler flow.
• To construct explicit solutions to the Euler equations with complex dynamical features, such as quasiperiodicity or transitions between steady states, using geometric and Lie-theoretic methods.
• To demonstrate that the Euler equation is 'Turing universal' in the sense that it can simulate arbitrary finite-dimensional ODEs with energy-conserving quadratic nonlinearities.

Proposed method

• Use of Hodge theory to project the nonlinear term $-\frac{1}{2}(\nabla_u u_2 + \nabla_u u_1)$ onto divergence-free vector fields, yielding the symmetric bilinear operator $B_E(u_1, u_2)$.
• Construction of a right-invariant vector field $U(y)$ on the special orthogonal group $SO(n)$ using a skew-adjoint map $S(y) \in \mathfrak{so}(n)$ derived from the bilinear form $B$.
• Definition of a scalar function $F(y,z)(Q) = \langle y, Qz \rangle_{\mathbb{R}^n}$ on $SO(n)$ to relate the ODE dynamics to the fluid equations.
• Proof that the resulting vector field $U(y)$ satisfies the incompressibility condition $\mathrm{div}_g U(y) = 0$ due to invariance under Haar measure.
• Verification that the Euler equation holds via the identity $U(B(y,y)) + \nabla_{U(y)} U(y) = -\mathrm{grad}_g P(y)$, using the skew-adjointness of $S(y)$ and the cancellation condition.
• Reduction of the overdetermined system of $n^2 r$ equations to $n r$ functions by exploiting high symmetry in the construction, particularly via the group structure of $SO(n)$.

Experimental results

Research questions

• RQ1 Under what conditions can a symmetric quadratic ODE $\partial_t y = B(y,y)$ be embedded into the incompressible Euler equations on a compact Riemannian manifold?
• RQ2 Is the cancellation condition $\langle B(y,y), y \rangle = 0$ both necessary and sufficient for such an embedding to exist?
• RQ3 Can the Euler equations on a compact manifold simulate arbitrary finite-dimensional dynamical systems with energy-conserving quadratic nonlinearities?
• RQ4 Does the existence of such embeddings imply a form of 'Turing universality' for the Euler flow?
• RQ5 How can one explicitly construct the manifold, vector field, and pressure map to realize a given ODE within the Euler framework?

Key findings

• A symmetric bilinear map $B: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ can be embedded into the incompressible Euler equations on some compact Riemannian manifold if and only if there exists a positive definite inner product on $\mathbb{R}^n$ such that $\langle B(y,y), y \rangle = 0$ for all $y \in \mathbb{R}^n$.
• The cancellation condition $\langle B(y,y), y \rangle = 0$ is equivalent to the existence of a linear map $S: \mathbb{R}^n \to \mathfrak{so}(n)$ such that $B(y,y) = S(y)(y)$ for all $y \in \mathbb{R}^n$, providing a geometric realization of the quadratic form.
• The embedding is explicitly constructed on the manifold $M = SO(n)$, where the velocity field $U(y)$ is the right-invariant vector field generated by $S(y)$, ensuring incompressibility and correct dynamics.
• The pressure field $P(y)$ is constructed as a quadratic function of $u$ via Hodge projection, and the resulting system satisfies the Euler equations exactly on the time interval of existence of the ODE solution.
• The construction demonstrates that the Euler equation on a compact manifold can simulate any finite-dimensional quadratic ODE with energy conservation, including quasiperiodic and transition dynamics.
• The method evades overdetermination by leveraging the symmetry of the Lie group $SO(n)$, allowing a non-trivial solution to the overdetermined system of $n^2 r$ equations with only $n r$ independent functions.