Higher dimensional affine fluids and geodesics of \(\mathsf{SL}(n)\)

2021 Conference on Geometric Analysis and Hyperbolic Equations — Guangxi University, Nanning, China — 2021.12.8

Higher dimensional affine fluids and geodesics of \(\mathsf{SL}(n)\)

Willie Wai Yeung Wong
Michigan State University
2021 Conference on Geometric Analysis and Hyperbolic Equations

Slides available :   https://slides-n-notes.qnlw.info

Credits page


  1. From Fluids to Geometry
  2. Preliminary Analyses
  3. Prior Results
  4. Our results

1. From Fluids to Geometry

Euler-Arnol'd Equations

Fix a compact Riemannian manifold $(M,g)$. Denote by $\mathrm{Sdiff}(M)$ the set of all volume-preserving diffeomorphisms of $M$.

In Lagrangian coord., incompressible flow $\quad\cong\quad$ path $t\to \Phi_t$ in $\mathrm{Sdiff}(M)$.

Action (no potential energy, purely kinetic): \[ S[\Phi] = \int_0^T \int_M g|_{\Phi_t(y)}(\partial_t \Phi_t(y), \partial_t \Phi_t(y)) ~dy~dt.\] Inner integral defines a Riemannian metric on $\mathrm{Sdiff}(M)$.
Tangent space $\cong$ divergence-free vector fields; ELE are the Euler Equations

Free Boundary

Affinely constrained flows

Geodesic motion on $\mathsf{SL}(n)$

\[ \begin{gathered} S = \int \left[ \mathrm{vol}(\Omega_0) |\dot{\gamma}(t)|^2 + \mathrm{tr}\left( \dot{A}(t) I_{\Omega_0} \dot{A}(t)^{\mathsf{T}} \right) \right]~dt \newline I_{\Omega_0} = \int_{\Omega_0} y y^T ~ \mathrm{d}y \end{gathered}\]

When is a constraint not a constraint?


2. Preliminary Analyses

Discrete symmetry

Continuous symmetries

Polar decomposition

Every $A\in \mathsf{SL}(n)$ has a unique factorization as $OP$ (or $PO$): $ \begin{cases} O\in \mathsf{SO}(n) \newline P\in \mathring{S}_+(n)\end{cases}$

$\mathsf{SL}(n)$ foliated by cosets

So the geodesics are easy to describe, right?

Problem: Left actions generate right-invariant vector fields and vice versa

... so even with our conserved quantities, the motions on the two factors do not split.

Some definitions


Call $\mathsf{SO}(n)\subsetneq \mathsf{SL}(n)$ the "throat": it describes the points closest to the origin in $M(n)$.

Second fundamental form

Sign convention: $\mathrm{II}(X,X) \gt 0$ if it curves away from the origin.

Second f.f. defined via normal pointing away from origin of \(M(n)\).

Geodesic equation

Let $X,Y\in \mathsf{T}_A \mathsf{SL}(n)$, then \[ \mathrm{II}(X,Y) = \frac{\mathrm{tr}(A^{-1}X A^{-1}Y)}{|A^{-1}|} \] and the geodesic equation reads \[ \ddot{A} = \frac{ \mathrm{tr}( A^{-1} \dot{A} A^{-1} \dot{A})}{\mathrm{tr}(A^{-1} A^{-T})} A^{-T}.\]

3. Prior Results

$n = 2$ is special: analysis

$n = 2$ geodesic classification

$n = 2$ is special: geometry

$n = 3$ prior results

Analyses of some special explicit solutions and their asymptotics

Sideris (ARMA 2017)

$n$-independent prior results

Sideris (ARMA 2017)

Significance of $\mathrm{II}(\dot{A}, \dot{A}) \gt 0$

Fluid Questions

4. Our Results

Higher dimensions are more curvy

Linear solutions are linearly stable

Bounded geodesics abound in $n \geq 4$

Plot of pulsating motion
Numerical simulation for $n = 6$, for generic data when axes are paired; plots are the three semi-major axial lengths.

Refinement of Virial argument

\[ \frac{d^2}{dt^2} |A|^2 = \frac{|\dot{A}|^2}{|A|} + \frac{n}{|A|^2} \mathrm{II}(\dot{A}, \dot{A}).\]

Lemma. When $n = 2$, we have $\frac{d^2}{dt^2} |A(t)|^2 \leq 0 \implies |A(t)| = |\mathrm{Id}|$.

Corollary. All geodesics except for the throat rotation are unbounded.

Emphatically false when $n \geq 3$.

Virial revisited

Theorem. [sufficient conditions for unboundedness]

  1. Solution is tangent to $\mathring{S}_+(n)$ (or another coset)
  2. Solution has vanishing total angular momentum
  3. Solution has vanishing total vorticity

New formulation of equations

Given solution $A(t)$, define \[ \beta = A^{-1}A^{-\mathsf{T}}, \quad \omega = A^{\mathsf{T}} \dot{A} + \dot{A}^{\mathsf{T}} A, \quad \zeta = A^{\mathsf{T}} \dot{A} - \dot{A}^{\mathsf{T}} A \]

Above is the vorticity version; angular momentum version moves the transpose to the other factor.

Cannot do both simultaneously.

New formulation of equations

Solves problem with Killing vector field not pointing in the fibre direction.

\begin{align*} \frac{d}{dt} \beta & = -\beta \omega \beta \newline \frac{d}{dt}\omega & = \frac12(\omega + \zeta)^{\mathsf{T}}\beta(\omega + \zeta) + \underbrace{\frac{\mathrm{tr}\omega\beta\omega\beta}{2\mathrm{tr}\beta}}_{\geq 0} I + \underbrace{\frac{\mathrm{tr}\zeta\beta\zeta\beta}{2\mathrm{tr}\beta}}_{\leq 0} I \newline \frac{d}{dt}\zeta &= 0 \end{align*}

Compatibility condition: $\quad \mathrm{tr}\beta\omega = 0$.

Preserves "block diagonal structure"

Block diagonal solutions

Assume $\beta, \omega,\zeta$ decompose into $2\times 2$ blocks (with one $1\times 1$ block when $n$ is odd) along the diagonal.

A boundedness criterion

Theorem. Let $n$ be even. If the initial data of $\beta,\omega$ are built from $2\times 2$ blocks that are pure-trace, and if $\zeta$ has no vanishing bocks, then the solution is bounded.

When $n = 2$, hypothesis only possible with rotation at throat.

"Swirling and shear flows"

Theorem. If the initial data has the form \[ \beta_0 = \begin{pmatrix} b_1 \mathrm{Id}_{2m} \newline & b_2 \mathrm{Id}_{n - 2m}\end{pmatrix}, \quad \omega_0 = \begin{pmatrix} w_1 \mathrm{Id}_{2m} \newline & w_2 \mathrm{Id}_{n-2m} \end{pmatrix} \] and writing $\varepsilon$ for the antisymmetric $2\times 2$ matrix \[ \zeta_0 = \begin{pmatrix} z (\underbrace{\varepsilon \oplus \cdots \oplus \varepsilon}_{m \text{ copies}}) \newline & 0 \end{pmatrix};\] then the corresponding solution is unbounded, and asymptotically linear.

Sideris (ARMA 2017) described $n = 3$ and $m = 1$ with explicit integration

Further generalization?

Plot generalized swirling flow.
Numerical simulation of a generalized swirling and shear flow with three sets of axes, instead of two.

Application: Asymptotically linear solutions can be unstable


Further questions

Thank you!