Semi-invariant metrics on groups of diffeomorphisms

Klas Modin

\mathrm{Diff}(M)
P^\infty(M)

Joint work with

Martin Bauer

Florida State University

Outline

  • Background and motivation
  • Main result
  • Pitfalls in the Ebin–Marsden-type analysis
  • From local to global
  • From Banach to Fréchet
  • Outlook

Background

Arnold

(1966)

foundation of geometric hydrodynamics

Ebin & Marsden (1970)

local well-posedness + more

Many authors

\[\mathrm{Diff}(M)\] and higher order Sobolev metrics + global results

\mathrm{Diff}(M)
\mathfrak{X}(M)
v = \dot\varphi\circ\varphi^{-1}

Lagrangian to Eulerian

Geodesic equation: smooth 2nd order ODE on Sobolev completion \(\mathrm{Diff}^s(M) \)

Basic question:

Does the analysis survive for \(\mathrm{Diff}_{\mu}(M)\)-invariant metrics?

Sobolev-type Riemannian metrics fulfilling

\langle \dot\varphi,\dot\varphi\rangle_{\varphi} = \langle \dot\varphi\circ\eta,\dot\varphi\circ\eta\rangle_{\varphi\circ\eta}\quad \forall\, \eta \in \mathrm{Diff}_{\mu}(M)

Motivations?

\mathrm{Diff}(M)
P^\infty(M)
\mathrm{Id}
\rho_0
\rho_1
\pi(\eta)=\eta_*(\rho_0 dx)

Moser 1965:

Principal bundle

\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)

\(L^2\) metric on \(\mathrm{Diff}(M)\)

\overline{\mathcal{G}}_\eta(\dot\eta,\dot\eta) = \int_{M}\left\vert \dot\eta \right\vert^2 \mu_0

Induces Otto metric

{\mathcal{G}}_\rho(\dot\rho,\dot\rho) \Rightarrow d_W^2(\mu_0,\mu_1)
\mathrm{Hor}

Open problem: higher order versions

Higher order Otto calculus

Shallow water equations

\left\{ \begin{aligned} & v_t + \nabla_v v + \nabla h = 0 \\ & h_t + \mathrm{div}(h v) = 0 \end{aligned} \right.
\displaystyle E(v,h) = \frac{1}{2}\int_M \left( h | v |^2 + h^2 \right)\mu
\displaystyle L(\varphi,\dot\varphi) = \frac{1}{2}\int_M \left( | \dot\varphi |^2 + \det(D\varphi^{-1})^2 \right)\mu

Energy functional

Lagrangian on \(T\mathrm{Diff}(M)\)

h
v

Improved model: Green-Naghdi

\left\{ \begin{aligned} & v_t + \nabla_v v + \nabla h = -\frac{1}{3 h}\Big(\frac{\partial}{\partial t} + \nabla_v \Big)\nabla^* (h^3 \nabla v) \\ & h_t + \mathrm{div}(h v) = 0 \end{aligned} \right.

Extra terms from Taylor expansion \(\rightarrow\) higher order metrics

\displaystyle E(v,h) = \frac{1}{2}\int_M \left( h | v |^2 + \frac{h^3}{3}|\nabla v|^2+ h^2 \right)\mu
h
v

\(H^1\) metric

Reduced Lagrangian

T\mathrm{Diff}(M)/\mathrm{Diff}_\mu \simeq\mathfrak{X}(M)\times P^\infty(M)

\(\mathrm{Diff}_\mu(M)\)-invariant Lagrangian \(\iff L(\varphi,\dot\varphi) = \ell(\dot\varphi\circ\varphi^{-1},\det(D\varphi^{-1}))\)

\ell\colon \mathfrak{X}(M)\times P^\infty(M)\to \mathbb{R}
\ell(v,\rho) = \langle A(\rho)v,v \rangle_{L^2} - V(\rho)

Inertia tensor

potential energy

Governing equations

\displaystyle \left. \begin{aligned} & m_t + \nabla_v m + (\mathrm{div}\,u)m + (\nabla u)^\top m - \rho \nabla \frac{\delta \ell}{\delta \rho}=0 \\ & \rho_t + \mathrm{div}(\rho v) = 0 \\ & m = A(\rho)v \end{aligned} \right.

Difference to EPDiff

Assumptions on \(A(\rho)\)

  1. The map \[ (v,\rho)\mapsto A(\rho)v\] is a smooth differential operator of order \(2k-2\) in \(\rho\) and \(2k\) in \(v\)
  2. For any \(\rho\in P^\infty(M)\) \[ u\mapsto A(\rho)u\] is linear, positive, elliptic
  3. Let \[B(v,\rho)\colon C^\infty(M)\to \mathfrak{X}(M),\quad \dot\rho \mapsto \frac{\delta A(\rho)v}{\delta\rho}\dot\rho\] Then its \(L^2\)-adjoint \[(\rho,v,u)\mapsto B(\rho,v)^*u \] is a smooth differential operator of order \(2k-2\) in \(\rho\) and order \(2k-1\) in \(v\) and \(u\)

Example and counter-example

\displaystyle G_{\rho}(v,v) = \sum_{i=0}^k\int_M a_i\circ\rho |\nabla^i v|^2 \mu

\(a_i\colon \mathbb{R}_{>0}\to\mathbb{R}_{\geq 0}\) and either

  1. \(k=1\), \(a_0>0\), \(a_1 = \text{const}> 0\)
  2. \(k\geq 2\), \(a_0>0\), \(a_k>0\)
\displaystyle G_{\rho}(v,v) = \int_M \frac{1}{\rho} |L(\rho v)|^2 \mu

(However, nice convexity properties a la Brenier-Benamou)

Example

Counter example

Local well-posedness

Theorem

\(M\) compact + assumption + \(s>d/2+2k\)

\(\Rightarrow\) local well-posedness for geodesics on Sobolev completion \(\mathrm{Diff}^s(M)\)

Corollary

Potential \(V\colon P^{s-1}\to \mathbb{R}\) such that \( \delta V/\delta\rho\) (non-linear) differential operator  of order \(2k-2\)

\(\Rightarrow\) local well-posedness  on \(\mathrm{Diff}^s(M)\)

necessary

Failure of local to global result

\varepsilon
\mathrm{id}
u_0
\eta(1)
\dot\eta(1)
u_1 = \dot\eta(1)\circ\eta(1)^{-1}
u_1
\eta(2)

But for semi-invariant metrics \(\epsilon\) depends on \(\rho\)

\mathrm{Diff}(M)

Can one expect global existence?

h = \rho
x

Wave-breaking is expected to happen in shallow water models

A "cheap" global result

Theorem

\(M\) compact, \(k>d/2+1\), \(a_0 >C_1>0\), and \(a_k>C_2>0 \).

Then:

  1. \((\mathrm{Diff}^k(M),G)\) complete metric space
  2. \((\mathrm{Diff}^k(M),G)\)  geodesically complete (global geodesics)
\displaystyle G_{\rho}(v,v) = \sum_{i=0}^k\int_M a_i\circ\rho |\nabla^i v|^2 \mu

Proof: \(G\) uniformly stronger than a strong right-invariant metric

What about no-loss-no-gain?

Theorem

\(F\) a \(\mathrm{Diff}_{\mu}\)-equivariant vector field on \(T\mathrm{Diff}(M)\) that extends to a smooth vector field on \(T\mathrm{Diff}^s(M)\) for all \(s>s_0\).

If \((\varphi_0,v_0)\in T\mathrm{Diff}^{s+1}(M)\) then \(J_{s+1}(\varphi_0,v_0) = J_s(\varphi_0,v_0)\).

\(s\to\infty\) gives result on Fréchet Lie group \(\mathrm{Diff}(M)\)

Proof: Read Ebin & Marsden [1970] carefully

Outlook

  • More sophisticated global results
     
  • Sectional curvature, Fredholm properties, etc
     
  • Gradient flows a la Otto

THANKS!

Reference:

M. Bauer and K. Modin. Semi-invariant Riemannian metrics in hydrodynamics, Calc. Var. PDE, 2020
[link] [arXiv]