Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
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 \(A(\rho)\)
- The map \[ (v,\rho)\mapsto A(\rho)v\] is a smooth differential operator of order \(2k-2\) in \(\rho\) and \(2k\) in \(v\)
- For any \(\rho\in P^\infty(M)\) \[ u\mapsto A(\rho)u\] is linear, positive, elliptic
- 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
- \(k=1\), \(a_0>0\), \(a_1 = \text{const}> 0\)
- \(k\geq 2\), \(a_0>0\), \(a_k>0\)
\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:
- \((\mathrm{Diff}^k(M),G)\) complete metric space
- \((\mathrm{Diff}^k(M),G)\) 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!
Semi-invariant Riemannian metrics on diffeomorphisms
By Klas Modin
Semi-invariant Riemannian metrics on diffeomorphisms
Presentation given 2018-12 at BIRS in Banff
