Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Semi-invariant metrics on groups of diffeomorphisms
Klas Modin
Diff(M)
\mathrm{Diff}(M)
P∞(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
Diff(M) and higher order Sobolev metrics + global results
Diff(M)
\mathrm{Diff}(M)
X(M)
\mathfrak{X}(M)
v=φ˙∘φ−1
v = \dot\varphi\circ\varphi^{-1}
Lagrangian to Eulerian
Geodesic equation: smooth 2nd order ODE on Sobolev completion Diffs(M)
Basic question:
Does the analysis survive for Diffμ(M)-invariant metrics?
Sobolev-type Riemannian metrics fulfilling
⟨φ˙,φ˙⟩φ=⟨φ˙∘η,φ˙∘η⟩φ∘η∀η∈Diffμ(M)
\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?
Diff(M)
\mathrm{Diff}(M)
P∞(M)
P^\infty(M)
Id
\mathrm{Id}
ρ0
\rho_0
ρ1
\rho_1
π(η)=η∗(ρ0dx)
\pi(\eta)=\eta_*(\rho_0 dx)
Moser 1965:
Principal bundle
Diff(M)/Diffμ0(M)
\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)
L2 metric on Diff(M)
Gη(η˙,η˙)=∫M∣η˙∣2μ0
\overline{\mathcal{G}}_\eta(\dot\eta,\dot\eta) = \int_{M}\left\vert \dot\eta \right\vert^2 \mu_0
Induces Otto metric
Gρ(ρ˙,ρ˙)⇒dW2(μ0,μ1)
{\mathcal{G}}_\rho(\dot\rho,\dot\rho) \Rightarrow d_W^2(\mu_0,\mu_1)
Hor
\mathrm{Hor}
Open problem: higher order versions
Higher order Otto calculus
Shallow water equations
{vt+∇vv+∇h=0ht+div(hv)=0
\left\{
\begin{aligned}
& v_t + \nabla_v v + \nabla h = 0 \\
& h_t + \mathrm{div}(h v) = 0
\end{aligned}
\right.
E(v,h)=21∫M(h∣v∣2+h2)μ
\displaystyle E(v,h) = \frac{1}{2}\int_M \left( h | v |^2 + h^2 \right)\mu
L(φ,φ˙)=21∫M(∣φ˙∣2+det(Dφ−1)2)μ
\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 TDiff(M)
h
h
v
v
Improved model: Green-Naghdi
⎩⎨⎧vt+∇vv+∇h=−3h1(∂t∂+∇v)∇∗(h3∇v)ht+div(hv)=0
\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 → higher order metrics
E(v,h)=21∫M(h∣v∣2+3h3∣∇v∣2+h2)μ
\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
h
v
v
H1 metric
Reduced Lagrangian
TDiff(M)/Diffμ≃X(M)×P∞(M)
T\mathrm{Diff}(M)/\mathrm{Diff}_\mu \simeq\mathfrak{X}(M)\times P^\infty(M)
Diffμ(M)-invariant Lagrangian ⟺L(φ,φ˙)=ℓ(φ˙∘φ−1,det(Dφ−1))
ℓ:X(M)×P∞(M)→R
\ell\colon \mathfrak{X}(M)\times P^\infty(M)\to \mathbb{R}
ℓ(v,ρ)=⟨A(ρ)v,v⟩L2−V(ρ)
\ell(v,\rho) = \langle A(\rho)v,v \rangle_{L^2} - V(\rho)
Inertia tensor
potential energy
Governing equations
mt+∇vm+(divu)m+(∇u)⊤m−ρ∇δρδℓ=0ρt+div(ρv)=0m=A(ρ)v
\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(ρ)
- The map (v,ρ)↦A(ρ)v is a smooth differential operator of order 2k−2 in ρ and 2k in v
- For any ρ∈P∞(M) u↦A(ρ)u is linear, positive, elliptic
- Let B(v,ρ):C∞(M)→X(M),ρ˙↦δρδA(ρ)vρ˙ Then its L2-adjoint (ρ,v,u)↦B(ρ,v)∗u is a smooth differential operator of order 2k−2 in ρ and order 2k−1 in v and u
Example and counter-example
Gρ(v,v)=i=0∑k∫Mai∘ρ∣∇iv∣2μ
\displaystyle G_{\rho}(v,v) = \sum_{i=0}^k\int_M a_i\circ\rho |\nabla^i v|^2 \mu
ai:R>0→R≥0 and either
- k=1, a0>0, a1=const>0
- k≥2, a0>0, ak>0
Gρ(v,v)=∫Mρ1∣L(ρv)∣2μ
\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
⇒ local well-posedness for geodesics on Sobolev completion Diffs(M)
Corollary
Potential V:Ps−1→R such that δV/δρ (non-linear) differential operator of order 2k−2
⇒ local well-posedness on Diffs(M)
necessary
Failure of local to global result
ε
\varepsilon
id
\mathrm{id}
u0
u_0
η(1)
\eta(1)
η˙(1)
\dot\eta(1)
u1=η˙(1)∘η(1)−1
u_1 = \dot\eta(1)\circ\eta(1)^{-1}
u1
u_1
η(2)
\eta(2)
But for semi-invariant metrics ϵ depends on ρ
Diff(M)
\mathrm{Diff}(M)
Can one expect global existence?
h=ρ
h = \rho
x
x
Wave-breaking is expected to happen in shallow water models
A "cheap" global result
Theorem
M compact, k>d/2+1, a0>C1>0, and ak>C2>0.
Then:
- (Diffk(M),G) complete metric space
- (Diffk(M),G) geodesically complete (global geodesics)
Gρ(v,v)=i=0∑k∫Mai∘ρ∣∇iv∣2μ
\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 Diffμ-equivariant vector field on TDiff(M) that extends to a smooth vector field on TDiffs(M) for all s>s0.
If (φ0,v0)∈TDiffs+1(M) then Js+1(φ0,v0)=Js(φ0,v0).
s→∞ gives result on Fréchet Lie group 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 metrics on groups of diffeomorphisms Klas Modin D i f f ( M ) \mathrm{Diff}(M) P ∞ ( M ) P^\infty(M)
Semi-invariant Riemannian metrics on diffeomorphisms
By Klas Modin
Semi-invariant Riemannian metrics on diffeomorphisms
Presentation given 2018-12 at BIRS in Banff
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