Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
The Structure of Groups of Diffeomorphisms
Klas Modin
Geometry of the Euler equations
v˙+∇vv=−∇p
\dot v + \nabla_v v = -\nabla p
divv=0
\operatorname{div}v = 0
Euler's equations describe Riemannian geodesics on
Diffμ(M)={φ∈Diff(M)∣∣Dφ∣=1}
\operatorname{Diff}_{\mu}(M) = \{ \varphi\in\operatorname{Diff}(M) \mid |D\varphi| = 1 \}
But how?
Riemannian metrics on Lie groups
V∈TgG
V \in T_g G
g
g
⟨V,V⟩g=
\langle V,V\rangle_g =
?
?
G
G
Riemannian metrics on Lie groups
V⋅g−1∈g
V\cdot g^{-1} \in \mathfrak{g}
e
e
⟨V,V⟩g=
\langle V,V\rangle_g =
⟨V⋅g−1,V⋅g−1⟩e
\langle V\cdot g^{-1},V\cdot g^{-1}\rangle_e
G
G
"Right-invariant" Riemannian metric determined by inner product on g
Simple example: free rigid body
G=SO(3),g=so(3)≃R3
G = SO(3), \quad \mathfrak{g} = \mathfrak{so}(3)\simeq \mathbb{R}^3
Inner product: moments of inertia tensor
Arnold's example: Euler equations
G=Diffμ(M),g=Xμ(M)={v∣divv=0}
G = \operatorname{Diff}_\mu(M), \quad \mathfrak{g} = \mathfrak{X}_\mu(M) = \{ v\mid \operatorname{div}v=0 \}
Inner product:
⟨v,v⟩L2=∫M∣v∣2μ
\displaystyle\langle v,v\rangle_{L^2} = \int_M \lvert v \rvert^2 \mu
Arnold's theorem: γ(t)∈Diffμ(M) geodesic curve ⇒ vector field v(t)=γ˙(t)∘γ(t)−1 fulfills Euler's equations
But in what sense is Diffμ(M) a Lie group?
Manifold structure of Diffs(M)
Diffs(M)
\operatorname{Diff}^s(M)
U⊂Hs
U \subset H^s
V⊂Hs
V \subset H^s
Thm [Palais, Omori, Ebin, Ebin and Marsden]
Diffs(M) is smooth Hilbert manifold if s>dim(M)/2+1
Diffμs(M) is a submanifold
What about Lie group structure?
Thm [Ebin]
Diffs(M) topological group if s>dim(M)/2+1
Group structure:
(φ,η)↦φ∘η
(\varphi,\eta)\mapsto \varphi\circ \eta
φ↦φ−1
\varphi\mapsto \varphi^{-1}
Failure of smoothness (illustration):
Compute derivative of left translation Lφ(η)=φ∘η
TηLφ⋅V=Dφ∘η⋅V
T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
Hs
H^s
What about Lie group structure?
Thm [Ebin]
Diffs(M) topological group if s>dim(M)/2+1
Group structure:
(φ,η)↦φ∘η
(\varphi,\eta)\mapsto \varphi\circ \eta
φ↦φ−1
\varphi\mapsto \varphi^{-1}
Failure of smoothness (illustration):
Compute derivative of left translation Lφ(η)=φ∘η
TηLφ⋅V=Dφ∘η⋅V
T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
Hs
H^s
What about Lie group structure?
Thm [Ebin]
Diffs(M) topological group if s>dim(M)/2+1
Group structure:
(φ,η)↦φ∘η
(\varphi,\eta)\mapsto \varphi\circ \eta
φ↦φ−1
\varphi\mapsto \varphi^{-1}
Failure of smoothness (illustration):
Compute derivative of left translation Lφ(η)=φ∘η
TηLφ⋅V=Dφ∘η⋅V
T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
Hs
H^s
What about Lie group structure?
Thm [Ebin]
Diffs(M) topological group if s>dim(M)/2+1
Group structure:
(φ,η)↦φ∘η
(\varphi,\eta)\mapsto \varphi\circ \eta
φ↦φ−1
\varphi\mapsto \varphi^{-1}
Failure of smoothness (illustration):
Compute derivative of left translation Lφ(η)=φ∘η
TηLφ⋅V=Dφ∘η⋅V
T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
Hs−1
H^{s-1}
"Fix" the Lie group structure
Option 1
Diff(M)
\operatorname{Diff}(M)
U⊂C∞
U \subset C^\infty
V⊂C∞
V \subset C^\infty
Use Fréchet manifolds instead
- Pros: proper Lie group structure
- Cons: no Picard iterations
Option 2
Right translation Rφ(η)=η∘φ
TηRφ⋅V=V∘φ
T_\eta R_\varphi\cdot V = V\circ\varphi
Hs
H^s
"Fix" the Lie group structure
Option 1
Diff(M)
\operatorname{Diff}(M)
U⊂C∞
U \subset C^\infty
V⊂C∞
V \subset C^\infty
Use Fréchet manifolds instead
- Pros: proper Lie group structure
- Cons: no Picard iterations
Option 2
Right translation Rφ(η)=η∘φ
TηRφ⋅V=V∘φ
T_\eta R_\varphi\cdot V = V\circ\varphi
Hs
H^s
"Fix" the Lie group structure
Option 1
Diff(M)
\operatorname{Diff}(M)
U⊂C∞
U \subset C^\infty
V⊂C∞
V \subset C^\infty
Use Fréchet manifolds instead
- Pros: proper Lie group structure
- Cons: no Picard iterations
Option 2
Right translation Rφ(η)=η∘φ
TηRφ⋅V=V∘φ
T_\eta R_\varphi\cdot V = V\circ\varphi
Hs
H^s
So, let's work only with right translation
Crazy idea, but it works!
Simple example of how it works
v˙−v˙xx+3vvx−2vxvxx−vvxxx=0
\dot v - \dot v_{xx} + 3 v v_x - 2v_x v_{xx} - v v_{xxx} = 0
Camassa-Holm equation
Av˙+vAvx+2vxAv=0,
A \dot v + v Av_x + 2 v_x A v = 0,
A=1−∂xx2
A = 1 - \partial_{xx}^2
v(t,⋅)∈Hs(S1),s>2
v(t,\cdot) \in H^s(S^1), \quad s>2
Why PDE not ODE?
Idea: [Ebin and Marsden] maybe geodesic equation on TDiffs(S1) is an ODE
Hs−2
H^{s-2}
Hs−2
H^{s-2}
Hs−3
H^{s-3}
Camassa-Holm as geodesic equation
Diffs(S1)
\operatorname{Diff}^s(S^1)
Hs(S1)
H^s(S^1)
v(t)
v(t)
Hs(S1)∋v(t)↦(φ˙(t)v(t)∘φ(t),φ(t))∈TDiffs(S1)
H^s(S^1) \ni v(t) \mapsto (\underbrace{v(t)\circ\varphi(t)}_{\dot\varphi(t)},\varphi(t))\in T\operatorname{Diff}^s(S^1)
φ˙(t)=v(t)∘φ(t)
\dot\varphi(t) = v(t)\circ\varphi(t)
Camassa-Holm as geodesic equation
Diffs(S1)
\operatorname{Diff}^s(S^1)
Hs(S1)
H^s(S^1)
φ˙(t)
\dot\varphi(t)
Hs(S1)∋v(t)↦(φ˙(t)v(t)∘φ(t),φ(t))∈TDiffs(S1)
H^s(S^1) \ni v(t) \mapsto (\underbrace{v(t)\circ\varphi(t)}_{\dot\varphi(t)},\varphi(t))\in T\operatorname{Diff}^s(S^1)
φ˙(t)=v(t)∘φ(t)
\dot\varphi(t) = v(t)\circ\varphi(t)
φ¨=v˙∘φ+vx∘φφ˙
\ddot\varphi = \dot v\circ\varphi + v_x\circ\varphi \, \dot\varphi
=(v˙+vxv)∘φ=(A−1(−vAvx−2vxAv+A(vxv)))∘φ
= (\dot v + v_x v)\circ\varphi = \Big( A^{-1}(-v Av_x - 2 v_x A v + A(v_x v)) \Big)\circ\varphi
=(A−1(−vAvx+vAvx+Hs−2-terms))∘φ
= \Big( A^{-1}(-v Av_x + v A v_x + H^{s-2}\text{-terms}) \Big)\circ\varphi
Camassa-Holm as geodesic equation
Diffs(S1)
\operatorname{Diff}^s(S^1)
Hs(S1)
H^s(S^1)
φ˙(t)
\dot\varphi(t)
φ˙(t)=v(t)∘φ(t)
\dot\varphi(t) = v(t)\circ\varphi(t)
φ¨=v˙∘φ+vx∘φφ˙
\ddot\varphi = \dot v\circ\varphi + v_x\circ\varphi \, \dot\varphi
=(v˙+vxv)∘φ=(A−1(−vAvx−2vxAv+A(vxv)))∘φ
= (\dot v + v_x v)\circ\varphi = \Big( A^{-1}(-v Av_x - 2 v_x A v + A(v_x v)) \Big)\circ\varphi
=(A−1(−vAvx+vAvx+Hs−2-terms))∘φ
= \Big( A^{-1}(-v Av_x + v A v_x + H^{s-2}\text{-terms}) \Big)\circ\varphi
Are we done?
No! RHS must be smooth as function of φ,φ˙ but v=φ˙∘φ−1
B(v,v)gffdffd
\underbrace{\phantom{gffdffd}}_{B(v,v)}
Second piece of magic
φ¨=A~φ−1(B~φ(φ˙,φ˙)),B:Hs×Hs→Hs−2
\ddot\varphi = \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi)), \quad B:H^s \times H^s\to H^{s-2}
P:Hs→Hs−2,P~φ:TφDiffs(S1)→Tφs−2Diffs(S1)
P: H^s \to H^{s-2}, \; \tilde P_\varphi: T_\varphi\operatorname{Diff}^s(S^1)\to T^{s-2}_\varphi\operatorname{Diff}^s(S^1)
P~φV=(P(V∘φ−1))∘φ
\tilde P_\varphi V = (P(V\circ\varphi^{-1}))\circ\varphi
Geodesic equation, again
Lemma: Mapping TDiffs(S1)→Ts−1Diffs(S1) given by (φ,φ˙)↦(∂x(φ˙∘φ−1))∘φ is smooth
Second piece of magic
φ¨=A~φ−1(B~φ(φ˙,φ˙)),B:Hs×Hs→Hs−2
\ddot\varphi = \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi)), \quad B:H^s \times H^s\to H^{s-2}
Geodesic equation, again
Lemma: Mapping TDiffs(S1)→Ts−1Diffs(S1) given by (φ,φ˙)↦(∂x(φ˙∘φ−1))∘φ is smooth
Proof:
(∂x(φ˙∘φ−1))∘φ=φ˙x((∂xφ−1)∘φ)=φxφ˙x
\displaystyle (\partial_x (\dot\varphi\circ\varphi^{-1}))\circ\varphi = \dot\varphi_x((\partial_x\varphi^{-1})\circ\varphi) = \frac{\dot\varphi_x}{\varphi_x}
Second piece of magic
φ¨=A~φ−1(B~φ(φ˙,φ˙)),B:Hs×Hs→Hs−2
\ddot\varphi = \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi)), \quad B:H^s \times H^s\to H^{s-2}
Geodesic equation, again
Lemma: Mapping TDiffs(S1)→Ts−1Diffs(S1) given by (φ,φ˙)↦(∂x(φ˙∘φ−1))∘φ is smooth
Thm: Spray TDiffs(S1)→Ts−2Diffs(S1) given by (φ,φ˙)↦A~φ−1(B~φ(φ˙,φ˙)) is smooth
That's it!
Examples of more recent work
- Replace A by pseudo-differential operator
[Bauer, Bruveris, Cismas, Escher, Kolev (2020)]
- Extend to semi-invariant metrics on Diffs(M) (with connections to optimal transport)
[Bauer and Modin (2020)]
- Extend to Euler equations perturbed by noise (SPDE)
[Maurelli, Modin, Schmeding (2020)]
Thank you!
The Structure of Groups of Diffeomorphisms Klas Modin slides.com/kmodin/diffeos
Groups of diffeomorphisms
By Klas Modin
Groups of diffeomorphisms
Online-presentation given 2020-10 at Hebrew University Analysis Seminar.
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