IAS Workshop on Symplectic Dynamics 2: Wednesday

IAS School of Mathematics

Simonyi Hall

Workshop web page

Happy Einstein Birthday!

Albert Einstein

Wednesday: 2012-03-13

  • 9:00 - 10:00 Wilfrid Gangbo, Georgia Institute of Technology, “Lifting absolutely continuous curves from P(Td) to P2(Rd)” abstract
  • 10:15 - 11:15 Jonatan Lenells, Baylor University, “Geometry of diffeomorphism groupos, complete integrability and optimal transport” abstract
  • 11:30 - 12:30 David Ebin, SUNY, “Groups of diffeomorphisms and geodesics on them” abstract
  • 2:30 - 3:30 Susan Friedlander, University of Southern California, “Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even”. abstract

Wilfrid Gangbo: Lifting absolutely continuous curves from P(Td) to P2(Rd)

Wilfrid Gangbo

(chalk talk; joint work with A. Tudorascu)

This work extends earlier work on the space of probability measures on the torus P(T) to analogous results on P(Td). The earlier work used the embedding P(T)L2(0,1) but we don’t have this embedding in the higher dimensional case.

Let P2(Rd) be the set of Borel measures on Rd with finite second moment |x|2μ(dx)<. We say μ0μ1 if and only if Fdμ0=Fdμ1 FC(Td), FC(Rd), F(x+z)=F(x),zZd.

I define P(T)=P(Td)/.

Let γ be a measure on Rd×Rd which satisfy π1 # γ=μ0 and π2 # γ=μ1. More generally, we write W22(μ0,μ1)=infγRd×Rd|xy|2γ(dx,dy).

Problem: Data: v:(0,T)×TdRd and tσtP(Td). Assume that tσt+(σv)=0 (in the sense of distributions). Can we find t^σtP2(Rd) and ˆv:(0,T)×TdRd such that tˆσ+(ˆσˆv)=0 in the sense of distributions. Here ^σtσt.

Such a lift becomes important if I want to associate the rotation number. I want to write ddtTdxdσt=Tdvtdσt$.

Weak KAM: (A small fraction of what is known) M=Td. Let h:TMR. Let w(z0,z1)=z0(Jz1) where J is the usual matrix satisfying J2=Id. Let Xh denote the associated Hamiltonian vector field: ˙ϕ=Xh(ϕ),ϕ0=(x0,p0).

The associated flow is denoted ϕt=(xt,pt).

Existence of weak Lagrangian Tori: h:RdR is the effective Hamiltonian, cRd.

  1.  uC(td) with h(x,c+u)=h(c) (viscosity)
  2. uc(x)=cx+u(t). (uc is invariant under ϕ.)
  3.  x0Td v0 such that if (xt,pt)=ϕt then  T u(xT)u(x0)=T0[l(x,˙x)+c˙x+h(c)]dt
    limt^xtt=h(c).
General Fact: M compact.

Specific to finite d:

Given xW1,2(0,T,Td) and take two lifts ˆx,ˆyW1,2,Rd. We then find that ^xt^yt=nZd

because we have limt^xtt=limt^ytt

Obstacle in infinite dimensions:

Let M0=P(Td) and in the sense of distributions we have tσ+(σv)=0.

If (σw)=0 then v+w is another velocity.

(wash board…)

Let μP2(Rd) and define the tangent space TμP2(Rd) and also the space TμP(Td). These are defined with L2 closures.

Pseudo symplectic form: ….going faster and I’m not keeping up with the typing.

Theorem (Gangbo-Kun-Pacuni 2011):

  1. Ω is a closed skew symmetric nondegenerate 2-form.
  2.  XH such that dH=Ω(XH,).
  3. ˙f=XH(f)tf+x(vf)=v(f(V+Wρ)).
This is a nonlinear Vlasov equation.

I want to state the analog of the weak KAM theorem in our context.

Theorem Let H be the effective Hamiltonian of H restricted to Rd and let cRd.

  1.  U:P(Td)R such that (in viscosity sense) H(μ,c+wH)=H(c).
  2. Given σ0P(Td)  v0:TdRd such that if f0=σ0δv0 then ft=σtδvt,
    U(σt)U(σ0)=T0[L(σt,vt)+Rdvtcdσt+H(c)]dt.
  3. We also have |1TT0dtRdvtdσt+H(c)|constT.
Corollary: If (ˆσ,^vt) is an appropriate lift then limT1TT0(Rdxdσt)dt=H(c).

Questions:

Mather: This has connections with fluids?

Answer: Kinetic theory. Consider the system 2tx=1NNj=1W(xixj)V(xi). When we consider the N limit, we can move the weak KAM theory from this N particle system to the infinite particle case by moving to the setting of measures. This framework lets us prove convergence of discrete models to the PDE case.

Jonatan Lenells: Geometry of diffeomorphism groups, complete integrability and optimal transport

Jonatan Lennells

(pdf slides; Happy Π day!; Einstein’s birthday)

(joint work with B. Khesin, G. Misiolek, S. Preston)

Outline

  • A new equation
  • Geometry of Diff(M)
  • A sphere
  • Optimal Transport
  • Geometric Statistics

A new equation

ρt+uρ+12ρ2=Mρ2dμ2μ(M).
  • M is a compact Riemannian manifold.
  • μ(M) is the volume of M.
  • This is an exciting equation because it is completely integrable for any M.
  • This is a geodesic equation on Diff(M)/Diffμ(M).
  • Describes ˙H1-optimal transport
  • reduces to the Hunter-Saxton equation for M=S1. (derived in the context of liquid crystals in the early 90s.)
Euler-Arnold Equations. Summary of those ideas.

abc-metric. You can add some other terms to the original L2 inner product involving L2 inner products involvling codifferentials and the musical isomorphism. A lot of different equations arise as you take different values of the parameters. Writing down the associated abc Euler-Arnold equation generates a big equation which can be specialized into various equations. To obtain the ˙H1 metric, we cancel away the terms associated with factors a and c. We simplify by setting a=0,b=14,c=0.

The equation induced by these choices has some degeneracy issues. These can be resolved by quotienting out part of the phase space. The function u is not uniquely determined but its coset is uniquely determined. (Similar issues arise in Hunter-Saxton.) The equation we are considering here is a geodesic equation on a (quotiented) Diffeomorphism group.

Jacobian determinant.

A sphere

Theorem (Khesin-Misiolek-Lennels-Preston): The map which takes the coset [η] to its associated Jacoobian Jacμη is an isometry onto a subset of the sphere.

This isometry lets them transport all the questions about the geodesic equation on this complicated Diff phase space into corresponding questions about geodesics on the sphere. Since we understand the sphere well, we can conjugate results there using the mapping to obtain explicit solution formulae for the geodesic equation. Magical integrability!

Preceding works.

Khesin-Misiolek 2003: Showed Hunter-Saxton may be viewed within the Euler-Arnold framework.

Lennels 2006: Recognized the image of the map as a portion of the sphere.

Optimal Transport

Optimal Transport. Wasserstein distance between two probability measures.

Moser 1965, Ebin-Marsden 1970, Otto 2001, also Benamou-Brenier.

The ˙H1 optimal distance induces what they call the spherical Hellinger distance since it resembles the Hellinger distance used in probability theory

Geometric Statistics

Statistical model.

Fisher-Rao information metric.

Theorem (KMLP): The ˙H1 metric coincides with the Fisher-Rao metric when restricted to any k-dimensional submanifold of the (quotiented) Diff.

This is another reason why we think this metric is important. It arises from many different points of view.

Summary

  • We found a new integrable PDE.
  • The PDE is a geodesic equation on a quotieted diff with ˙H1 metric.
  • A sphere
  • One can understand what is going on using the optimal transportation point of view using the ˙H1 metric.
  • This metric coincides with a basic metric arising in geometric statistics.

Open problems

  • Global weak solutions of the PDE? All solutions break in finite time because you hi the boundary of the diffeomorphism coset. However, there is no problem when you view the dynamics on the sphere. The motion along the great circle may be continued. This type of development has taken place already in the context of the Hunter-Saxton equation. The process appears to be more complicated in this more general context since the Jacobian can vanish.
  • Transfer results from geometric statistics into this diffeomorphism quotient. Then reinterpret these objects in the setting of PDE. Amari-Nagaoka 2000 alpha-connections, dual connections.
  • Develop an optimal transport theory based on the ˙H1 theory.
  • Find a Lax pair.
  • Find a bi-Hamiltonian structure.
  • Analyze the associated two-component equation (c.f. Lennels-Zhao 2011). There is a 2-component Hunter-Saxton so that object suggests we might find a corresponding generalization. This has been observed by LZ.

David Ebin: Groups of diffeomorphisms and geodesics on them

David Ebin

(joint work with Stephen Preston)

Maps from a manifold to itself. Discussion of various topologies of such maps.

  • Volume preserving maps. Diffeomorphisms (Volumorphisms)
  • Even dimensional manifolds with a symplectic form. We can consider the maps which preserve the symplectic form. (Symplectomorphisms)
  • For odd dimensional manifolds, we can consider maps which preserve the contact form. (Contactomorphisms)
In all these cases, we can discuss the geodesics…..ack low battery.

Boothby-Wang fibration.

Susan Friedlander: Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even

Susan Friedlander

(joint work with Vlad Vicol, Walter Rusin, Francisco Gancedo, Weiran Sun)

Homage to Oscar Wilde…

Amain theme is that there is a difference in behavior of solutions in Active Scalar Equations when the associated Fourier multiplier is even versus odd.

Active Scalar Equations; Incompressible Fluids

θt+uθ=0.
u=0.

u=O[θ], PDO

Rd or Td. Even or odd Fourier multiplier symbol. The results I’ll describe are not influenced by the presence of a physical boundary. The emphasis will be on examining the influence of the operator O on the properties of the PDE.

Consider uj=iTijθ, u=. Here Tij is a d×d Calderon-Zygmund operators.

  • ODD Symbol: Locally well-posed in Sobolev spaces. Commutator in energy estimates. **Chae et. al, Friedlander-Vicol.
  • EVEN Symbol: Lipschitz ill-posed in Sobolev spaces. Friedlander-Vicol; Nonuniqueness for L-weak solutions. Techniques from convext integration. Shuydkoy.
Recent reviews of results for certain active scalar equations. “Regularity and blowup for active scalars” Kiselev 2010.

SQG Equation: u=Rθ, symbol i(k2,k1)|k|.

Constantin-Majda-Tabak 1994, Resnick 1995 (Chicago thesis; unpublished), Wu, Cordoba, Chae, Iyer, Ju, Fefferman

The SQG equation had been known in the geophysics community before its introduction to the mathematical community by Constantin et.al.

Local existence for smooth initial data BUT global existence of smooth solutions is OPEN (just as it is open for 3D Euler). Cordoba and Fefferman have ruled out the existence of certain solution scenarios.

“Modified” SQG Equation (Okhitani)

Insert a power of (Δ)1/2=Λ in the map θu so that u=Λβ2θ
where $1 < \beta \leq 2. Chae, Constantin, Cordoba, Ganceda, Wu 2011. Local existence of smooth solutions in Hs, global existence of weak solutions.

Note: result holds more generally when the symbol is ODD and order 1.

IPM equation: singular integral operator with EVEN symbol

Darcy’s law.

u=RR1θ.

Cordoba-Gancedo-Orive 2007

Regular initial data, local existence, weak solutions, SQG and IPM present different behaviours. Global existence of smooth solutions is OPEN.

Even symbol: (k1k2|k|2,k21|k|2).

There is very different behaviors among these equations for rough data. “Patch type initial data”

SIPM equations (even, unbounded)

u=RR1Λβθ

Friedlander-Gancedo-Sun-Vicol (2012)

  • Locally Lipschitz ill-posed in Hs, s>2. Proved for 0<β2 in Td×[0,]
  • Locally well-posed for some “patch-type” weak solutions. Proved for 0<β<1 in R2.
(Discussion: the notions of “wellposedness” changes between the previous two bullet points.)

Symbol: k1k|k|β2

Magnetogeostrophic (MG) equations

Friedlander-Vicol 2011

Long symbol M, even, unbounded, 3D. u=Mθ. Here M is a vector operator that defines a 3-vector u.

Cauchy problem is ill-posed in Hadamard sense in Sobolev spaces. There is no Lipschitz solution map.

Ill-posedness: singular, even, symbol

Active scalar equation. Special direction with index d, often associated with gravity. A list of many conditions on the dth component Sd of the Fourier multiplier operator….allowing them to build eigenfunctions to show Lipschitz failure.

Definition: Locally Lipschitz (X,Y) well-posed.

$$| \theta_1 (\cdot, t) - \theta_1 (\cdot, t)|X \leq K | \theta1 (\cdot, 0) -\theta_2 (\cdot, 0) |_Y. $$

The spaces X,Y are often chosen to be (Hr,HS).

Theorem: Under the many assumptions on Sd, the active scalar equation is Lipschitz (Hr,Hs)-illposed for any r>R,sr+1.

Linear problem

Linearize around θ0=sinmxd. Write out a Fourier series. Crank out a recurrence relation.

Continued fractions, characteristic equation. These ideas where used by Michalkin and Sinai to show unstable eigenvalues for the shear flow for Navier-Stokes equations.

Ill-posedness of the nonlinear problem

Follows a proof by contradiction.

Effects of dissipation: MG

Dissipation: ν(Delta)1/2

Using De Giorgi techniques, Caffarelli-Vasseur proved critical SQG. What can we say about the MG equation?

  • Case 1/2Mγ<1: LWP in Hs, for s>52+(12γ). Well-prepared initial data.
  • Case 0<γ<1/2: Diffusion is too weak to overcome the continued fraction construction.
  • Case γ=1/2: Unique global solution when the initial data and source are small in a suitable sense, then there exists a unique golbal solution. However, if the data are large in this respect then we can run the ill-posedness construction. This reveals a very precise dichotomy.
 

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