Geometric aspects of hydrodynamic blowup

Stephen C. PrestonUniversity of Colorado

Stephen.Preston@colorado.edumath.colorado.edu/˜prestos

February 26, 2010

Consider an ideal fluid in T3.

Its velocity field is u : [0,T )× T3 → R3.

ut(t, x) + (u ·∇u)(t, x) = −∇p(t, x)

div u = 0; ∆p = − div (u ·∇u).

Its flow is given by η : [0,T )× T3 → T3.

ηt(t, x) = u�t, η(t, x)

�; η(0, x) = x .

Incompressibility can be expressed as η∗µ = µ whereµ = dx

1 ∧ dx2 ∧ dx

3 is the volume form.

This is equivalent to detDη(t, x) ≡ 1 for all t and x .

We expect any fluid to be described by diffeomorphisms of T3,denoted by D(T3).

The incompressible fluids are a subset described by

Dµ(T3) = {η ∈ D(T3) : η∗µ = µ}.

The shape of this subset depends on the smoothness we consider.

� One extreme: volume-preserving maps, where η is onlyassumed to be in L

2 and µ(η[U]) = µ(U) for every Borel setU.

� Other extreme: all maps are assumed C∞.

In between, we can consider η and η−1 to be in Hs where s > 5

2 .

Need this to get η ∈ C1. Also ensures that Dµ is a C

∞

submanifold of D.

The geometry we are interested in is that generated by the Arnold

distance:

d(η, ξ) = inf{� 1

0

��

T3

����∂γ(t, x)

∂t

����2

µ dt,

among curves γ : [0, 1] → Dµ(T3) with γ(0) = η and γ(1) = ξ.

Inspired by the formula for finitely many particles.

Imagine Dµ(T3) as a curved surface inside D(T3).

Careful:

� Need objects at least Hs to get smoothness.

� But geometrically only L2 matters.

Intuitively we might imagine a smooth interior and rough boundary.

Geodesics are curves that locally minimize length.

On Dµ(T3), the geodesics satisfy ηtt(t, x) = −∇p�t, η(t, x)

�; i.e.,

the acceleration is perpendicular to the submanifold.

Variations of geodesics are called Jacobi fields. They satisfy theJacobi equation

D̃2J

∂t2+ R̃(J, ηt)ηt = 0,

where R̃ is the curvature operator.

Intuitively we think of this as d2Jdt2 + R(t)J(t) = 0. The sign of R

controls the stability.

Figure: R ≡ 1 Figure: R ≡ 0 Figure: R ≡ −1

Curvature formula:

If u and v are divergence-free, then

�R̃(u, v)v , u�L2 =

�

T3

��B(u, u),B(v , v)� − �B(u, v),B(u, v)�

�µ,

whereB(u, v) = ∇∆−1 div (∇uv).

For a fixed u, the curvature satisfies

−1

2supx∈Tn

�

i ,j

∂iuj∂jui ≤�R̃(u, v)v , u�L2

|v |2L2

≤ supx∈Tn

maxi ,j

∂i∂jp,

where ∇p = B(u, u).

This is sharp when n = 3. (The right side is not sharp whenn = 2.) So curvature is basically a C

1 object.

Roughly the curvature is more negative than positive.

For example, take a basis of curl eigenfields ξm on T3. Then forevery divergence-free v we have

�R̃(ξm, v)v , ξm�L2 ≤ 0.

However there is enough positive curvature to get conjugate points.

Conjugate points are points η(a) and η(b) such that some Jacobifield J has J(a) = J(b) = 0.

If η(a) and η(b) are conjugate, then η is not length-minimizing.

Conversely if η is not infinitesimally length-minimizing (in any

space), then η(a) and η(b) are conjugate.

A fluid rigidly rotating on the surface of the sphere.

The red arrows represent a Jacobi field along this rotation, which iszero at the initial and final times.

So the original configuration is conjugate to the final configurationalong this geodesic.

To analyze the Jacobi equation: set J = y ◦ η. Then

yt + [u, y ] = z ; zt + P(u ·∇z + z ·∇u) = 0.

The first equation is equivalent to ∂∂t

�η(t)−1

∗ y(t)�

= η(t)−1∗ z(t).

So set y(t) = η(t)∗v(t). Then using conservation of the vorticity2-form ω = du

�, we get

∂

∂tP

�Λ(t, x)

∂v

∂t(t, x)

�+

∂

∂tP

�ιv(t,x)ω0(x)

��= 0,

where Λ(t, x) = Dη(t, x)†Dη(t, x), and P is the Leray projectiononto divergence-free vector fields.

Λ(t, x) represents the stretching rate of vectors based at x whichare transported by η.

� In two dimensions ω0 is a function, and (ιvω0)� = ω0�v , i.e.,the vorticity function times a rotation of v . HenceP(ι(�∇g)ω0)� = −P(ω0∇g) = P(g∇ω0) is compact.

� In three dimensions ω0 is a vector field, and (ιvω0)� = ω0 × v ,i.e., the cross product. This operator is generally not compact.

In three dimensions, if the initial condition v(0, x) is supportednear x , then the solution is close to the solution of

d

dt

�Λ(t, x)

dv

dt

�+ ω0(x)× dv

dt= 0.

So we can solve this ODE at each point in time, look for solutionssatisfying v(a) = v(b) = 0, and use them to find conjugate pointson Dµ(T3).

In two dimensions this doesn’t work. Trying to localize near apoint yields

d

dt

�Λ(t, x)

dv

dt

�= 0,

so there is no way to detect conjugate points using a localcriterion. This is related to the fact that conjugate points cannotcluster along a geodesic (due to Fredholmness of the exponentialmap [Ebin, Misio�lek, P.]).

Although there are conjugate points in Dµ(T2), they are generallyharder to find. (The first examples were found in the 1990s byMisio�lek.)

The fact that vorticity and the stretching matrix are the importantterms in the Jacobi equation suggests that Jacobi field propertiesare related to the Beale-Kato-Majda criterion for blowup.

We’d like to say: if for some Lagrangian path t �→ η(t, x) we have� T0 |ω

�t, η(t, x)

�| dt = +∞, then there are infinitely many

conjugate point pairs {η(tn), η(tn+1)} for a sequence tn � T .This almost works.

If it doesn’t work, we can show that there must be orthonormalvectors {e1, e2, e3} at x , with e3 parallel to ω0(x), such that thecomponents Λij = �ei ,Λ(t, x)ej� and Λij = �ei ,Λ(t, x)−1

ej� satisfy

limt→T

� t0 Λ11(τ) dτ

� t0 Λ22(τ) dτ

= 0 and

� T

0

Λ33(t)

Λ22(t)dt < ∞.

Roughly speaking, this says that the largest and smallesteigenvectors of the stretching matrix are basically transported bythe flow.

Simplifications:

� Project orthogonally to the vorticity vector ω0(x) = |ω0(x)|e3,so the reduced stretching matrix is

�Φ11 Φ12

Φ12 Φ22

�=

1√Λ33

�Λ11Λ33 − Λ2

13 Λ12Λ33 − Λ23Λ13

Λ12Λ33 − Λ23Λ13 Λ22Λ33 − Λ323

�.

Note that detΦ(s) ≡ 1.

� Rescale the time variable by definings =

� t0

��ω�τ, η(τ, x)

��� dτ , which maps [0,T ) to [0,∞).

We end up with:

d

ds

��Φ11(s) Φ12(s)Φ12(s) Φ22(s)

�dY

ds

�+

�0 −11 0

�dY

ds= 0,

with Y (sn) = 0 and Y�(sn) = id, and trying to find sn+1 > 0 such

that det Y (sn+1) = 0.

Even simpler: set W (s) = Φ(s)Y �(s).

Then

dW

ds+

�0 −11 0

� �Φ22(s) −Φ12(s)−Φ12(s) Φ11(s)

�W (s) = 0.

Alternatively W�(s) + JY

�(s) = 0, where J =

�0 −11 0

�, so that

W (s)−W (sn) = JY (s).

Since det W (s) ≡ 1, we see that Tr W (s) = 2 if and only ifdet Y (s) = 0.

Linear ordinary differential equation! Just like old times.

Write Φ in terms of its eigenvalues eλ and e

−λ as

Φ =

�coshλ + sinh λ cos 2γ sinhλ sin 2γ

sinhλ sin 2γ coshλ− sinhλ cos 2γ

�.

Write

W = coshψ

�cos α sinα− sinα cos α

�− sinhψ

�− cos β sinβsinβ cos β

�.

Then we get equations for α, β, and ψ.

We have Tr W (s) = 2 cosh ψ(s) cos α(s). So lims→∞ α(s) = ∞ensures that Tr W (s) = 2 infinitely many times.

We have

dα

ds= coshλ− sinhλ tanhψ cos (α + β + 2γ).

So if α(s) does not approach ∞, then we get conditions on λ andγ; basically λ →∞ and γ → 0.

In other words, the eigenvectors of the reduced stretching matrixconverge, with one stretching and one compressing.

What’s the point?

If there is blowup along a Lagrangian path, then either:

� there are infinitely many conjugate point pairs {η(tn), η(tn+1)}for a sequence tn � T , which means the geodesic in Dµ(T3)fails to be minimizing on successively smaller intervals. This isa geometric condition, which makes sense even in the veryweak space of L

2 bijections preserving Borel measures.Techniques of Brenier/Ambrosio/Figalli/Shnirelman for weak(L2) geometry should be useful for analyzing this.

� or, the eigenvectors of the stretching matrix tend to align withsome constant vectors based at x . Dongho Chae hasproduced conditions on the stretching matrix near blowupwhich should be helpful here.

There’s still more to do. We can get some results without assuminganything about the components of Λ, but we can get more if weassume some monotonicity or that certain limits of ratios exist.

For example, if the reduced stretching matrix

Φ(s) =

�e2ks 00 e

−2ks

�for some k > 0, then we can solve the ODE

explicitly. We get infinitely many conjugate pairs if and only ifk < 1.

However this Φ(s) satisfies the conclusion of the previous theoremfor every k > 0.

So we expect sharper results on the components of Φ.

We should also eliminate the assumption that there is an x ∈ T3

such that� T0 |ω

�t, η(t, x)

�| dt = ∞. The technique of

approximating Jacobi fields near a single Lagrangian path probablyextends to approximate them near any path at all, so we should beable to find a length-reducing perturbation based around thevorticity-maximizer. Still need to check this, though.

Numerical results on the stretching matrixΛ(t, x) = Dη(t, x)†Dη(t, x) would be much appreciated.

Numerical results on sectional curvature (as Koji Ohkitani hasdone) would also help.

Thanks to the organizers, and thanks to you!