Penrose condition around rough velocity profiles · 2018-09-24 · Physicalmodels...

Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof

Penrose condition around rough velocity profiles

Aymeric Baradat

CMLS, Ecole Polytechnique

24/09/2018

Aymeric Baradat Penrose condition around rough velocity profiles 1 / 29


Outline

Physical models

The Penrose condition and known results

The measure-valued setting

Main result

Ideas of proof



Physical models



Main result

Ideas of proof



The Vlasov equation

Consider the following Vlasov equation:

(V )

∂t f (t, x , v) + v ·∇x f (t, x , v)−∇xU(t, x)·∇v f (t, x , v) = 0,

U(t, x) = A ·∫

Φ(v)f (t, x , v) dv ,

f |t=0 = f0,

where• t ≥ 0, x ∈ Td , v ∈ Rd ,• Φ is a smooth function and A is a multiplier operator.



The kinetic Euler equation (kEu)

The potential is the pressure field (U = p), prescribed through theconstraint ∫

f (t, x , v) dv ≡ 1.

One gets

−∆p(t, x) = div div∫

v ⊗ v f (t, x , v) dv .

Φ(v) = v ⊗ v and A = (−∆)−1 div div.Brenier 89, Grenier 95.• Euler-Lagrange equation of the Brenier model (Brenier 93,

Ambrosio-Figalli 09).• In 1D, equivalent to the 2D hydrostatic Euler equation

(Zakharov 80).





f (t, x , v) dv ≡ 1.

One gets


v ⊗ v f (t, x , v) dv .

Φ(v) = v ⊗ v and A = (−∆)−1 div div.

Brenier 89, Grenier 95.• Euler-Lagrange equation of the Brenier model (Brenier 93,


(Zakharov 80).





f (t, x , v) dv ≡ 1.

One gets


v ⊗ v f (t, x , v) dv .

Φ(v) = v ⊗ v and A = (−∆)−1 div div.Brenier 89, Grenier 95.

• Euler-Lagrange equation of the Brenier model (Brenier 93,Ambrosio-Figalli 09).• In 1D, equivalent to the 2D hydrostatic Euler equation

(Zakharov 80).





f (t, x , v) dv ≡ 1.

One gets


v ⊗ v f (t, x , v) dv .


Ambrosio-Figalli 09).

• In 1D, equivalent to the 2D hydrostatic Euler equation(Zakharov 80).





f (t, x , v) dv ≡ 1.

One gets


v ⊗ v f (t, x , v) dv .



(Zakharov 80).



The Vlasov-Benney equation (VB)

The potential is the density

U(t, x) = ρ(t, x) =

∫f (t, x , v) dv .

Φ ≡ 1, A = Id.

Han-Kwan 11, Jabin-Nouri 11, Bardos-Nouri 12



The Vlasov-Benney equation (VB)

The potential is the density

U(t, x) = ρ(t, x) =

∫f (t, x , v) dv .

Φ ≡ 1, A = Id.Han-Kwan 11, Jabin-Nouri 11, Bardos-Nouri 12



Physical models



Main result

Ideas of proof



Linearization around stationary solutions

In each case, f (t, x , v) = µ(v) (smooth for the moment) is a sta-tionary solution with ∇xU = 0.

The linearization of (V ) leads to

(L)

∂t f (t, x , v) + v · ∇x f (t, x , v)−∇xU(t, x) · ∇vµ(v) = 0,

U(t, x) = A ·∫

Φ(v)f (t, x , v) dv ,

f |t=0 = f0,

Use the ansatz (exponential growing mode or EGM)

f (t, x , v) = g(v) exp(in · x) exp(λt),

where n ∈ Zd is the frequency, and λ ∈ C with <(λ) > 0 is thegrowing rate.




In each case, f (t, x , v) = µ(v) (smooth for the moment) is a sta-tionary solution with ∇xU = 0.The linearization of (V ) leads to

(L)


U(t, x) = A ·∫

Φ(v)f (t, x , v) dv ,

f |t=0 = f0,







In each case, f (t, x , v) = µ(v) (smooth for the moment) is a sta-tionary solution with ∇xU = 0.The linearization of (V ) leads to

(L)


U(t, x) = A ·∫

Φ(v)f (t, x , v) dv ,

f |t=0 = f0,






Unstability condition

Proposition (Penrose 60)

Let µ be a smooth profile. Equation (L) admits an EGM offrequency n ∈ Zd and growing rate λ if and only if

(Pen)

∫in · ∇vµ(v)

λ+ in · vdv =

{0 in (kEu),

1 in (VB).

If there exist EGMs, we say that µ is unstable. Else, we say that µis stable.



Unstability condition

Proposition (Penrose 60)

Let µ be a smooth profile. Equation (L) admits an EGM offrequency n ∈ Zd and growing rate λ if and only if

(Pen)

∫in · ∇vµ(v)

λ+ in · vdv =

{0 in (kEu),

1 in (VB).

If there exist EGMs, we say that µ is unstable. Else, we say that µis stable.



Shape of stable and unstable profiles

In dimension 1:

v

µ(v)

0

STABLE

v

µ(v)

0

UNSTABLE



Unbounded spectrum

Proposition

If µ is unstable, if (n, λ) satisfies (Pen) and if k ∈ N∗, then(kn, kλ) also satisfies (Pen).

As a consequence, we define

γ0 := sup(n,λ) satisfying (Pen)

<(λ)

|n|.

EGMs of frequency n grow like

exp(γ0|n|t).



Unbounded spectrum

Proposition




<(λ)

|n|.


exp(γ0|n|t).



Unbounded spectrum

Proposition




<(λ)

|n|.


exp(γ0|n|t).



Ill-posedness for (kEu) and (VB)

Theorem (Han-Kwan, Nguyen 16)

Let µ be analytic, Penrose unstable and satisfying cancellationconditions. For all s ∈ N, α ∈ (0, 1], there exist solutions f k up toTk with Tk → 0 and

‖f k − µ‖L2([0,Tk )×Td )

‖f k0 − µ‖αHs

−→k→+∞

+∞.



Physical models



Main result

Ideas of proof



Measure-valued solutions

A measure-valued solution to (V ) will be an

f : [0,T )× Td −→M+(Rd),

with• for all ϕ ∈ C∞c (Rd) ∪ {Φ}, the macroscopic observable

〈f , ϕ〉 : (t, x) 7−→∫ϕ(v)f (t, x , dv)

is smooth,• for all ϕ ∈ C∞c (Rd),

∂t〈f , ϕ〉+ divx〈f , vϕ〉+∇xU · 〈f ,∇vϕ〉 = 0,U = A · 〈f ,Φ〉,f |t=0 = f0.





f : [0,T )× Td −→M+(Rd),


〈f , ϕ〉 : (t, x) 7−→∫ϕ(v)f (t, x , dv)

is smooth,

• for all ϕ ∈ C∞c (Rd),∂t〈f , ϕ〉+ divx〈f , vϕ〉+∇xU · 〈f ,∇vϕ〉 = 0,

U = A · 〈f ,Φ〉,f |t=0 = f0.





f : [0,T )× Td −→M+(Rd),


〈f , ϕ〉 : (t, x) 7−→∫ϕ(v)f (t, x , dv)

is smooth,• for all ϕ ∈ C∞c (Rd),

∂t〈f , ϕ〉+ divx〈f , vϕ〉+∇xU · 〈f ,∇vϕ〉 = 0,U = A · 〈f ,Φ〉,f |t=0 = f0.



The Penrose condition still makes sense

Any µ ∈M+(Rd) with∫|Φ(v)| dµ(v) < +∞

provides a stationary solution.

It is said to be unstable if for somen ∈ Zd and λ with positive real part,∫

|n|2

(λ+ in · v)2 dµ(v) =

{0 in (kEu),

1 in (VB).

Remark: a Dirac mass is stable, a superposition of a finite numberDirac masses is unstable.





provides a stationary solution. It is said to be unstable if for somen ∈ Zd and λ with positive real part,∫

|n|2

(λ+ in · v)2 dµ(v) =

{0 in (kEu),

1 in (VB).






provides a stationary solution. It is said to be unstable if for somen ∈ Zd and λ with positive real part,∫

|n|2

(λ+ in · v)2 dµ(v) =

{0 in (kEu),

1 in (VB).




Multifluid formulation

Pictures from Frans Ebersohn, PEPL, University of Michigan.




This suggests looking for solutionsthat can be written as superpositionsof smooth graphs (with density).

v

x

In other terms, we look for solutions f such that for all ϕ ∈ C∞c (Rd),

〈f , ϕ〉(t, x) =

∫ϕ(vα(t, x))ρα(t, x) dm(α)

for some smooth densities (ρα), some smooth velocity fields (vα)and some measure m on a set of indices I = {α}.Convention:• dm(α): total mass of particles located on graph α,•

∫ρα dx = 1.





v

x


〈f , ϕ〉(t, x) =


for some smooth densities (ρα), some smooth velocity fields (vα)and some measure m on a set of indices I = {α}.

Convention:• dm(α): total mass of particles located on graph α,•

∫ρα dx = 1.





v

x


〈f , ϕ〉(t, x) =


for some smooth densities (ρα), some smooth velocity fields (vα)and some measure m on a set of indices I = {α}.Convention:• dm(α): total mass of particles located on graph α,•∫ρα dx = 1.




If µ ∈M+(Rd) is a measure-valued stationary solution, it is of thisform with I = Rd and m = µ.

v

x

• dµ(w) is the mass of particles withvelocity w ,• for all w ∈ Rd , define vws (x) ≡ w

(function of the flat graph correspondingto velocity w),• for all w ∈ Rd , define ρws (x) ≡ 1 (in the

stationary case, the particles are uniformlydistributed on the graphs).

Then for all test function ϕ,∫ϕ(w) dµ(w) =

∫ϕ(vws (x))ρws (x) dµ(w).





v

x










v

x

• dµ(w) is the mass of particles withvelocity w ,

• for all w ∈ Rd , define vws (x) ≡ w(function of the flat graph correspondingto velocity w),• for all w ∈ Rd , define ρws (x) ≡ 1 (in the








v

x


(function of the flat graph correspondingto velocity w),

• for all w ∈ Rd , define ρws (x) ≡ 1 (in thestationary case, the particles are uniformlydistributed on the graphs).







v

x










v

x









Then, if (ρ, v) = (ρw (t, x), vw (t, x)) is in the neighbourhood of thestationary solution (ρs , v s), it represents our physical model if:

• the densities are transported by the corresponding velocities:

∀w ∈ Rd , ∂tρw + div(ρwvw ) = 0,

• the acceleration of each particle is induced by the samepotential U:

∀w ∈ Rd , ∂tvw + (vw · ∇)vw = −∇U,

• U is obtained by applying A to a macroscopic observable:

U = A ·∫

Φ(vw )ρw dµ(w).




Then, if (ρ, v) = (ρw (t, x), vw (t, x)) is in the neighbourhood of thestationary solution (ρs , v s), it represents our physical model if:• the densities are transported by the corresponding velocities:



∀w ∈ Rd , ∂tvw + (vw · ∇)vw = −∇U,


U = A ·∫

Φ(vw )ρw dµ(w).







∀w ∈ Rd , ∂tvw + (vw · ∇)vw = −∇U,


U = A ·∫

Φ(vw )ρw dµ(w).







∀w ∈ Rd , ∂tvw + (vw · ∇)vw = −∇U,


U = A ·∫

Φ(vw )ρw dµ(w).




We get the following system: for all w ∈ Rd ,

(MFµ)

∂tρw + div(ρwvw ) = 0,

∂tvw + (vw · ∇)vw = −∇U,

U = A ·∫

Φ(vw )ρw dµ(w),

ρw |t=0 = ρw0 and vw |t=0 = vw0 .

See Grenier 95, Brenier 97...




Proposition

If (ρ, v) is a smooth solution to (MFµ) on [0,T ), take f definedfor all test function ϕ, for all t ∈ [0,T ) and for all x ∈ Td by

〈f , ϕ〉(t, x) =

∫ϕ(vw (t, x))ρw (t, x) dµ(w).

Then f is a measured-valued solution to (V ).



Penrose in the multifluid formulation

Linearizing (MFµ) around (ρs , v s), we recover the Penrose condition.

Proposition

Let µ ∈M+ with∫|Φ| dµ < +∞. The linearization of (MFµ)

around (ρs , v s) admits an EGM of frequency n ∈ Zd and growingrate λ with <(λ) > 0 if and only if∫

|n|2

(λ+ in · v)2 dµ(v) =

{0 in (kEu),

1 in (VB).



Penrose in the multifluid formulation

Linearizing (MFµ) around (ρs , v s), we recover the Penrose condition.

Proposition

Let µ ∈M+ with∫|Φ| dµ < +∞. The linearization of (MFµ)

around (ρs , v s) admits an EGM of frequency n ∈ Zd and growingrate λ with <(λ) > 0 if and only if∫

|n|2

(λ+ in · v)2 dµ(v) =

{0 in (kEu),

1 in (VB).



Physical models



Main result

Ideas of proof



Ill-posedness for (kEu) and (VB), multifluid formulation

Theorem (B. in preparation)

Take µ an unstable profile, s ∈ N and α ∈ (0, 1].

Then there exist(Tk) ∈ (R∗+)N tending to zero and (ρk

0 , vk0)k∈N a family of initial

data such that:• for all k , there is a solution (ρk , vk) to (MFµ) starting from

(ρk0 , v

k0) up to time Tk (we denote by Uk the corresponding

potential, and by Us the stationary potential),• we have:

‖Uk − Us‖L1([0,Tk )×Td )

supw∈Rd

{‖ρk,w0 − 1‖αW s,∞ + ‖vk,w0 − w‖αW s,∞

} −→k→+∞

+∞.





Take µ an unstable profile, s ∈ N and α ∈ (0, 1].Then there exist(Tk) ∈ (R∗+)N tending to zero and (ρk


data such that:

• for all k , there is a solution (ρk , vk) to (MFµ) starting from(ρk

0 , vk0) up to time Tk (we denote by Uk the corresponding


‖Uk − Us‖L1([0,Tk )×Td )

supw∈Rd

{‖ρk,w0 − 1‖αW s,∞ + ‖vk,w0 − w‖αW s,∞

} −→k→+∞

+∞.








(ρk0 , v


potential, and by Us the stationary potential),

• we have:

‖Uk − Us‖L1([0,Tk )×Td )

supw∈Rd

{‖ρk,w0 − 1‖αW s,∞ + ‖vk,w0 − w‖αW s,∞

} −→k→+∞

+∞.








(ρk0 , v



‖Uk − Us‖L1([0,Tk )×Td )

supw∈Rd

{‖ρk,w0 − 1‖αW s,∞ + ‖vk,w0 − w‖αW s,∞

} −→k→+∞

+∞.



Ill-posedness for (kEu) and (VB), kinetic formulation

Corollary

Take µ an unstable profile, ϕ1, . . . , ϕN ∈ C∞c (Rd), s ∈ N andα ∈ (0, 1].

Then there exists, (Tk) ∈ (R∗+)N tending to 0 and (f k0 ) afamily of measure-valued initial data such that:• for all k , there is a measure-valued solution f k to (V ) starting

from f k0 up to time Tk (we denote by Uk the correspondingpotential, and by Us the stationary potential),• we have:

‖Uk − Us‖L1([0,Tk )×Td )∑Ni=1 ‖〈f k0 , ϕi 〉 − 〈µ, ϕi 〉‖αW s,∞(Td )

−→k→+∞

+∞.




Corollary

Take µ an unstable profile, ϕ1, . . . , ϕN ∈ C∞c (Rd), s ∈ N andα ∈ (0, 1].Then there exists, (Tk) ∈ (R∗+)N tending to 0 and (f k0 ) afamily of measure-valued initial data such that:

• for all k , there is a measure-valued solution f k to (V ) startingfrom f k0 up to time Tk (we denote by Uk the correspondingpotential, and by Us the stationary potential),• we have:


−→k→+∞

+∞.




Corollary

Take µ an unstable profile, ϕ1, . . . , ϕN ∈ C∞c (Rd), s ∈ N andα ∈ (0, 1].Then there exists, (Tk) ∈ (R∗+)N tending to 0 and (f k0 ) afamily of measure-valued initial data such that:• for all k , there is a measure-valued solution f k to (V ) starting

from f k0 up to time Tk (we denote by Uk the correspondingpotential, and by Us the stationary potential),

• we have:


−→k→+∞

+∞.




Corollary

Take µ an unstable profile, ϕ1, . . . , ϕN ∈ C∞c (Rd), s ∈ N andα ∈ (0, 1].Then there exists, (Tk) ∈ (R∗+)N tending to 0 and (f k0 ) afamily of measure-valued initial data such that:• for all k , there is a measure-valued solution f k to (V ) starting

from f k0 up to time Tk (we denote by Uk the correspondingpotential, and by Us the stationary potential),• we have:


−→k→+∞

+∞.



Physical models



Main result

Ideas of proof



Analytic framework

If ρw (x) =∑n∈Zd

αn(w) exp(in·x) and vw (x) =∑n∈Zd

βn(w) exp(in·x),

we work with the norms

∀δ > 0, ‖ρ, v‖δ =∑n∈Zd

{‖αn‖∞ + ‖βn‖∞

}exp(δ|n|).

Used for PDEs in the 90’s in various contexts:• Foias, Temam 89, Titi, Doelman 93: speed of convergence of

Galerkin approximations for semilinear parabolic PDEs,• Caflisch 90, Safonov 95: Cauchy-Kovalevskaya theorems for

quasilinear PDEs.



Analytic framework

If ρw (x) =∑n∈Zd

αn(w) exp(in·x) and vw (x) =∑n∈Zd

βn(w) exp(in·x),

we work with the norms

∀δ > 0, ‖ρ, v‖δ =∑n∈Zd

{‖αn‖∞ + ‖βn‖∞

}exp(δ|n|).

Used for PDEs in the 90’s in various contexts:• Foias, Temam 89, Titi, Doelman 93: speed of convergence of

Galerkin approximations for semilinear parabolic PDEs,• Caflisch 90, Safonov 95: Cauchy-Kovalevskaya theorems for

quasilinear PDEs.



Sketch of proof


<(λ)

|n|.

• Prove a sharp estimate for the semigroup of the linearizedoperator: for all Γ > γ0, for all δ > 0, for all (ρ0, v0):

‖St(ρ0, v0)‖δ−Γt ≤ C‖ρ0, v0‖δ.

• Derive a Cauchy-Kovalevskaya theorem to build solutions nearthe EGMs up to time δ/Γ.• Take solutions around more and more unstable EGMs.



Sketch of proof


<(λ)

|n|.


‖St(ρ0, v0)‖δ−Γt ≤ C‖ρ0, v0‖δ.




Sketch of proof


<(λ)

|n|.


‖St(ρ0, v0)‖δ−Γt ≤ C‖ρ0, v0‖δ.

• Derive a Cauchy-Kovalevskaya theorem to build solutions nearthe EGMs up to time δ/Γ.

• Take solutions around more and more unstable EGMs.



Sketch of proof


<(λ)

|n|.


‖St(ρ0, v0)‖δ−Γt ≤ C‖ρ0, v0‖δ.




Thank you!


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Penrose condition around rough velocity profiles · 2018-09-24 · Physicalmodels...

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