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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
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Outline
Physical models
The Penrose condition and known results
The measure-valued setting
Main result
Ideas of proof
Aymeric Baradat Penrose condition around rough velocity profiles 2 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Physical models
The Penrose condition and known results
The measure-valued setting
Main result
Ideas of proof
Aymeric Baradat Penrose condition around rough velocity profiles 3 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 4 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 5 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 5 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 5 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 5 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 5 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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
Aymeric Baradat Penrose condition around rough velocity profiles 6 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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
Aymeric Baradat Penrose condition around rough velocity profiles 6 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Physical models
The Penrose condition and known results
The measure-valued setting
Main result
Ideas of proof
Aymeric Baradat Penrose condition around rough velocity profiles 7 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 8 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 8 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 8 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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.
Aymeric Baradat Penrose condition around rough velocity profiles 9 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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.
Aymeric Baradat Penrose condition around rough velocity profiles 9 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Shape of stable and unstable profiles
In dimension 1:
v
µ(v)
0
STABLE
v
µ(v)
0
UNSTABLE
Aymeric Baradat Penrose condition around rough velocity profiles 10 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 11 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 11 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 11 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 12 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Physical models
The Penrose condition and known results
The measure-valued setting
Main result
Ideas of proof
Aymeric Baradat Penrose condition around rough velocity profiles 13 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 14 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 14 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 14 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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.
Aymeric Baradat Penrose condition around rough velocity profiles 15 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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.
Aymeric Baradat Penrose condition around rough velocity profiles 15 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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.
Aymeric Baradat Penrose condition around rough velocity profiles 15 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
Pictures from Frans Ebersohn, PEPL, University of Michigan.
Aymeric Baradat Penrose condition around rough velocity profiles 16 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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.
Aymeric Baradat Penrose condition around rough velocity profiles 17 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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.
Aymeric Baradat Penrose condition around rough velocity profiles 17 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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.
Aymeric Baradat Penrose condition around rough velocity profiles 17 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 18 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 18 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 18 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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 thestationary 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).
Aymeric Baradat Penrose condition around rough velocity profiles 18 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 18 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 18 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 19 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 19 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 19 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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).
Aymeric Baradat Penrose condition around rough velocity profiles 19 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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...
Aymeric Baradat Penrose condition around rough velocity profiles 20 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Multifluid formulation
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 ).
Aymeric Baradat Penrose condition around rough velocity profiles 21 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 22 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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).
Aymeric Baradat Penrose condition around rough velocity profiles 22 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Physical models
The Penrose condition and known results
The measure-valued setting
Main result
Ideas of proof
Aymeric Baradat Penrose condition around rough velocity profiles 23 / 29
Physical models The Penrose condition and known results The measure-valued setting 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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 24 / 29
Physical models The Penrose condition and known results The measure-valued setting 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(ρk
0 , vk0) 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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 24 / 29
Physical models The Penrose condition and known results The measure-valued setting 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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 24 / 29
Physical models The Penrose condition and known results The measure-valued setting 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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 24 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 25 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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 ) startingfrom 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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 25 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 25 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
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→+∞
+∞.
Aymeric Baradat Penrose condition around rough velocity profiles 25 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Physical models
The Penrose condition and known results
The measure-valued setting
Main result
Ideas of proof
Aymeric Baradat Penrose condition around rough velocity profiles 26 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 27 / 29
Physical models The Penrose condition and known results The measure-valued setting 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.
Aymeric Baradat Penrose condition around rough velocity profiles 27 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Sketch of proof
γ0 := sup(n,λ) satisfying (Pen)
<(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.
Aymeric Baradat Penrose condition around rough velocity profiles 28 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Sketch of proof
γ0 := sup(n,λ) satisfying (Pen)
<(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.
Aymeric Baradat Penrose condition around rough velocity profiles 28 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Sketch of proof
γ0 := sup(n,λ) satisfying (Pen)
<(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.
Aymeric Baradat Penrose condition around rough velocity profiles 28 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Sketch of proof
γ0 := sup(n,λ) satisfying (Pen)
<(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.
Aymeric Baradat Penrose condition around rough velocity profiles 28 / 29
Physical models The Penrose condition and known results The measure-valued setting Main result Ideas of proof
Thank you!
Aymeric Baradat Penrose condition around rough velocity profiles 29 / 29