Lecture 1: Main Models & Basics of Wasserstein...
Transcript of Lecture 1: Main Models & Basics of Wasserstein...
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Lecture 1: Main Models & Basics ofWasserstein Distance
J. A. Carrillo
ICREA - Universitat Autònoma de Barcelona
Methods and Models of Kinetic Theory
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Leading examples
Nonlinear Diffusions.-
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
Inelastic Dissipative Models: Nonlinear friction equations.- 1:
∂f∂t
+ v∂f∂x
=∂
∂v
»ZR(v− w)|v− w|γ(|v− w|)f (x, w, t) dw f (x, v, t)
–Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.-
∂f∂t
+ (v · ∇x)f =Qe(f , f )
→ Conservation of mass and center of mass/mean velocity.
→ Spreading (diffusions) versus Concentration (dissipative models).
1Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat.
Phys. (1998).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Leading examples
Nonlinear Diffusions.-
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
Inelastic Dissipative Models: Nonlinear friction equations.- 1:
∂f∂t
+ v∂f∂x
=∂
∂v
»ZR(v− w)|v− w|γ(|v− w|)f (x, w, t) dw f (x, v, t)
–Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.-
∂f∂t
+ (v · ∇x)f =Qe(f , f )
→ Conservation of mass and center of mass/mean velocity.
→ Spreading (diffusions) versus Concentration (dissipative models).
1Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat.
Phys. (1998).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Leading examples
Nonlinear Diffusions.-
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
Inelastic Dissipative Models: Nonlinear friction equations.- 1:
∂f∂t
+ v∂f∂x
=∂
∂v
»ZR(v− w)|v− w|γ(|v− w|)f (x, w, t) dw f (x, v, t)
–Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.-
∂f∂t
+ (v · ∇x)f =Qe(f , f )
→ Conservation of mass and center of mass/mean velocity.
→ Spreading (diffusions) versus Concentration (dissipative models).
1Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat.
Phys. (1998).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Leading examples
Nonlinear Diffusions.-
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
Inelastic Dissipative Models: Nonlinear friction equations.- 1:
∂f∂t
+ v∂f∂x
=∂
∂v
»ZR(v− w)|v− w|γ(|v− w|)f (x, w, t) dw f (x, v, t)
–Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.-
∂f∂t
+ (v · ∇x)f =Qe(f , f )
→ Conservation of mass and center of mass/mean velocity.
→ Spreading (diffusions) versus Concentration (dissipative models).
1Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat.
Phys. (1998).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Leading examples
Nonlinear Diffusions.-
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
Inelastic Dissipative Models: Nonlinear friction equations.- 1:
∂f∂t
+ v∂f∂x
=∂
∂v
»ZR(v− w)|v− w|γ(|v− w|)f (x, w, t) dw f (x, v, t)
–Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.-
∂f∂t
+ (v · ∇x)f =Qe(f , f )
→ Conservation of mass and center of mass/mean velocity.
→ Spreading (diffusions) versus Concentration (dissipative models).
1Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat.
Phys. (1998).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
→ Self-similar solution: Barenblatt profile2.- An explicit self-similar solution thatis integrable for m > (d − 2)/d:
Bm(x, t) = t−d/λ
„C − 2m
1− m|x|2
t2/λ
«1/(1−m)
+
for m 6= 1 where λ = d(m− 1) + 2 and C > 0 is determined to have unit mass.It verifies that Bm(x, t) converges weakly-* as measures towards δ0 as t → 0+.
2Zeldovich, Ya. B., Barenblatt, G. I. Doklady, USSR Academy of Sciences (1958).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
→ Asymptotic behaviour: Comparison methods.- Friedman & Kamin (1980)(completed by Vázquez (1997)). Given an initial data in the class
X0 = u0 ∈ L1(Rd) : u0 ≥ 0 ,
then, for any (d − 2)/d < m
limt→∞
‖u(·, t)−Bm(·, t)‖L1 = 0, and limt→∞
td/λ‖u(·, t)−Bm(·, t)‖L∞ = 0.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
∂ρ
∂t= div(xρ +∇ρm), (x ∈ Rd, t > 0)
ρ(x, t = 0) = ρ0(x) ≥ 0, (x ∈ Rd)
??
ρ(x, t) = edtu(etx,1λ
(eλt − 1))
λ = d(m− 1) + 2
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
u(x, t = 0) = u0(x) ≥ 0, (x ∈ Rd)3
→ Exponential decay to equilibria translates into algebraic decay to Barenblattprofiles.
3J. A. Carrillo, G. Toscani, Indiana Math. Univ. J. (2000); F. Otto, Comm. PDE (2001); J. Dolbeault, M. del Pino, J. Math. Pures Appl. (2002); J.L.
Vázquez, J. Evol. Eq. 2003.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
∂ρ
∂t= div(xρ +∇ρm), (x ∈ Rd, t > 0)
ρ(x, t = 0) = ρ0(x) ≥ 0, (x ∈ Rd)
??
ρ(x, t) = edtu(etx,1λ
(eλt − 1))
λ = d(m− 1) + 2
∂u∂t
= ∆um, (x ∈ Rd, t > 0)
u(x, t = 0) = u0(x) ≥ 0, (x ∈ Rd)3
→ Exponential decay to equilibria translates into algebraic decay to Barenblattprofiles.
3J. A. Carrillo, G. Toscani, Indiana Math. Univ. J. (2000); F. Otto, Comm. PDE (2001); J. Dolbeault, M. del Pino, J. Math. Pures Appl. (2002); J.L.
Vázquez, J. Evol. Eq. 2003.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
→ Generic finite speed of propagation/moving free boundary for degeneratediffusion equations and creation of thick tails for fast diffusion equations.
→ No better rate of convergence can be established under the generality u0 aprobability density.At which rate does this self-similarity take over?
→ General nonlinear diffusion equation:8><>:∂u∂t
= ∆P(u), (x ∈ Rd, t > 0),
u(x, t = 0) = u0(x) ≥ 0 (x ∈ Rd)
No explicit source-type solutions, so...What is the typical asymptotic profile?
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
→ Generic finite speed of propagation/moving free boundary for degeneratediffusion equations and creation of thick tails for fast diffusion equations.
→ No better rate of convergence can be established under the generality u0 aprobability density.At which rate does this self-similarity take over?
→ General nonlinear diffusion equation:8><>:∂u∂t
= ∆P(u), (x ∈ Rd, t > 0),
u(x, t = 0) = u0(x) ≥ 0 (x ∈ Rd)
No explicit source-type solutions, so...What is the typical asymptotic profile?
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Nonlinear diffusions
Nonlinear diffusions
→ Generic finite speed of propagation/moving free boundary for degeneratediffusion equations and creation of thick tails for fast diffusion equations.
→ No better rate of convergence can be established under the generality u0 aprobability density.At which rate does this self-similarity take over?
→ General nonlinear diffusion equation:8><>:∂u∂t
= ∆P(u), (x ∈ Rd, t > 0),
u(x, t = 0) = u0(x) ≥ 0 (x ∈ Rd)
No explicit source-type solutions, so...What is the typical asymptotic profile?
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Rapid Granular Flows
Pattern formation in a vertically oscillated granular layer. 4
4Bizon, C., Shattuck, M. D., Swift, J.B., Swinney, H.L., Phys. Rev. E (1999); Carrillo, J.A., Poschel, T., Salueña, C., in preparation (2006).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Rapid Granular Flows
Schock waves in supersonic sand. 5
5Rericha, E., Bizon, C., Shattuck, M. D., Swinney, H.L., Phys. Rev. Letters (2002).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Inelastic Collisions
Binary Collisions:
Spheres of diameter r > 0. Given (x, v) and (x− rn, w), where n ∈ S2 is the unitvector along the impact direction, the post-collisional velocities are found assumingconservation of momentum and a loose of normal relative velocity after the collision:
(v′ − w′) · n = −e((v− w) · n)
where 0 < e ≤ 1 is called the restitution coefficient.
Postcollisional velocities:
v′ =12(v + w) +
u′
2
w′ =12(v + w)− u′
2
where u′ = u− (1 + e)(u · n)n, u = v− w and u′ = v′ − w′.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Inelastic Collisions
Binary Collisions:
Spheres of diameter r > 0. Given (x, v) and (x− rn, w), where n ∈ S2 is the unitvector along the impact direction, the post-collisional velocities are found assumingconservation of momentum and a loose of normal relative velocity after the collision:
(v′ − w′) · n = −e((v− w) · n)
where 0 < e ≤ 1 is called the restitution coefficient.
Postcollisional velocities:
v′ =12(v + w) +
u′
2
w′ =12(v + w)− u′
2
where u′ = u− (1 + e)(u · n)n, u = v− w and u′ = v′ − w′.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Inelastic Collisions
Postcollisional velocities: NewParameterization
v′ =12(v + w) +
u′
2
w′ =12(v + w)− u′
2
where
u′ =1− e
4u +
1 + e4|u|σ,
u = v− w and u′ = v′ − w′.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Boltzmann equation for granular gasesHypothesis:
Binary, localized in t and x and inelastic collisions.
Molecular chaos.
Boltzmann equation for inelastic particles:
∂f∂t
+(v ·∇x)f =Qe(f , f )=1
4π
ZR3
ZS2+
((v−w) · n)
»1e2 f (v∗)f (w∗)− f (v)f (w)
–dndw.a
Weak form of the Boltzmann equation:
< ϕ, Qe(f , f ) >=1
4π
ZR3
ZR3
ZS2|v− w|f (v)f (w)
hϕ(v′)− ϕ(v)
idσ dv dw
aJenkins, J. T., Richman, M. W., Arch. Rat. Mech. Anal. (1985).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Boltzmann equation for granular gasesHypothesis:
Binary, localized in t and x and inelastic collisions.
Molecular chaos.
Boltzmann equation for inelastic particles:
∂f∂t
+(v ·∇x)f =Qe(f , f )=1
4π
ZR3
ZS2+
((v−w) · n)
»1e2 f (v∗)f (w∗)− f (v)f (w)
–dndw.a
Weak form of the Boltzmann equation:
< ϕ, Qe(f , f ) >=1
4π
ZR3
ZR3
ZS2|v− w|f (v)f (w)
hϕ(v′)− ϕ(v)
idσ dv dw
aJenkins, J. T., Richman, M. W., Arch. Rat. Mech. Anal. (1985).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Boltzmann equation for granular gasesHypothesis:
Binary, localized in t and x and inelastic collisions.
Molecular chaos.
Boltzmann equation for inelastic particles:
∂f∂t
+(v ·∇x)f =Qe(f , f )=1
4π
ZR3
ZS2+
((v−w) · n)
»1e2 f (v∗)f (w∗)− f (v)f (w)
–dndw.a
Weak form of the Boltzmann equation:
< ϕ, Qe(f , f ) >=1
4π
ZR3
ZR3
ZS2|v− w|f (v)f (w)
hϕ(v′)− ϕ(v)
idσ dv dw
aJenkins, J. T., Richman, M. W., Arch. Rat. Mech. Anal. (1985).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Boltzmann equation for granular gasesHypothesis:
Binary, localized in t and x and inelastic collisions.
Molecular chaos.
Boltzmann equation for inelastic particles:
∂f∂t
+(v ·∇x)f =Qe(f , f )=1
4π
ZR3
ZS2+
((v−w) · n)
»1e2 f (v∗)f (w∗)− f (v)f (w)
–dndw.a
Weak form of the Boltzmann equation:
< ϕ, Qe(f , f ) >=1
4π
ZR3
ZR3
ZS2|v− w|f (v)f (w)
hϕ(v′)− ϕ(v)
idσ dv dw
aJenkins, J. T., Richman, M. W., Arch. Rat. Mech. Anal. (1985).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Basic Properties6
∂f∂t
= Qe(f , f )
Mass and momentum are preserved while energy decreases:
|v′|2 + |w′|2 − |v|2 − |w|2 =1− e2
2|u|2
Let us fix unit mass and zero mean velocity for the rest.
Temperature cools off down to 0 as t →∞ (Haff’s law):
dθ
dt≤ − 1− e2
4θ
32 ,
What are the typical asymptotic profiles? Do self-similar solutions exist?If so At which rate does this self-similarity take over?
6J.A. Carrillo, C. Cercignani, I.M. Gamba, Phys. Rev. E (2000); C. Cercignani, R. Illner, C. Stoica, J. Stat. Phys. (2001); A. Bobylev, C.
Cercignani, J. Stat. Phys. (2002); I.M. Gamba, V. Panferov, C. Villani, Comm. Math. Phys. (2004); C. Mouhot, S. Mischer, Martínez-Ricard, J. Stat.
Phys. (2006).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Basic Properties6
∂f∂t
= Qe(f , f )
Mass and momentum are preserved while energy decreases:
|v′|2 + |w′|2 − |v|2 − |w|2 =1− e2
2|u|2
Let us fix unit mass and zero mean velocity for the rest.
Temperature cools off down to 0 as t →∞ (Haff’s law):
dθ
dt≤ − 1− e2
4θ
32 ,
What are the typical asymptotic profiles? Do self-similar solutions exist?If so At which rate does this self-similarity take over?
6J.A. Carrillo, C. Cercignani, I.M. Gamba, Phys. Rev. E (2000); C. Cercignani, R. Illner, C. Stoica, J. Stat. Phys. (2001); A. Bobylev, C.
Cercignani, J. Stat. Phys. (2002); I.M. Gamba, V. Panferov, C. Villani, Comm. Math. Phys. (2004); C. Mouhot, S. Mischer, Martínez-Ricard, J. Stat.
Phys. (2006).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Simplified Granular media model
Another dissipative models (toy models) keeping the same properties: conservationof mass, mean velocity and dissipation of energy are of the form:
∂f∂t
+ (v · ∇x)f = I(f , f )
where
I(f , f ) = divv
»ZR3
(v− w)|v− w|f (x, w, t) dw f (x, v, t)–
.
This dissipative collision in the one dimensional case simplifies to7:
I(f , f ) =∂
∂v
»ZR(v− w)|v− w|f (x, w, t) dw f (x, v, t)
–
7Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Simplified Granular media model
Another dissipative models (toy models) keeping the same properties: conservationof mass, mean velocity and dissipation of energy are of the form:
∂f∂t
+ (v · ∇x)f = I(f , f )
where
I(f , f ) = divv
»ZR3
(v− w)|v− w|f (x, w, t) dw f (x, v, t)–
.
This dissipative collision in the one dimensional case simplifies to7:
I(f , f ) =∂
∂v
»ZR(v− w)|v− w|f (x, w, t) dw f (x, v, t)
–
7Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Simplified Granular media model
Another dissipative models (toy models) keeping the same properties: conservationof mass, mean velocity and dissipation of energy are of the form:
∂f∂t
+ (v · ∇x)f = I(f , f )
where
I(f , f ) = divv
»ZR3
(v− w)|v− w|f (x, w, t) dw f (x, v, t)–
.
This dissipative collision in the one dimensional case simplifies to7:
I(f , f ) =∂
∂v
»ZR(v− w)|v− w|f (x, w, t) dw f (x, v, t)
–
7Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Summary: Spreading versus ConcentrationDiffusive models - Spreading:
∂u∂t
= ∆P(u), (x ∈ Rd, t > 0)
Typical asymptotic profiles for P(u) = um (Barenblatt):
Bm(x, t) = θm(t)−d/2Bm(θm(t)−1/2x, to,m)
with θm(t) ∞.
Homogeneous Kinetic Dissipative models - Concentration:
∂f∂t
=
8><>:Qe(f , f )
divv
»f (v, t)
ZRd∇W(v− w)f (w, t) dw
–Typical asymptotic profiles (Homogeneous Cooling States), if they exist:
fhc(v, t) = ρ θ− 3
2hc (t) g∞((v− u) θ
− 12
hc (t))
with θhc(t) 0.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Kinetic models for granular gases
Summary: Spreading versus ConcentrationDiffusive models - Spreading:
∂u∂t
= ∆P(u), (x ∈ Rd, t > 0)
Typical asymptotic profiles for P(u) = um (Barenblatt):
Bm(x, t) = θm(t)−d/2Bm(θm(t)−1/2x, to,m)
with θm(t) ∞.
Homogeneous Kinetic Dissipative models - Concentration:
∂f∂t
=
8><>:Qe(f , f )
divv
»f (v, t)
ZRd∇W(v− w)f (w, t) dw
–Typical asymptotic profiles (Homogeneous Cooling States), if they exist:
fhc(v, t) = ρ θ− 3
2hc (t) g∞((v− u) θ
− 12
hc (t))
with θhc(t) 0.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance8
Transporting measures:
Given T : Rd −→ Rd mesurable, we say that T transports µ ∈ P onto ν ∈ P or thatν is the push-forward of µ through T , ν = T#µ, if ν[K] := µ[T−1(K)] for allmesurable sets K ⊂ Rd, equivalentlyZ
Rdϕ dν =
ZRd
(ϕ T) dµ
for all ϕ ∈ Co(Rd).
Random variables:
Say that X is a random variable with law given by µ, is to sayX : (Ω,A, P) −→ (Rd,Bd) is a mesurable map such that X#P = µ, i.e.,Z
Rdϕ(x) dµ =
ZΩ
(ϕ X) dP = E [ϕ(X)] .
Coming back to the transport of measures: if X and Y are random variables with lawµ and ν respectively, then ν = T#µ is equivalent to Y = T(X).
8C. Villani, AMS Graduate Texts (2003).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance8
Transporting measures:
Given T : Rd −→ Rd mesurable, we say that T transports µ ∈ P onto ν ∈ P or thatν is the push-forward of µ through T , ν = T#µ, if ν[K] := µ[T−1(K)] for allmesurable sets K ⊂ Rd, equivalentlyZ
Rdϕ dν =
ZRd
(ϕ T) dµ
for all ϕ ∈ Co(Rd).
Random variables:
Say that X is a random variable with law given by µ, is to sayX : (Ω,A, P) −→ (Rd,Bd) is a mesurable map such that X#P = µ, i.e.,Z
Rdϕ(x) dµ =
ZΩ
(ϕ X) dP = E [ϕ(X)] .
Coming back to the transport of measures: if X and Y are random variables with lawµ and ν respectively, then ν = T#µ is equivalent to Y = T(X).
8C. Villani, AMS Graduate Texts (2003).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance8
Transporting measures:
Given T : Rd −→ Rd mesurable, we say that T transports µ ∈ P onto ν ∈ P or thatν is the push-forward of µ through T , ν = T#µ, if ν[K] := µ[T−1(K)] for allmesurable sets K ⊂ Rd, equivalentlyZ
Rdϕ dν =
ZRd
(ϕ T) dµ
for all ϕ ∈ Co(Rd).
Random variables:
Say that X is a random variable with law given by µ, is to sayX : (Ω,A, P) −→ (Rd,Bd) is a mesurable map such that X#P = µ, i.e.,Z
Rdϕ(x) dµ =
ZΩ
(ϕ X) dP = E [ϕ(X)] .
Coming back to the transport of measures: if X and Y are random variables with lawµ and ν respectively, then ν = T#µ is equivalent to Y = T(X).
8C. Villani, AMS Graduate Texts (2003).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance8
Transporting measures:
Given T : Rd −→ Rd mesurable, we say that T transports µ ∈ P onto ν ∈ P or thatν is the push-forward of µ through T , ν = T#µ, if ν[K] := µ[T−1(K)] for allmesurable sets K ⊂ Rd, equivalentlyZ
Rdϕ dν =
ZRd
(ϕ T) dµ
for all ϕ ∈ Co(Rd).
Random variables:
Say that X is a random variable with law given by µ, is to sayX : (Ω,A, P) −→ (Rd,Bd) is a mesurable map such that X#P = µ, i.e.,Z
Rdϕ(x) dµ =
ZΩ
(ϕ X) dP = E [ϕ(X)] .
Coming back to the transport of measures: if X and Y are random variables with lawµ and ν respectively, then ν = T#µ is equivalent to Y = T(X).
8C. Villani, AMS Graduate Texts (2003).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance
Euclidean Wasserstein Distance:
W2(µ, ν)= infπ
ZZRd×Rd
|x− y|2 dπ(x, y)ff1/2
= inf(X,Y)
nE
h|X − Y|2
io1/2
where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ P2(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.
Monge’s optimal mass transport problem:
Find
I := infT
ZRd|x− T(x)|2 dµ(x); ν = T#µ
ff1/2
.
Take γT = (1Rd × T)#µ as transference plan π.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance
Euclidean Wasserstein Distance:
W2(µ, ν)= infπ
ZZRd×Rd
|x− y|2 dπ(x, y)ff1/2
= inf(X,Y)
nE
h|X − Y|2
io1/2
where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ P2(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.
Monge’s optimal mass transport problem:
Find
I := infT
ZRd|x− T(x)|2 dµ(x); ν = T#µ
ff1/2
.
Take γT = (1Rd × T)#µ as transference plan π.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance
Euclidean Wasserstein Distance:
W2(µ, ν)= infπ
ZZRd×Rd
|x− y|2 dπ(x, y)ff1/2
= inf(X,Y)
nE
h|X − Y|2
io1/2
where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ P2(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.
Monge’s optimal mass transport problem:
Find
I := infT
ZRd|x− T(x)|2 dµ(x); ν = T#µ
ff1/2
.
Take γT = (1Rd × T)#µ as transference plan π.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance
Wasserstein Distances:
Given 1 ≤ p < ∞, we define
Wp(µ, ν)= infπ
ZZRd×Rd
|x− y|p dπ(x, y)ff1/p
= inf(X,Y)
E [|X − Y|p]1/p
where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ Pp(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.
We define the∞-Wasserstein distance as
W∞(µ, ν) = limp∞
Wp(µ, ν) = infTesssup |x− T(x)|, x ∈ supp(µ) with ν = T#µ
for measures with all moments bounded.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance
Wasserstein Distances:
Given 1 ≤ p < ∞, we define
Wp(µ, ν)= infπ
ZZRd×Rd
|x− y|p dπ(x, y)ff1/p
= inf(X,Y)
E [|X − Y|p]1/p
where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ Pp(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.
We define the∞-Wasserstein distance as
W∞(µ, ν) = limp∞
Wp(µ, ν) = infTesssup |x− T(x)|, x ∈ supp(µ) with ν = T#µ
for measures with all moments bounded.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Definition
Definition of the distance
Wasserstein Distances:
Given 1 ≤ p < ∞, we define
Wp(µ, ν)= infπ
ZZRd×Rd
|x− y|p dπ(x, y)ff1/p
= inf(X,Y)
E [|X − Y|p]1/p
where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ Pp(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.
We define the∞-Wasserstein distance as
W∞(µ, ν) = limp∞
Wp(µ, ν) = infTesssup |x− T(x)|, x ∈ supp(µ) with ν = T#µ
for measures with all moments bounded.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Euclidean Wasserstein Distance
Basic Properties
1 Convexity: f1, f2, g1, g2 ∈ P2(Rd) and α ∈ [0, 1], then
W22 (αf1 + (1− α)f2, αg1 + (1− α)g2) ≤ αW2
2 (f1, g1) + (1− α)W22 (f2, g2).
2 Relation to Temperature: f ∈ P2(Rd) and a ∈ R3, then
W22 (f , δa) =
ZRd|v− a|2df (v).
3 Scaling: Given f in P2(Rd) and θ > 0, let us define
Sθ[f ] = θd/2f (θ1/2v)
for absolutely continuous measures with respect to Lebesgue or itscorresponding definition by duality for general measures; then for any f and gin P2(Rd), we have
W2(Sθ[f ],Sθ[g]) = θ−1/2 W2(f , g).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Euclidean Wasserstein Distance
Basic Properties
1 Convexity: f1, f2, g1, g2 ∈ P2(Rd) and α ∈ [0, 1], then
W22 (αf1 + (1− α)f2, αg1 + (1− α)g2) ≤ αW2
2 (f1, g1) + (1− α)W22 (f2, g2).
2 Relation to Temperature: f ∈ P2(Rd) and a ∈ R3, then
W22 (f , δa) =
ZRd|v− a|2df (v).
3 Scaling: Given f in P2(Rd) and θ > 0, let us define
Sθ[f ] = θd/2f (θ1/2v)
for absolutely continuous measures with respect to Lebesgue or itscorresponding definition by duality for general measures; then for any f and gin P2(Rd), we have
W2(Sθ[f ],Sθ[g]) = θ−1/2 W2(f , g).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Euclidean Wasserstein Distance
Basic Properties
1 Convexity: f1, f2, g1, g2 ∈ P2(Rd) and α ∈ [0, 1], then
W22 (αf1 + (1− α)f2, αg1 + (1− α)g2) ≤ αW2
2 (f1, g1) + (1− α)W22 (f2, g2).
2 Relation to Temperature: f ∈ P2(Rd) and a ∈ R3, then
W22 (f , δa) =
ZRd|v− a|2df (v).
3 Scaling: Given f in P2(Rd) and θ > 0, let us define
Sθ[f ] = θd/2f (θ1/2v)
for absolutely continuous measures with respect to Lebesgue or itscorresponding definition by duality for general measures; then for any f and gin P2(Rd), we have
W2(Sθ[f ],Sθ[g]) = θ−1/2 W2(f , g).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Euclidean Wasserstein Distance
Convergence Properties
1 Convergence of measures: W2(fn, f ) → 0 is equivalent to fn f weakly-* asmeasures and convergence of second moments.
2 Weak lower semicontinuity: Given fn f and gn g weakly-* as measures,then
W2(f , g) ≤ lim infn→∞
W2(fn, gn).
3 Completeness: The space P2(Rd) endowed with the distance W2 is a completemetric space. Moreover, the set
Mθ =
µ ∈ P2(Rd) such that
ZR3|v|2df (v) = θ
ff,
i.e. the sphere of radius√
θ in P2(Rd), is a complete metric space.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Euclidean Wasserstein Distance
Convergence Properties
1 Convergence of measures: W2(fn, f ) → 0 is equivalent to fn f weakly-* asmeasures and convergence of second moments.
2 Weak lower semicontinuity: Given fn f and gn g weakly-* as measures,then
W2(f , g) ≤ lim infn→∞
W2(fn, gn).
3 Completeness: The space P2(Rd) endowed with the distance W2 is a completemetric space. Moreover, the set
Mθ =
µ ∈ P2(Rd) such that
ZR3|v|2df (v) = θ
ff,
i.e. the sphere of radius√
θ in P2(Rd), is a complete metric space.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Euclidean Wasserstein Distance
Convergence Properties
1 Convergence of measures: W2(fn, f ) → 0 is equivalent to fn f weakly-* asmeasures and convergence of second moments.
2 Weak lower semicontinuity: Given fn f and gn g weakly-* as measures,then
W2(f , g) ≤ lim infn→∞
W2(fn, gn).
3 Completeness: The space P2(Rd) endowed with the distance W2 is a completemetric space. Moreover, the set
Mθ =
µ ∈ P2(Rd) such that
ZR3|v|2df (v) = θ
ff,
i.e. the sphere of radius√
θ in P2(Rd), is a complete metric space.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Wasserstein Distances
Properties
1 Convexity of Wpp
2 Scaling: for any f and g in Pp(Rd), we have
Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).
3 Completeness and weak lower semicontinuity.
4 Convergence⇐⇒ weak-* convergence as measures and convergence ofp-moments.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Wasserstein Distances
Properties
1 Convexity of Wpp
2 Scaling: for any f and g in Pp(Rd), we have
Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).
3 Completeness and weak lower semicontinuity.
4 Convergence⇐⇒ weak-* convergence as measures and convergence ofp-moments.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Wasserstein Distances
Properties
1 Convexity of Wpp
2 Scaling: for any f and g in Pp(Rd), we have
Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).
3 Completeness and weak lower semicontinuity.
4 Convergence⇐⇒ weak-* convergence as measures and convergence ofp-moments.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Wasserstein Distances
Properties
1 Convexity of Wpp
2 Scaling: for any f and g in Pp(Rd), we have
Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).
3 Completeness and weak lower semicontinuity.
4 Convergence⇐⇒ weak-* convergence as measures and convergence ofp-moments.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
One dimensional Case
Distribution functions:
In one dimension, denoting by F(x) the distribution function of µ,
F(x) =
Z x
−∞dµ,
we can define its pseudo inverse:
F−1(η) = infx : F(x) > η for η ∈ (0, 1),
we have F−1 : ((0, 1),B1), dη) −→ (R,B1) is a random variable with law µ, i.e.,F−1#dη = µ Z
Rϕ(x) dµ =
Z 1
0ϕ(F−1(η)) dη = E [ϕ(X)] .
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
One dimensional Case
Distribution functions:
In one dimension, denoting by F(x) the distribution function of µ,
F(x) =
Z x
−∞dµ,
we can define its pseudo inverse:
F−1(η) = infx : F(x) > η for η ∈ (0, 1),
we have F−1 : ((0, 1),B1), dη) −→ (R,B1) is a random variable with law µ, i.e.,F−1#dη = µ Z
Rϕ(x) dµ =
Z 1
0ϕ(F−1(η)) dη = E [ϕ(X)] .
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
One dimensional Case
Wasserstein distance:
Given µ and ν ∈ P2(Rd) with distribution functions F(x) and G(y) then,
W22 (µ, ν) =
ZZRd×Rd
|x− y|2 dH(x, y)
where H(x, y) = min(F(x), G(y)) is a joint distribution function and the aboveintegral is defined in the Riemann-Stieljes sense.a
In one dimension, it is easy to check that given two measures µ and ν withdistribution functions F(x) and G(y) then, (F−1 × G−1)#dη has joint distributionfunction H(x, y) = min(F(x), G(y)). Therefore, in one dimension, the optimal planis given by πopt(x, y) = (F−1 × G−1)#dη, and thus
W2(µ, ν) =
„Z 1
0[F−1(η)− G−1(η)]2 dη
«1/2
aW. Hoeffding (1940); M. Fréchet (1951); A. Pulvirenti, G. Toscani, Annali Mat. Pura Appl. (1996).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
One dimensional Case
Wasserstein distance:
Given µ and ν ∈ P2(Rd) with distribution functions F(x) and G(y) then,
W22 (µ, ν) =
ZZRd×Rd
|x− y|2 dH(x, y)
where H(x, y) = min(F(x), G(y)) is a joint distribution function and the aboveintegral is defined in the Riemann-Stieljes sense.a
In one dimension, it is easy to check that given two measures µ and ν withdistribution functions F(x) and G(y) then, (F−1 × G−1)#dη has joint distributionfunction H(x, y) = min(F(x), G(y)). Therefore, in one dimension, the optimal planis given by πopt(x, y) = (F−1 × G−1)#dη, and thus
W2(µ, ν) =
„Z 1
0[F−1(η)− G−1(η)]2 dη
«1/2
aW. Hoeffding (1940); M. Fréchet (1951); A. Pulvirenti, G. Toscani, Annali Mat. Pura Appl. (1996).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
One dimensional Case
Wasserstein distance:
Given µ and ν ∈ Pp(R), 1 ≤ p < ∞ then
Wp(µ, ν) =
„Z 1
0[F−1(η)− G−1(η)]p dη
«1/p
= ‖F−1 − G−1‖Lp(R)
andW∞(µ, ν) = ‖F−1 − G−1‖L∞(R).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Relation 1D-anyD
Euclidean Wasserstein distance:
If the probability measures µij on R are the successive one-dimensional marginals of
the measure µi on Rd, for i = 1, 2 and j = 1, . . . , d, then
dXj=1
W22 (µ
1j , µ
2j ) ≤ W2
2 (µ1, µ2).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Relation 1D-anyD
Proof.
Let π be a measure on Rdx × Rd
y with marginals µ1 and µ2, optimal in the sense that
W22 (µ
1, µ2) =
ZZRd×Rd
|x− y|2 dπ(x, y).
Then its marginal πj on Rvj × Rwj has itself marginals µ1j and µ2
j , so
W22 (µ
1j , µ
2j ) ≤
ZZR×R
|vj − wj|2 dπj(vj, wj).
The inequality follows by noting thatdX
j=1
|vj − wj|2 = |v− w|2.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Relation 1D-anyD
Proof.
Let π be a measure on Rdx × Rd
y with marginals µ1 and µ2, optimal in the sense that
W22 (µ
1, µ2) =
ZZRd×Rd
|x− y|2 dπ(x, y).
Then its marginal πj on Rvj × Rwj has itself marginals µ1j and µ2
j , so
W22 (µ
1j , µ
2j ) ≤
ZZR×R
|vj − wj|2 dπj(vj, wj).
The inequality follows by noting thatdX
j=1
|vj − wj|2 = |v− w|2.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Properties
Relation 1D-anyD
Proof.
Let π be a measure on Rdx × Rd
y with marginals µ1 and µ2, optimal in the sense that
W22 (µ
1, µ2) =
ZZRd×Rd
|x− y|2 dπ(x, y).
Then its marginal πj on Rvj × Rwj has itself marginals µ1j and µ2
j , so
W22 (µ
1j , µ
2j ) ≤
ZZR×R
|vj − wj|2 dπj(vj, wj).
The inequality follows by noting thatdX
j=1
|vj − wj|2 = |v− w|2.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions9
The flows of nonlinear diffusions are contractions in Wasserstein distance.
ut = (P(u))xx
u(x, 0) = u0(x) ≥ 0 ∈ L1(R)
with P : R+0 → R, continuous, nondecreasing with P(0) = 0.
Diccionary for F−1(η):
∂F−1
∂t(η, t) = −
»1u
–(F−1(η, t), t)
∂F∂t
(F−1(η, t), t)
∂F−1
∂η(η, t) =
»1u
–(F−1(η, t), t) ,
∂2F−1
∂η2 (η, t) = −h ux
u3
i(F−1(η, t), t)
Equation for F−1:
∂F−1
∂t(η, t) = − ∂
∂η
»P
»„∂F−1
∂η
«−1––,
9H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions9
The flows of nonlinear diffusions are contractions in Wasserstein distance.
ut = (P(u))xx
u(x, 0) = u0(x) ≥ 0 ∈ L1(R)
with P : R+0 → R, continuous, nondecreasing with P(0) = 0.
Diccionary for F−1(η):
∂F−1
∂t(η, t) = −
»1u
–(F−1(η, t), t)
∂F∂t
(F−1(η, t), t)
∂F−1
∂η(η, t) =
»1u
–(F−1(η, t), t) ,
∂2F−1
∂η2 (η, t) = −h ux
u3
i(F−1(η, t), t)
Equation for F−1:
∂F−1
∂t(η, t) = − ∂
∂η
»P
»„∂F−1
∂η
«−1––,
9H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions9
The flows of nonlinear diffusions are contractions in Wasserstein distance.
ut = (P(u))xx
u(x, 0) = u0(x) ≥ 0 ∈ L1(R)
with P : R+0 → R, continuous, nondecreasing with P(0) = 0.
Diccionary for F−1(η):
∂F−1
∂t(η, t) = −
»1u
–(F−1(η, t), t)
∂F∂t
(F−1(η, t), t)
∂F−1
∂η(η, t) =
»1u
–(F−1(η, t), t) ,
∂2F−1
∂η2 (η, t) = −h ux
u3
i(F−1(η, t), t)
Equation for F−1:
∂F−1
∂t(η, t) = − ∂
∂η
»P
»„∂F−1
∂η
«−1––,
9H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions10
Contraction for nonlinear diffusions:
Given any two solutions of the nonlinear diffusion equation
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for all 1 ≤ p ≤ ∞.
Proof.
Assume 1 < p < ∞, the other ones by approximation in p, then
1p(p− 1)
ddt
Z 1
0|F−1 − G−1|pdη =Z 1
0|F−1 − G−1|p−2(F−1
η − G−1η )
P
»“F−1
η
”−1–− P
»“G−1
η
”−1–ff
dη ≤ 0
10H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions10
Contraction for nonlinear diffusions:
Given any two solutions of the nonlinear diffusion equation
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for all 1 ≤ p ≤ ∞.
Proof.
Assume 1 < p < ∞, the other ones by approximation in p, then
1p(p− 1)
ddt
Z 1
0|F−1 − G−1|pdη =Z 1
0|F−1 − G−1|p−2(F−1
η − G−1η )
P
»“F−1
η
”−1–− P
»“G−1
η
”−1–ff
dη ≤ 0
10H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions11
Strict Contraction for nonlinear diffusions with confining potential:
Given any two probability density solutions of the nonlinear diffusion equation
ρt =
„dVdx
(t, x)ρ«
x
+ (P(ρ))xx
where V(t, x) is a smooth strictly uniform convex potential ( d2Vdx2 (t, x) ≥ α > 0), then
Wp(ρ1(t), ρ2(t)) ≤ Wp(ρ1(0), ρ2(0)) e−αt
for all 1 ≤ p ≤ ∞.
Proof.
Exercise;∂F−1
∂t(η, t) = −dV
dx(t, F−1)− ∂
∂η
»P
»„∂F−1
∂η
«−1––,
The case of V(x) = x2/2 and P(ρ) = ρm comes by the scaling property of theWasserstein distances and the previous non strict contraction.
11H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions11
Strict Contraction for nonlinear diffusions with confining potential:
Given any two probability density solutions of the nonlinear diffusion equation
ρt =
„dVdx
(t, x)ρ«
x
+ (P(ρ))xx
where V(t, x) is a smooth strictly uniform convex potential ( d2Vdx2 (t, x) ≥ α > 0), then
Wp(ρ1(t), ρ2(t)) ≤ Wp(ρ1(0), ρ2(0)) e−αt
for all 1 ≤ p ≤ ∞.
Proof.
Exercise;∂F−1
∂t(η, t) = −dV
dx(t, F−1)− ∂
∂η
»P
»„∂F−1
∂η
«−1––,
The case of V(x) = x2/2 and P(ρ) = ρm comes by the scaling property of theWasserstein distances and the previous non strict contraction.
11H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions
Convergence towards equilibrium:
All probability solutions of the equation
ρt =
„dVdx
(t, x)ρ«
x
+ (P(ρ))xx
converges exponentially fast towards the unique equilibrium ρ∞ characterized by
V(x) + h(ρ∞(x)) = C ρh′(ρ) = P′(ρ).
Here, P is strictly increasing for being h invertible.
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Diffusive Models
1D Nonlinear Diffusions12
Uniform Finite Speed of Propagation:
Given any two probability density solutions of the degenerate nonlinear diffusionequation
ρt = (ρm)xx ,
we have the following control of the "relative speed" of the support spreading
| infsuppρ1(t) − infsuppρ2(t)| ≤ W∞(ρ1(0), ρ2(0))
| supsuppρ1(t) − supsuppρ2(t)| ≤ W∞(ρ1(0), ρ2(0))
for any t > 0.
12J.A. Carrillo, M.P. Gualdani, G. Toscani, CRAS (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Dissipative Models
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Dissipative Models
1D Dissipative Kinetic Equations13
Given the 1D granular model
∂f∂t
=∂
∂v
»ZR(v− w)|v− w|γ f (w, t) dw f (v, t)
–
Equation for F−1:
∂F−1
∂t(η, t) = −
Z 1
0|F−1(η, t)− F−1(p, t)|γ(F−1(η, t)− F−1(p, t)) dp .
After some manipulations (good exercise)
ddt
W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ
implying
W2(f (t), g(t)) ≤„
2γ−1
γt + 2γ−1W2(f (0), g(0))−γ
«1/γ
.
13H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Dissipative Models
1D Dissipative Kinetic Equations13
Given the 1D granular model
∂f∂t
=∂
∂v
»ZR(v− w)|v− w|γ f (w, t) dw f (v, t)
–
Equation for F−1:
∂F−1
∂t(η, t) = −
Z 1
0|F−1(η, t)− F−1(p, t)|γ(F−1(η, t)− F−1(p, t)) dp .
After some manipulations (good exercise)
ddt
W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ
implying
W2(f (t), g(t)) ≤„
2γ−1
γt + 2γ−1W2(f (0), g(0))−γ
«1/γ
.
13H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Dissipative Models
1D Dissipative Kinetic Equations13
Given the 1D granular model
∂f∂t
=∂
∂v
»ZR(v− w)|v− w|γ f (w, t) dw f (v, t)
–
Equation for F−1:
∂F−1
∂t(η, t) = −
Z 1
0|F−1(η, t)− F−1(p, t)|γ(F−1(η, t)− F−1(p, t)) dp .
After some manipulations (good exercise)
ddt
W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ
implying
W2(f (t), g(t)) ≤„
2γ−1
γt + 2γ−1W2(f (0), g(0))−γ
«1/γ
.
13H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Dissipative Models
1D Dissipative Kinetic Equations13
Given the 1D granular model
∂f∂t
=∂
∂v
»ZR(v− w)|v− w|γ f (w, t) dw f (v, t)
–
Equation for F−1:
∂F−1
∂t(η, t) = −
Z 1
0|F−1(η, t)− F−1(p, t)|γ(F−1(η, t)− F−1(p, t)) dp .
After some manipulations (good exercise)
ddt
W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ
implying
W2(f (t), g(t)) ≤„
2γ−1
γt + 2γ−1W2(f (0), g(0))−γ
«1/γ
.
13H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Outline
1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases
2 Wasserstein Distance: BasicsDefinitionProperties
3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Heat Equation14
Contraction for the Heat Equation:
Given any two probability density solutions u1(x, t), u2(x, t) of the heat equation,then
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for t ≥ 0, 1 ≤ p ≤ ∞.
Proof.
Let γo the optimal transference plan between u1(0) y u2(0) for Wp, 1 ≤ p < ∞.The solutions are u1(t) = K(t, x) ∗ u1(0) and u2(t) = K(t, x) ∗ u2(0).Let us define γt by duality:Z
Rd×Rdϕ(x, y) dγt(x, y) =
ZRd×Rd
ZRd
ϕ(x + z, y + z)K(t, z) dz dγo(x, y)
Check that γt is a transference plan between u1(t) and u2(t), and thus
Wpp (u1(t), u2(t))≤
ZRd×Rd
|x−y|p dγt(x, y)=
ZRd×Rd
|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).
14J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Heat Equation14
Contraction for the Heat Equation:
Given any two probability density solutions u1(x, t), u2(x, t) of the heat equation,then
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for t ≥ 0, 1 ≤ p ≤ ∞.
Proof.
Let γo the optimal transference plan between u1(0) y u2(0) for Wp, 1 ≤ p < ∞.The solutions are u1(t) = K(t, x) ∗ u1(0) and u2(t) = K(t, x) ∗ u2(0).Let us define γt by duality:Z
Rd×Rdϕ(x, y) dγt(x, y) =
ZRd×Rd
ZRd
ϕ(x + z, y + z)K(t, z) dz dγo(x, y)
Check that γt is a transference plan between u1(t) and u2(t), and thus
Wpp (u1(t), u2(t))≤
ZRd×Rd
|x−y|p dγt(x, y)=
ZRd×Rd
|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).
14J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Heat Equation14
Contraction for the Heat Equation:
Given any two probability density solutions u1(x, t), u2(x, t) of the heat equation,then
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for t ≥ 0, 1 ≤ p ≤ ∞.
Proof.
Let γo the optimal transference plan between u1(0) y u2(0) for Wp, 1 ≤ p < ∞.The solutions are u1(t) = K(t, x) ∗ u1(0) and u2(t) = K(t, x) ∗ u2(0).Let us define γt by duality:Z
Rd×Rdϕ(x, y) dγt(x, y) =
ZRd×Rd
ZRd
ϕ(x + z, y + z)K(t, z) dz dγo(x, y)
Check that γt is a transference plan between u1(t) and u2(t), and thus
Wpp (u1(t), u2(t))≤
ZRd×Rd
|x−y|p dγt(x, y)=
ZRd×Rd
|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).
14J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Heat Equation14
Contraction for the Heat Equation:
Given any two probability density solutions u1(x, t), u2(x, t) of the heat equation,then
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for t ≥ 0, 1 ≤ p ≤ ∞.
Proof.
Let γo the optimal transference plan between u1(0) y u2(0) for Wp, 1 ≤ p < ∞.The solutions are u1(t) = K(t, x) ∗ u1(0) and u2(t) = K(t, x) ∗ u2(0).Let us define γt by duality:Z
Rd×Rdϕ(x, y) dγt(x, y) =
ZRd×Rd
ZRd
ϕ(x + z, y + z)K(t, z) dz dγo(x, y)
Check that γt is a transference plan between u1(t) and u2(t), and thus
Wpp (u1(t), u2(t))≤
ZRd×Rd
|x−y|p dγt(x, y)=
ZRd×Rd
|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).
14J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Heat Equation14
Contraction for the Heat Equation:
Given any two probability density solutions u1(x, t), u2(x, t) of the heat equation,then
Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))
for t ≥ 0, 1 ≤ p ≤ ∞.
Proof.
Let γo the optimal transference plan between u1(0) y u2(0) for Wp, 1 ≤ p < ∞.The solutions are u1(t) = K(t, x) ∗ u1(0) and u2(t) = K(t, x) ∗ u2(0).Let us define γt by duality:Z
Rd×Rdϕ(x, y) dγt(x, y) =
ZRd×Rd
ZRd
ϕ(x + z, y + z)K(t, z) dz dγo(x, y)
Check that γt is a transference plan between u1(t) and u2(t), and thus
Wpp (u1(t), u2(t))≤
ZRd×Rd
|x−y|p dγt(x, y)=
ZRd×Rd
|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).
14J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Homogeneous Boltzmann Equation15
Convergence Towards Monokinetic State:
Given f (v, t) ∈ P2(R3) with zero mean velocity then
W2(f (t), δ0) ≤ C(1 + t)−2
para t ≥ 0.
Proof.
Evolution of temperature:
< |v|2, Qe(f , f ) >=1− e2
4
ZR3
ZR3|v− w|3f (v)f (w) dv dw
and thus using Jensen’s and Hölder’s inequality, we get the Haff’s law:
dθ
dt≤ − 1− e2
4θ
32 .
Using the relation to temperature of the Euclidean Wasserstein distance, weconclude.
15J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Homogeneous Boltzmann Equation15
Convergence Towards Monokinetic State:
Given f (v, t) ∈ P2(R3) with zero mean velocity then
W2(f (t), δ0) ≤ C(1 + t)−2
para t ≥ 0.
Proof.
Evolution of temperature:
< |v|2, Qe(f , f ) >=1− e2
4
ZR3
ZR3|v− w|3f (v)f (w) dv dw
and thus using Jensen’s and Hölder’s inequality, we get the Haff’s law:
dθ
dt≤ − 1− e2
4θ
32 .
Using the relation to temperature of the Euclidean Wasserstein distance, weconclude.
15J.A. Carrillo, Boletín SEMA (2004).
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Presentation of models Wasserstein Distance: Basics Contractivity in 1D
Towards any dimension
Homogeneous Boltzmann Equation15
Convergence Towards Monokinetic State:
Given f (v, t) ∈ P2(R3) with zero mean velocity then
W2(f (t), δ0) ≤ C(1 + t)−2
para t ≥ 0.
Proof.
Evolution of temperature:
< |v|2, Qe(f , f ) >=1− e2
4
ZR3
ZR3|v− w|3f (v)f (w) dv dw
and thus using Jensen’s and Hölder’s inequality, we get the Haff’s law:
dθ
dt≤ − 1− e2
4θ
32 .
Using the relation to temperature of the Euclidean Wasserstein distance, weconclude.
15J.A. Carrillo, Boletín SEMA (2004).