Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Global attractors for gradient flows
in metric spaces
Riccarda Rossi
(University of Brescia)
in collaboration with
Antonio Segatti (WIAS, Berlin),
Ulisse Stefanelli (IMATI–CNR, Pavia)
Recent advances in Free Boundary Problems and related topics
Levico, September 14–16 2006
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Evolution PDEs of diffusive type and the
Wasserstein metric
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Evolution PDEs of diffusive type and the
Wasserstein metric
δL
∂t ρ − div ρ∇( ) = 0
δρ
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(x, t) ∈ Rn × (0, +∞),
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Evolution PDEs of diffusive type and the
Wasserstein metric
δL
∂t ρ − div ρ∇( ) = 0
δρ
(x, t) ∈ Rn × (0, +∞),
Z
L(ρ) :=
L(x, ρ(x), ∇ρ(x)) dx
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(Integral functional)
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Evolution PDEs of diffusive type and the
Wasserstein metric
δL
∂t ρ − div ρ∇( ) = 0
δρ
(x, t) ∈ Rn × (0, +∞),
Z
L(ρ) :=
L(x, ρ(x), ∇ρ(x)) dx
(Integral functional)
Rn
L = L(x, ρ, ∇ρ) : Rn × (0, +∞) × Rn → R
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(Lagrangian)
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Evolution PDEs of diffusive type and the
Wasserstein metric
δL
∂t ρ − div ρ∇( ) = 0
δρ
(x, t) ∈ Rn × (0, +∞),
Z
L(ρ) :=
L(x, ρ(x), ∇ρ(x)) dx
(Integral functional)
Rn
δL
= ∂ρ L(x, ρ, ∇ρ) − div(∂∇ρ L(x, ρ, ∇ρ))
δρ
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Evolution PDEs of diffusive type and the
Wasserstein metric

δL

∂
ρ
−
div
ρ∇(
)
= 0 (x, t) ∈ Rn × (0, +∞),

t

δρ
R
n
t) ≥ 0,
Rn ρ(x, t) dx = 1 ∀ (x, t) ∈ R × (0, +∞),
ρ(x,
R


2
Rn |x| ρ(x, t) dx < +∞ ∀ t ≥ 0,
Z
L(ρ) :=
L(x, ρ(x), ∇ρ(x)) dx
(Integral functional)
Rn
δL
= ∂ρ L(x, ρ, ∇ρ) − div(∂∇ρ L(x, ρ, ∇ρ))
δρ
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Evolution PDEs of diffusive type and the
Wasserstein metric

δL

∂
ρ
−
div
ρ∇(
)
= 0 (x, t) ∈ Rn × (0, +∞),

t

δρ
R
n
t) ≥ 0,
Rn ρ(x, t) dx = 1 ∀ (x, t) ∈ R × (0, +∞),
ρ(x,
R


2
Rn |x| ρ(x, t) dx < +∞ ∀ t ≥ 0,
(PDE)
For t fixed, identify ρ(·, t) with the probability measure µt := ρ(·, t)dx
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Evolution PDEs of diffusive type and the
Wasserstein metric

δL

∂
ρ
−
div
ρ∇(
)
= 0 (x, t) ∈ Rn × (0, +∞),

t

δρ
R
n
t) ≥ 0,
Rn ρ(x, t) dx = 1 ∀ (x, t) ∈ R × (0, +∞),
ρ(x,
R


2
Rn |x| ρ(x, t) dx < +∞ ∀ t ≥ 0,
(PDE)
For t fixed, identify ρ(·, t) with the probability measure µt := ρ(·, t)dx
then L can be considered defined on P2 (Rn ) (the space of probability measures on Rn with finite second moment)
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Evolution PDEs of diffusive type and the
Wasserstein metric

δL

∂
ρ
−
div
ρ∇(
)
= 0 (x, t) ∈ Rn × (0, +∞),

t

δρ
R
n
t) ≥ 0,
Rn ρ(x, t) dx = 1 ∀ (x, t) ∈ R × (0, +∞),
ρ(x,
R


2
Rn |x| ρ(x, t) dx < +∞ ∀ t ≥ 0,
(PDE)
Otto, Jordan & Kinderlehrer and Otto [’97–’01]
showed that (PDE) can be interpreted as
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Evolution PDEs of diffusive type and the
Wasserstein metric

δL

∂
ρ
−
div
ρ∇(
)
= 0 (x, t) ∈ Rn × (0, +∞),

t

δρ
R
n
t) ≥ 0,
Rn ρ(x, t) dx = 1 ∀ (x, t) ∈ R × (0, +∞),
ρ(x,
R


2
Rn |x| ρ(x, t) dx < +∞ ∀ t ≥ 0,
(PDE)
Otto, Jordan & Kinderlehrer and Otto [’97–’01]
showed that (PDE) can be interpreted as
the gradient flow of L in P2 (Rn )
w.r.t. the Wasserstein distance W2 on P2 (Rn )
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Examples
Ex.1: The potential energy functional
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Examples
Ex.1: The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
Ex.1: The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.2: The entropy functional
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.2: The entropy functional
Z
ρ(x) log(ρ(x)) dx
L2 (ρ) :=
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.2: The entropy functional
Z
L2 (ρ) :=
ρ(x) log(ρ(x)) dx,
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(
L2 (x, ρ, ∇ρ) = ρ log(ρ),
δL1
δρ = ∂ρ L1 (ρ) = log(ρ) + 1,
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.2: The entropy functional
Z
L2 (ρ) :=
ρ(x) log(ρ(x)) dx,
Rn
(
L2 (x, ρ, ∇ρ) = ρ log(ρ),
δL1
δρ = ∂ρ L1 (ρ) = log(ρ) + 1,
∂t ρ − div(ρ∇(log(ρ) + 1)) = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
The entropy functional
Z
L2 (ρ) :=
ρ(x) log(ρ(x)) dx,
Rn
The heat equation
(
L2 (x, ρ, ∇ρ) = ρ log(ρ),
δL1
δρ = ∂ρ L1 (ρ) = log(ρ) + 1,
∂t ρ − ∆ρ = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.3: The internal energy functional
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.3: The internal energy functional
Z
1
L3 (ρ) :=
ρm (x) dx, m 6= 1
m
−
1
n
R
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.3: The internal energy functional
Z
L3 (ρ) :=
Rn
1
ρm (x) dx,
m−1
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(
1
L3 (x, ρ, ∇ρ) = m−1
ρm ,
δL3
m
m−1 ,
δρ = ∂ρ L3 (ρ) = m−1 ρ
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.3: The internal energy functional
Z
L3 (ρ) :=
Rn
(
1
L3 (x, ρ, ∇ρ) = m−1
ρm ,
δL3
m
m−1 ,
δρ = ∂ρ L3 (ρ) = m−1 ρ
∇ρm
∂t ρ − div(ρ
)=0
ρ
1
ρm (x) dx,
m−1
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
The internal functional
Z
L3 (ρ) :=
Rn
1
ρm (x) dx,
m−1
The porous media equation
(
1
L3 (x, ρ, ∇ρ) = m−1
ρm ,
δL3
m
m−1 ,
δρ = ∂ρ L3 (ρ) = m−1 ρ
∂t ρ − ∆ρm = 0 Otto ’01
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.4: The (Entropy+ Potential) energy functional
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.4: The (Entropy+ Potential) energy functional
Z
(ρ(x) log(ρ(x)) + ρ(x)V (x)) dx,
L4 (ρ) :=
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.4: The (Entropy+ Potential) energy functional
Z
L4 (ρ) :=
(ρ log(ρ)+ρV ),
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(
L4 (x, ρ, ∇ρ) = ρ log(ρ) + ρV (x),
δL4
δρ = ∂ρ L4 (x, ρ) = log(ρ) + 1 + V (x),
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Ex.4: The (Entropy+ Potential) energy functional
Z
L4 (ρ) :=
(ρ log(ρ)+ρV ),
Rn
(
L4 (x, ρ, ∇ρ) = ρ log(ρ) + ρV (x),
δL4
δρ = ∂ρ L4 (x, ρ) = log(ρ) + 1 + V (x),
∂t ρ − div(ρ∇(log(ρ) + 1 + V )) = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Examples
The potential energy functional
Z
L1 (ρ) :=
V (x)ρ(x) dx,
Rn
The linear transport equation
(
L1 (x, ρ, ∇ρ) = L1 (x, ρ) = ρV (x),
δL1
δρ = ∂ρ L1 (x, ρ) = V (x),
∂t ρ − div(ρ∇V ) = 0
Entropy+Potential
Z
L4 (ρ) :=
(ρ log(ρ)+ρV ),
Rn
∂t ρ − ∆ρ − div(ρ∇V ) = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
The Fokker-Planck equation
(
L4 (x, ρ, ∇ρ) = ρ log(ρ) + ρV (x),
δL4
δρ = ∂ρ L4 (x, ρ) = log(ρ) + 1 + V (x),
Jordan-Kinderlehrer-Otto ’97
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
New insight
• This gradient flow approach has brought several developments
in:
I
approximation algorithms
I
asymptotic behaviour of solutions (new contraction and
energy estimates)
I
applications to functional inequalities.....
[Agueh, Brenier, Carlen, Carrillo, Dolbeault, Gangbo,
Ghoussoub,McCann, Otto, Vazquez, Villani..]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Towards gradient flows in metric spaces
• In [Gradient flows in metric and in the
Wasserstein spaces Ambrosio, Gigli, Savaré
’05], these existence, approximation, long-time
behaviour results were recovered for general
Gradient Flows in Metric Spaces
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Towards gradient flows in metric spaces
• In [Gradient flows in metric and in the
Wasserstein spaces Ambrosio, Gigli, Savaré
’05], these existence, approximation, long-time
behaviour results were recovered for general
Gradient Flows in Metric Spaces
• Metric spaces are a suitable framework for
rigorously interpreting diffusion PDE as gradient flows
in the Wasserstein spaces ( no linear structure!)
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Towards gradient flows in metric spaces
• In [Gradient flows in metric and in the
Wasserstein spaces Ambrosio, Gigli, Savaré
’05], these existence, approximation, long-time
behaviour results were recovered for general
Gradient Flows in Metric Spaces
• Metric spaces are a suitable framework for
rigorously interpreting diffusion PDE as gradient flows
in the Wasserstein spaces ( no linear structure!)
• The approach of [AGS’05] based on the theory of Minimizing
Movements & Curves of Maximal Slope [De Giorgi, Marino,
Tosques, Degiovanni, Ambrosio.. ’80∼’90]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Towards gradient flows in metric spaces
• In [Gradient flows in metric and in the
Wasserstein spaces Ambrosio, Gigli, Savaré
’05], these existence, approximation, long-time
behaviour results were recovered for general
Gradient Flows in Metric Spaces
• Metric spaces are a suitable framework for
rigorously interpreting diffusion PDE as gradient flows
in the Wasserstein spaces ( no linear structure!)
• [AGS’05]: ⇒ refined existence, approximation, long-time
behaviour for Curves of Maximal Slope & applications to gradient
flows in Wasserstein spaces
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Towards gradient flows in metric spaces
• In [Gradient flows in metric and in the
Wasserstein spaces Ambrosio, Gigli, Savaré
’05], these existence, approximation, long-time
behaviour results were recovered for general
Gradient Flows in Metric Spaces
• Metric spaces are a suitable framework for
rigorously interpreting diffusion PDE as gradient flows
in the Wasserstein spaces ( no linear structure!)
• [R.-Segatti-Stefanelli’06]: complement AGS’s results on the
long-time behaviour of Curves of Maximal Slope
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Data:
I
A complete metric space (X , d),
I
a proper functional φ : X → (−∞, +∞]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Data:
I
A complete metric space (X , d),
I
a proper functional φ : X → (−∞, +∞]
Problem:
How to formulate the gradient flow equation
“u 0 (t) = −∇φ(u(t))”,
Riccarda Rossi
Global attractors for gradient flows in metric spaces
t ∈ (0, T )
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Data:
I
A complete metric space (X , d),
I
a proper functional φ : X → (−∞, +∞]
Problem:
How to formulate the gradient flow equation
“u 0 (t) = −∇φ(u(t))”,
t ∈ (0, T )
in absence of a natural linear or differentiable structure on X ?
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Data:
I
A complete metric space (X , d),
I
a proper functional φ : X → (−∞, +∞]
Problem:
How to formulate the gradient flow equation
“u 0 (t) = −∇φ(u(t))”,
t ∈ (0, T )
in absence of a natural linear or differentiable structure on X ?
To get some insight, let us go back to the euclidean case...
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t))
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2hu 0 (t), ∇φ(u(t))i
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2hu 0 (t), ∇φ(u(t))i
d
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2 φ(u(t)) = 0
dt
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2hu 0 (t), ∇φ(u(t))i
d
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2 φ(u(t)) = 0
dt
So we get the equivalent formulation:
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2hu 0 (t), ∇φ(u(t))i
d
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2 φ(u(t)) = 0
dt
So we get the equivalent formulation:
1
1
d
φ(u(t)) = − |u 0 (t)|2 − |∇φ(u(t))|2
dt
2
2
This involves
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2hu 0 (t), ∇φ(u(t))i
d
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2 φ(u(t)) = 0
dt
So we get the equivalent formulation:
1
d
1
φ(u(t)) = − |u 0 (t)|2 − |∇φ(u(t))|2
dt
2
2
This involves the modulus of derivatives, rather than derivatives,
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Gradient flows in metric spaces: heuristics
Given a proper (differentiable) function φ : Rn → (−∞, +∞]
u 0 (t) = −∇φ(u(t)) ⇔ |u 0 (t) + ∇φ(u(t))|2 = 0
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2hu 0 (t), ∇φ(u(t))i
d
⇔ |u 0 (t)|2 + |∇φ(u(t))|2 + 2 φ(u(t)) = 0
dt
So we get the equivalent formulation:
1
d
1
φ(u(t)) = − |u 0 (t)|2 − |∇φ(u(t))|2
dt
2
2
This involves the modulus of derivatives, rather than derivatives,
hence it can make sense in the setting of a metric space!
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Given a curve u : (0, T ) → X ,
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Given a curve u : (0, T ) → X ,
u 0 (t)
|u 0 |(t) metric derivative:
|u 0 |(t) := lim
h→0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
d(u(t), u(t + h)
h
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Given a curve u : (0, T ) → X ,
u 0 (t)
|u 0 |(t) metric derivative:
|u 0 |(t) := lim
h→0
d(u(t), u(t + h)
h
⇒ definition of the space of absolutely continuous curves
(AC(0, T ; X ));
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Given a proper functional φ : X → (−∞, +∞],
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Given a proper functional φ : X → (−∞, +∞],
−∇φ(u)
|∂φ| (u) local slope:
|∂φ| (u) := lim sup
v →u
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(φ(u) − φ(v ))+
d(u, v )
u ∈ D(φ)
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Metric derivatives and slopes
• Setting: A complete metric space (X , d)
“Surrogates” of (the modulus of) derivatives
Given a proper functional φ : X → (−∞, +∞],
−∇φ(u)
|∂φ| (u) local slope:
|∂φ| (u) := lim sup
v →u
(φ(u) − φ(v ))+
d(u, v )
u ∈ D(φ)
The local slope in an upper gradient for φ, i.e. for any curve
v ∈ AC(0, T ; X ) s.t. φ ◦ u is absolutely continuous on (0, T )
d
φ(u(t)) ≤ |u 0 |(t) |∂φ| (u(t)) a.e. in (0, T ).
dt
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Definition of Curve of Maximal Slope
(2-)Curve of Maximal Slope
We say that an absolutely continuous curve u : (0, T ) → X is a
(2-)curve of maximal slope for φ if
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Definition of Curve of Maximal Slope
(2-)Curve of Maximal Slope
We say that an absolutely continuous curve u : (0, T ) → X is a
(2-)curve of maximal slope for φ if
1
1
d
φ(u(t)) = − |u 0 |2 (t) − |∂φ|2 (u(t))
dt
2
2
Riccarda Rossi
Global attractors for gradient flows in metric spaces
a.e. in (0, T ).
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Definition of p-Curve of Maximal Slope
Consider p, q ∈ (1, +∞) with
Riccarda Rossi
Global attractors for gradient flows in metric spaces
1
p
+
1
q
= 1.
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Definition of p-Curve of Maximal Slope
Consider p, q ∈ (1, +∞) with
1
p
+
1
q
= 1.
p-Curve of Maximal Slope
We say that an absolutely continuous curve u : (0, T ) → X is a
p-curve of maximal slope for φ if
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Definition of p-Curve of Maximal Slope
Consider p, q ∈ (1, +∞) with
1
p
+
1
q
= 1.
p-Curve of Maximal Slope
We say that an absolutely continuous curve u : (0, T ) → X is a
p-curve of maximal slope for φ if
d
1
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t))
dt
p
q
Riccarda Rossi
Global attractors for gradient flows in metric spaces
a.e. in (0, T ).
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Definition of p-Curve of Maximal Slope
Consider p, q ∈ (1, +∞) with
1
p
+
1
q
= 1.
p-Curve of Maximal Slope
We say that an absolutely continuous curve u : (0, T ) → X is a
p-curve of maximal slope for φ if
d
1
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t))
dt
p
q
a.e. in (0, T ).
2-curves of maximal slope in P2 (Rn ) lead (for a suitable φ) to the
linear transport equation
∂t ρ − div(ρ∇V ) = 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Definition of p-Curve of Maximal Slope
Consider p, q ∈ (1, +∞) with
1
p
+
1
q
= 1.
p-Curve of Maximal Slope
We say that an absolutely continuous curve u : (0, T ) → X is a
p-curve of maximal slope for φ if
d
1
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t))
dt
p
q
a.e. in (0, T ).
p-curves of maximal slope in Pp (Rn ) lead (for a suitable φ) to the
“doubly linear” transport equation
(
|r |q−2 r r 6= 0,
∂t ρ − ∇ · (ρjq (∇V )) = 0
jq (r ) :=
0
r = 0.
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
I
Given an initial datum u0 ∈ X , does there exist a p−curve of
maximal slope u on (0, T ) fulfilling u(0) = u0 ?
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
I
Given an initial datum u0 ∈ X , does there exist a p−curve of
maximal slope u on (0, T ) fulfilling u(0) = u0 ?
I
In [AGS’05] existence is proved by passing to the limit in an
approximation scheme by time discretization
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Then, for all u0 ∈ D(φ) there exists a p-curve of maximal slope u
for φ, fulfilling u(0) = u0 .
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Then, for all u0 ∈ D(φ) there exists a p-curve of maximal slope u
for φ, fulfilling u(0) = u0 .
A sufficient condition for |∂φ| to be lower semicontinuous is
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Then, for all u0 ∈ D(φ) there exists a p-curve of maximal slope u
for φ, fulfilling u(0) = u0 .
A sufficient condition for |∂φ| to be lower semicontinuous is
φ is λ-geodesically convex, for λ ∈ R
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Then, for all u0 ∈ D(φ) there exists a p-curve of maximal slope u
for φ, fulfilling u(0) = u0 .
A sufficient condition for |∂φ| to be lower semicontinuous is
∀v0 , v1 ∈ D(φ) ∃ (constant speed) geodesic γ, γ(0) = v0 , γ(1) = v1 ,
φ is λ-convex on γ.
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Then, for all u0 ∈ D(φ) there exists a p-curve of maximal slope u
for φ, fulfilling u(0) = u0 .
A sufficient condition for |∂φ| to be lower semicontinuous is
∀v0 , v1 ∈ D(φ) ∃ (constant speed) geodesic γ, γ(0) = v0 , γ(1) = v1 ,
λ
φ(γt ) ≤ (1 − t)φ(v0 ) + tφ(v1 ) − t(1 − t)d 2 (v0 , v1 ) ∀ t ∈ [0, 1].
2
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
An existence result
Theorem [Ambrosio-Gigli-Savaré ’05]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Then, for all u0 ∈ D(φ) there exists a p-curve of maximal slope u
for φ, fulfilling u(0) = u0 .
A sufficient condition for |∂φ| to be lower semicontinuous is
φ is λ-geodesically convex, for λ ∈ R
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Uniqueness for 2-curves of maximal slope
• Uniqueness proved only for p = 2!
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Uniqueness for 2-curves of maximal slope
• Uniqueness proved only for p = 2!
Theorem [Ambrosio-Gigli-Savaré ’05]
Main assumptions (simplified):
I
φ is λ-geodesically convex, λ ∈ R,
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Uniqueness for 2-curves of maximal slope
• Uniqueness proved only for p = 2!
Theorem [Ambrosio-Gigli-Savaré ’05]
Main assumptions (simplified):
I
φ is λ-geodesically convex, λ ∈ R,
I
a “structural property” of the metric space (X , d)
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Uniqueness for 2-curves of maximal slope
• Uniqueness proved only for p = 2!
Theorem [Ambrosio-Gigli-Savaré ’05]
Main assumptions (simplified):
I
φ is λ-geodesically convex, λ ∈ R,
I
(X , d) is the Wasserstein space (P2 (Rd ), W2 )
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Uniqueness for 2-curves of maximal slope
• Uniqueness proved only for p = 2!
Theorem [Ambrosio-Gigli-Savaré ’05]
Main assumptions (simplified):
I
φ is λ-geodesically convex, λ ∈ R,
(X , d) is the Wasserstein space (P2 (Rd ), W2 )
Then,
I
I
existence and uniqueness of the curve of maximal slope
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Uniqueness for 2-curves of maximal slope
• Uniqueness proved only for p = 2!
Theorem [Ambrosio-Gigli-Savaré ’05]
Main assumptions (simplified):
I
φ is λ-geodesically convex, λ ∈ R,
(X , d) is the Wasserstein space (P2 (Rd ), W2 )
Then,
I
I
existence and uniqueness of the curve of maximal slope
I
Generation of a λ-contracting semigroup
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for 2-curves of maximal slope
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for 2-curves of maximal slope
Main assumptions:
I
p=2
I
a “structural property” of the metric space (X , d)
I
φ is λ-geodesically convex, λ ≥ 0,
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for 2-curves of maximal slope
Main assumptions:
I
p=2
I
a “structural property” of the metric space (X , d)
I
φ is λ-geodesically convex, λ ≥ 0,
Theorem [Ambrosio-Gigli-Savaré ’05]
• λ > 0:
exponential convergence of the solution as t → +∞ to the
unique minimum point ū of φ:
d(u(t), ū) ≤ e −λt d(u0 , ū) ∀ t ≥ 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for 2-curves of maximal slope
Main assumptions:
I
p=2
I
a “structural property” of the metric space (X , d)
I
φ is λ-geodesically convex, λ ≥ 0,
Theorem [Ambrosio-Gigli-Savaré ’05]
• λ = 0 + φ has compact sublevels:
convergence to (an) equilibrium as t → +∞
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Our aim
“Fill in the gaps” in the study of the long-time behaviour of
p-curves of maximal slope:
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Our aim
“Fill in the gaps” in the study of the long-time behaviour of
p-curves of maximal slope:
Study the general case:
I
φ λ-geodesically convex, λ ∈ R
I
p general
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Our aim
“Fill in the gaps” in the study of the long-time behaviour of
p-curves of maximal slope:
Namely, we comprise the cases:
1. p = 2, λ < 0
uniqueness: YES
2. p 6= 2, λ ∈ R
uniqueness: NO
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Our point of view
Not the study of the convergence to equilibrium as t → +∞ of a
single trajectory
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Our point of view
Not the study of the convergence to equilibrium as t → +∞ of a
single trajectory
But the study of the long-time behaviour of a family of
trajectories (starting from a bounded set of initial data):
convergence to an invariant compact set (“attractor”)?
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Our point of view
Not the study of the convergence to equilibrium as t → +∞ of a
single trajectory
But the study of the long-time behaviour of a family of
trajectories (starting from a bounded set of initial data):
convergence to an invariant compact set (“attractor”)?
On the other hand, for p 6= 2 no uniqueness result, no
semigroup of solutions
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Our point of view
Not the study of the convergence to equilibrium as t → +∞ of a
single trajectory
But the study of the long-time behaviour of a family of
trajectories (starting from a bounded set of initial data):
convergence to an invariant compact set (“attractor”)?
On the other hand, for p 6= 2 no uniqueness result, no
semigroup of solutions
⇒ Need for a theory of global attractors for (autonomous)
dynamical systems without uniqueness
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Our point of view
Not the study of the convergence to equilibrium as t → +∞ of a
single trajectory
But the study of the long-time behaviour of a family of
trajectories (starting from a bounded set of initial data):
convergence to an invariant compact set (“attractor”)?
On the other hand, for p 6= 2 no uniqueness result, no
semigroup of solutions
⇒ Need for a theory of global attractors for (autonomous)
dynamical systems without uniqueness
Various possibilities: [Sell ’73,’96], [Chepyzhov & Vishik ’02],
[Melnik & Valero ’02], [Ball ’97,’04]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Our point of view
Not the study of the convergence to equilibrium as t → +∞ of a
single trajectory
But the study of the long-time behaviour of a family of
trajectories (starting from a bounded set of initial data):
convergence to an invariant compact set (“attractor”)?
On the other hand, for p 6= 2 no uniqueness result, no
semigroup of solutions
⇒ Need for a theory of global attractors for (autonomous)
dynamical systems without uniqueness
Various possibilities: [Sell ’73,’96], [Chepyzhov & Vishik ’02],
[Melnik & Valero ’02], [Ball ’97,’04]
In [R., Segatti, Stefanelli, Global attractors for curves of maximal slope, in
preparation]: Ball’s theory of generalized semiflows
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Generalized Semiflows: definition
Phase space: a metric space (X , dX )
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Existence of a Global Attractor
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Generalized Semiflows: definition
A generalized semiflow S on X
g : [0, +∞) → X (“solutions”), s. t.
is a family of maps
(Existence) ∀ g0 ∈ X ∃ at least one g ∈ S with g (0) = g0 ,
(Translation invariance) ∀ g ∈ S and τ ≥ 0, the map
g τ (·) := g (· + τ ) is in S,
(Concatenation) ∀ g , h ∈ S and t ≥ 0 with h(0) = g (t), then
z ∈ S, where
(
g (τ ) if 0 ≤ τ ≤ t,
z(τ ) :=
h(τ − t) if t < τ,
(U.s.c. w.r.t. initial data) If {gn } ⊂ S and gn (0) → g0 ,
∃ subsequence {gnk } and g ∈ S s.t. g (0) = g0 and
gnk (t) → g (t) for all t ≥ 0.
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Generalized Semiflows: dynamical system notions
Within this framework:
I
orbit of a solution/set
I
ω-limit of a solution/set
I
invariance under the semiflow of a set
I
attracting set (w.r.t. the Hausdorff semidistance of X )
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Generalized Semiflows: dynamical system notions
Within this framework:
I
orbit of a solution/set
I
ω-limit of a solution/set
I
invariance under the semiflow of a set
I
attracting set (w.r.t. the Hausdorff semidistance of X )
Definition
A set A ⊂ X is a global attractor for a generalized
semiflow S if:
♣ A is compact
♣ A is invariant under the semiflow
♣ A attracts the bounded sets of X (w.r.t. the
Hausdorff semidistance of X )
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
1
d
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t))
dt
p
q
Riccarda Rossi
Global attractors for gradient flows in metric spaces
for a.e. t ∈ (0, T ),
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
1
d
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t))
dt
p
q
for a.e. t ∈ (0, T ),
Choice of the phase space:
X = D(φ) ⊂ X ,
dX (u, v ) := d(u, v ) + |φ(u) − φ(v )|
Riccarda Rossi
Global attractors for gradient flows in metric spaces
∀ u, v ∈ X .
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
1
d
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t))
dt
p
q
for a.e. t ∈ (0, T ),
Choice of the phase space:
X = D(φ) ⊂ X ,
dX (u, v ) := d(u, v ) + |φ(u) − φ(v )|
∀ u, v ∈ X .
Choice of the solution notion: We consider the set S of the
locally absolutely continuous p-curves of maximal slope for φ
u : [0, +∞) → X .
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
1
d
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t)) for a.e. t ∈ (0, +∞),
dt
p
q
Choice of the phase space:
X = D(φ) ⊂ X ,
dX (u, v ) := d(u, v ) + |φ(u) − φ(v )|
∀ u, v ∈ X .
Choice of the solution notion: We consider the set S of the
locally absolutely continuous p-curves of maximal slope for φ
u : [0, +∞) → X .
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
1
d
1
φ(u(t)) = − |u 0 |p (t) − |∂φ|q (u(t)) for a.e. t ∈ (0, +∞),
dt
p
q
Choice of the phase space:
X = D(φ) ⊂ X ,
dX (u, v ) := d(u, v ) + |φ(u) − φ(v )|
∀ u, v ∈ X .
Choice of the solution notion: We consider the set S of the
locally absolutely continuous p-curves of maximal slope for φ
u : [0, +∞) → X .
I
I
¿ Is S a generalized semiflow?
¿ Does S possess a global attractor?
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 1 [R., Segatti, Stefanelli ’06]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 1 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 1 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
(the same assumptions of the existence theorem in [A.G.S. ’05])
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 1 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
(the same assumptions of the existence theorem in [A.G.S. ’05])
Then,
S is a generalized semiflow.
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
I
φ is continuous along sequences with bounded energies and
slopes
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
I
φ is continuous along sequences with bounded energies and
slopes
I
the set Z (S) the equilibrium points of S
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
I
φ is continuous along sequences with bounded energies and
slopes
I
the set Z (S) the equilibrium points of S
Z (S) = {ū ∈ D(φ) : |∂φ|(ū) = 0}
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
I
φ is continuous along sequences with bounded energies and
slopes
I
the set Z (S) the equilibrium points of S
is bounded in (X , dX ).
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
Long-time behaviour for p-curves of maximal slope
Theorem 2 [R., Segatti, Stefanelli ’06]
Suppose that
I
φ is lower semicontinuous
I
φ is coercive
I
the map v 7→ |∂φ|(v ) is lower semicontinuous
I
φ is continuous along sequences with bounded energies and
slopes
I
the set Z (S) the equilibrium points of S
is bounded in (X , dX ).
Then,
S admits a global attractor A.
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
The transport equation
n
∂t ρ − div (ρjq (∇V )) = 0 in R × (0, +∞)
Riccarda Rossi
Global attractors for gradient flows in metric spaces
(
p > 2,
1
1
p + q =1
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
The transport equation
n
∂t ρ − div (ρjq (∇V )) = 0 in R × (0, +∞)
(
p > 2,
1
1
p + q =1
• Consider the set S of solutions ρ arising from the gradient flow
in (Pp (Rn ), Wp ) of the functional
Z
ρV dx
V(ρ) :=
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
The transport equation
n
∂t ρ − div (ρjq (∇V )) = 0 in R × (0, +∞)
(
p > 2,
1
1
p + q =1
Assumptions on V
I
V : Rn → (−∞, +∞] proper and lower semicontinuous
I
∃ ε, M1 , M2 > 0 : V (x) ≥ M1 |x|p+ε − M2 for all x ∈ Rn
I
V is λ-convex, λ ≤ 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
The transport equation
n
∂t ρ − div (ρjq (∇V )) = 0 in R × (0, +∞)
(
p > 2,
1
1
p + q =1
Assumptions on V
I
V : Rn → (−∞, +∞] proper and lower semicontinuous
I
∃ ε, M1 , M2 > 0 : V (x) ≥ M1 |x|p+ε − M2 for all x ∈ Rn
I
V is λ-convex, λ ≤ 0
• The generalized semiflow S admits a global attractor
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Applications
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
The drift diffusion equation
∇(ρF 0 (ρ) − F (ρ))
∂t ρ − div ρjq (
+ ∇V ) = 0 in Rn × (0, +∞)
ρ
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
The drift diffusion equation
∇(ρF 0 (ρ) − F (ρ))
∂t ρ − div ρjq (
+ ∇V ) = 0 in Rn × (0, +∞)
ρ
• Consider the set S of solutions ρ arising from the gradient flow
in (Pp (Rn ), Wp ) of the functional
Z
F (ρ) + ρV dx
φ(ρ) :=
Rn
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
The drift diffusion equation
∇(ρF 0 (ρ) − F (ρ))
∂t ρ − div ρjq (
+ ∇V ) = 0 in Rn × (0, +∞)
ρ
Assumptions on V
I
V : Rn → (−∞, +∞] l.s.c.
I
∃ ε, M1 , M2 > 0 : V (x) ≥ M1 |x|p+ε − M2 for all x ∈ Rn
I
V is λ-convex, λ ≤ 0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
The drift diffusion equation
∇(ρF 0 (ρ) − F (ρ))
∂t ρ − div ρjq (
+ ∇V ) = 0 in Rn × (0, +∞)
ρ
Assumptions on F
I
F : [0, +∞) → (−∞, +∞] l.s.c., F (0) = 0
I
lim sups→+∞
I
the map s 7→ F (e −s )e s is convex and non increasing
I
F satisfies the doubling condition
F (s)
s
= +∞, lim inf s↓0
Riccarda Rossi
Global attractors for gradient flows in metric spaces
F (s)
sα
> −∞, α >
n
n+p
Evolution equations and the Wasserstein metric
Curves of Maximal Slope
Existence of a Global Attractor
Applications
The drift diffusion equation
∇(ρF 0 (ρ) − F (ρ))
∂t ρ − div ρjq (
+ ∇V ) = 0 in Rn × (0, +∞)
ρ
Assumptions on F
I
F : [0, +∞) → (−∞, +∞] l.s.c., F (0) = 0
I
lim sups→+∞
I
the map s 7→ F (e −s )e s is convex and non increasing
I
F satisfies the doubling condition
F (s)
s
= +∞, lim inf s↓0
F (s)
sα
> −∞, α >
• The generalized semiflow S admits a global attractor
Riccarda Rossi
Global attractors for gradient flows in metric spaces
n
n+p