Sticky particles with interaction Giuseppe Savar e - - PowerPoint PPT Presentation

Sticky particles Cumulative distribution Lagrangian representation

Sticky particles with interaction

Giuseppe Savar´ e

http://www.imati.cnr.it/∼savare

Department of Mathematics, University of Pavia, Italy

HYP2012 - Padova, June 26, 2012 Jointly with Y. Brenier, W. Gangbo, L. Natile, M. Westdickenberg

1

Sticky particles Cumulative distribution Lagrangian representation

Outline

1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative

distribution function

3 Monotone rearrangement and Lagrangian representation

2

Sticky particles Cumulative distribution Lagrangian representation

Outline

1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative

distribution function

3 Monotone rearrangement and Lagrangian representation

2

Sticky particles Cumulative distribution Lagrangian representation

Outline

1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative

distribution function

3 Monotone rearrangement and Lagrangian representation

2

Sticky particles Cumulative distribution Lagrangian representation

Outline

1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative

distribution function

3 Monotone rearrangement and Lagrangian representation

3

Sticky particles Cumulative distribution Lagrangian representation

A one-dimensional compressible fluid flow

Consider a simple hyperbolic system modeling a one-dimensional compressible and pressureless fluid flow under the influence of a force field that is generated by the fluid itself. The density-momentum couple (̺, ̺v) satisfy

• ∂t̺ + ∂x(̺v)= 0

∂t(̺v) + ∂x(̺v2)= f[̺] in [0, ∞) × R, for suitable initial data (̺, ̺v)(t = 0, ·) =: (̺0, ̺0v0).

◮ The continuity equation for the density ̺ (a nonnegative measure in

time and space describing the distribution of mass or electric charge) and the real-valued Eulerian velocity field v.

◮ The momentum equation for ̺v

The force field f[̺] is described by a signed measure depending on the mass distribution.

4

Sticky particles Cumulative distribution Lagrangian representation

The structure of the force field f

We will assume that f[̺] is absolutely continuous with respect to ̺: the typical simplest form of f is f[̺] = −̺ ∂xq̺, q̺(x) = V (x) +

• R

W(x − y) d̺(y) (*) for suitable C1 potential functions V, W with linearly growing derivatives. Another relevant example is the Euler-Poisson system, for which f[̺] = −̺ ∂xq̺ with q̺ solution of −∂2

xxq̺ = λ(̺ − σ),

q̺ admits the representation similar to (*) W(x) := λ 2 |x|, V (x) := −λ 2

• R

|x − y| dσ(y), but possibly non differentiable W if ̺ is not absolutely continuous. The Euler-Poisson equations in the attractive regime (with λ > 0 and positive convex potential W) is the one-dimensional caricature of a cosmological model for the universe at an early stage, describing the formation of galaxies.

5

Sticky particles Cumulative distribution Lagrangian representation

The structure of the force field f

We will assume that f[̺] is absolutely continuous with respect to ̺: the typical simplest form of f is f[̺] = −̺ ∂xq̺, q̺(x) = V (x) +

• R

W(x − y) d̺(y) (*) for suitable C1 potential functions V, W with linearly growing derivatives. Another relevant example is the Euler-Poisson system, for which f[̺] = −̺ ∂xq̺ with q̺ solution of −∂2

xxq̺ = λ(̺ − σ),

q̺ admits the representation similar to (*) W(x) := λ 2 |x|, V (x) := −λ 2

• R

|x − y| dσ(y), but possibly non differentiable W if ̺ is not absolutely continuous. The Euler-Poisson equations in the attractive regime (with λ > 0 and positive convex potential W) is the one-dimensional caricature of a cosmological model for the universe at an early stage, describing the formation of galaxies.

5

Sticky particles Cumulative distribution Lagrangian representation

The structure of the force field f

We will assume that f[̺] is absolutely continuous with respect to ̺: the typical simplest form of f is f[̺] = −̺ ∂xq̺, q̺(x) = V (x) +

• R

W(x − y) d̺(y) (*) for suitable C1 potential functions V, W with linearly growing derivatives. Another relevant example is the Euler-Poisson system, for which f[̺] = −̺ ∂xq̺ with q̺ solution of −∂2

xxq̺ = λ(̺ − σ),

q̺ admits the representation similar to (*) W(x) := λ 2 |x|, V (x) := −λ 2

• R

|x − y| dσ(y), but possibly non differentiable W if ̺ is not absolutely continuous. The Euler-Poisson equations in the attractive regime (with λ > 0 and positive convex potential W) is the one-dimensional caricature of a cosmological model for the universe at an early stage, describing the formation of galaxies.

5

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 v1 v2 v3 v4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 P4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 P3 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Starting point: motion of a finite number of particles.

Discrete particle model N particles Pi := (mi, xi, vi), i = 1, . . . , N, with positive mass mi satisfying N

i=1 mi = 1

• rdered positions x1 < x2 < . . . < xN−1 < xN,

and velocities vi. P1 P2 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d dtxi(t) = vi(t), d dtvi(t) = ai(x(t)). The first collision time t = t1 correspond to xj(t1) = xj+1(t1) = . . . = xk(t1) for some indices j < k. The particles Pj, Pj+1, . . . , Pk collapse and stick in a new particle P with mass m := mj + . . . + mk and “barycentric” velocity v := mjvj(t1) + mj+1vj+1(t1) + . . . + mkvk(t1) m

6

Sticky particles Cumulative distribution Lagrangian representation

Measure-theoretic description

We thus have: a (finite) sequence of collision times 0 < t1 < t2 < . . . in each interval [th, th+1) a finite number N h of (suitably relabeled) particles P1(t), · · · , PNh(t), Pi(t) := (mi, xi(t), vi(t)). We can introduce the measures ̺t :=

Nh

• i=1

miδxi(t) ∈ P(R), (̺ v)t :=

Nh

• i=1

mi vi(t) δxi(t) ∈ M(R) f[̺t] :=

Nh

• i=1

miai(t)δxi(t) ∈ M(R) if t ∈ [th, th+1). They satisfy the 1-dimensional pressureless Euler system in the sense

• f distributions
• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

7

Sticky particles Cumulative distribution Lagrangian representation

Measure-theoretic description

We thus have: a (finite) sequence of collision times 0 < t1 < t2 < . . . in each interval [th, th+1) a finite number N h of (suitably relabeled) particles P1(t), · · · , PNh(t), Pi(t) := (mi, xi(t), vi(t)). We can introduce the measures ̺t :=

Nh

• i=1

miδxi(t) ∈ P(R), (̺ v)t :=

Nh

• i=1

mi vi(t) δxi(t) ∈ M(R) f[̺t] :=

Nh

• i=1

miai(t)δxi(t) ∈ M(R) if t ∈ [th, th+1). They satisfy the 1-dimensional pressureless Euler system in the sense

• f distributions
• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

7

Sticky particles Cumulative distribution Lagrangian representation

Measure-theoretic description

We thus have: a (finite) sequence of collision times 0 < t1 < t2 < . . . in each interval [th, th+1) a finite number N h of (suitably relabeled) particles P1(t), · · · , PNh(t), Pi(t) := (mi, xi(t), vi(t)). We can introduce the measures ̺t :=

Nh

• i=1

miδxi(t) ∈ P(R), (̺ v)t :=

Nh

• i=1

mi vi(t) δxi(t) ∈ M(R) f[̺t] :=

Nh

• i=1

miai(t)δxi(t) ∈ M(R) if t ∈ [th, th+1). They satisfy the 1-dimensional pressureless Euler system in the sense

• f distributions
• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

7

Sticky particles Cumulative distribution Lagrangian representation

The various models

Motion driven by a potential V      d dtxi = vi, d dtvi = −∂xV (xi)

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = −̺∂xV Motion driven by an interaction potential W        d dtxi = vi, d dtvi = −

• j=i

mj∂xW(xi − xj)

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = −̺

• ̺ ∗ ∂xW
• Non smooth interaction: the Euler-Poisson system when

W(x) = ±λ|x|.

8

Sticky particles Cumulative distribution Lagrangian representation

The various models

Motion driven by a potential V      d dtxi = vi, d dtvi = −∂xV (xi)

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = −̺∂xV Motion driven by an interaction potential W        d dtxi = vi, d dtvi = −

• j=i

mj∂xW(xi − xj)

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = −̺

• ̺ ∗ ∂xW
• Non smooth interaction: the Euler-Poisson system when

W(x) = ±λ|x|.

8

Sticky particles Cumulative distribution Lagrangian representation

The various models

Motion driven by a potential V      d dtxi = vi, d dtvi = −∂xV (xi)

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = −̺∂xV Motion driven by an interaction potential W        d dtxi = vi, d dtvi = −

• j=i

mj∂xW(xi − xj)

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = −̺

• ̺ ∗ ∂xW
• Non smooth interaction: the Euler-Poisson system when

W(x) = ±λ|x|.

8

Sticky particles Cumulative distribution Lagrangian representation

Main problem: continuous limit

Consider a sequence of discrete initial data µn

0 := (̺n 0 , ̺n 0 vn 0 ) converging to

µ0 = (̺0, ̺0v0) in a suitable measure-theoretic sense and let µn

t = (̺n t , ̺n t vn t ) be the (discrete) solution of SPS.

Problem

◮ Prove that the limit µt = (̺t, ̺tvt) of the SPS µn t = (̺n t , ̺n t vn t ) as

n ↑ +∞ exists.

◮ Find a suitable characterization of µt ◮ Show that (̺t, ̺tvt) solves the pressureless Euler system

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

9

Sticky particles Cumulative distribution Lagrangian representation

Main problem: continuous limit

Consider a sequence of discrete initial data µn

0 := (̺n 0 , ̺n 0 vn 0 ) converging to

µ0 = (̺0, ̺0v0) in a suitable measure-theoretic sense and let µn

t = (̺n t , ̺n t vn t ) be the (discrete) solution of SPS.

Problem

◮ Prove that the limit µt = (̺t, ̺tvt) of the SPS µn t = (̺n t , ̺n t vn t ) as

n ↑ +∞ exists.

◮ Find a suitable characterization of µt ◮ Show that (̺t, ̺tvt) solves the pressureless Euler system

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

9

Sticky particles Cumulative distribution Lagrangian representation

Main problem: continuous limit

Consider a sequence of discrete initial data µn

0 := (̺n 0 , ̺n 0 vn 0 ) converging to

µ0 = (̺0, ̺0v0) in a suitable measure-theoretic sense and let µn

t = (̺n t , ̺n t vn t ) be the (discrete) solution of SPS.

Problem

◮ Prove that the limit µt = (̺t, ̺tvt) of the SPS µn t = (̺n t , ̺n t vn t ) as

n ↑ +∞ exists.

◮ Find a suitable characterization of µt ◮ Show that (̺t, ̺tvt) solves the pressureless Euler system

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

9

Sticky particles Cumulative distribution Lagrangian representation

Main problem: continuous limit

Consider a sequence of discrete initial data µn

0 := (̺n 0 , ̺n 0 vn 0 ) converging to

µ0 = (̺0, ̺0v0) in a suitable measure-theoretic sense and let µn

t = (̺n t , ̺n t vn t ) be the (discrete) solution of SPS.

Problem

◮ Prove that the limit µt = (̺t, ̺tvt) of the SPS µn t = (̺n t , ̺n t vn t ) as

n ↑ +∞ exists.

◮ Find a suitable characterization of µt ◮ Show that (̺t, ̺tvt) solves the pressureless Euler system

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], in R × (0, +∞); ̺|t=0 = ̺0, v|t=0 = v0,

9

Sticky particles Cumulative distribution Lagrangian representation

Outline

1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative

distribution function

3 Monotone rearrangement and Lagrangian representation

10

Sticky particles Cumulative distribution Lagrangian representation

Main contributions in the case f[̺] ≡ 0

• Existence and convergence

◮ Grenier ’95, E-Rykov-Sinai ’96: first existence and convergence result. ◮ Brenier-Grenier ’96: Characterization of the limit in terms of a

suitable scalar conservation law, uniqueness.

◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08: further

• refinements. Gangbo-Nguyen-Tudorascu ’09: Euler-Poisson.

Basic assumptions: ̺n

0 → ̺0 in the L2-Wasserstein distance,

vn

0 = v0 is given by a continuous function with (at most) linear growth.

In particular the result cover the case when ̺n

0 , ̺0 have a common

compact support and ̺n

0 → ̺0 weakly in the sense of distribution (or,

equivalently, in the duality with continuous functions).

• Pioneering ideas which lies (more or less explicitly) at the core of the

papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by

◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘

ı ’07 in a formal way.

• Different approaches and models:

◮ Bouchut-James ’95, Poupaud-Rascle ’97 ◮ Sobolevski˘

ı ’97, Boudin ’00: viscous regularization.

◮ Wolansky ’07: particles with finite size. 11

Sticky particles Cumulative distribution Lagrangian representation

Main contributions in the case f[̺] ≡ 0

• Existence and convergence

◮ Grenier ’95, E-Rykov-Sinai ’96: first existence and convergence result. ◮ Brenier-Grenier ’96: Characterization of the limit in terms of a

suitable scalar conservation law, uniqueness.

◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08: further

• refinements. Gangbo-Nguyen-Tudorascu ’09: Euler-Poisson.

Basic assumptions: ̺n

0 → ̺0 in the L2-Wasserstein distance,

vn

0 = v0 is given by a continuous function with (at most) linear growth.

In particular the result cover the case when ̺n

0 , ̺0 have a common

compact support and ̺n

0 → ̺0 weakly in the sense of distribution (or,

equivalently, in the duality with continuous functions).

• Pioneering ideas which lies (more or less explicitly) at the core of the

papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by

◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘

ı ’07 in a formal way.

• Different approaches and models:

◮ Bouchut-James ’95, Poupaud-Rascle ’97 ◮ Sobolevski˘

ı ’97, Boudin ’00: viscous regularization.

◮ Wolansky ’07: particles with finite size. 11

Sticky particles Cumulative distribution Lagrangian representation

Main contributions in the case f[̺] ≡ 0

• Existence and convergence

◮ Grenier ’95, E-Rykov-Sinai ’96: first existence and convergence result. ◮ Brenier-Grenier ’96: Characterization of the limit in terms of a

suitable scalar conservation law, uniqueness.

◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08: further

• refinements. Gangbo-Nguyen-Tudorascu ’09: Euler-Poisson.

Basic assumptions: ̺n

0 → ̺0 in the L2-Wasserstein distance,

vn

0 = v0 is given by a continuous function with (at most) linear growth.

In particular the result cover the case when ̺n

0 , ̺0 have a common

compact support and ̺n

0 → ̺0 weakly in the sense of distribution (or,

equivalently, in the duality with continuous functions).

• Pioneering ideas which lies (more or less explicitly) at the core of the

papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by

◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘

ı ’07 in a formal way.

• Different approaches and models:

◮ Bouchut-James ’95, Poupaud-Rascle ’97 ◮ Sobolevski˘

ı ’97, Boudin ’00: viscous regularization.

◮ Wolansky ’07: particles with finite size. 11

Sticky particles Cumulative distribution Lagrangian representation

Main contributions in the case f[̺] ≡ 0

• Existence and convergence

◮ Grenier ’95, E-Rykov-Sinai ’96: first existence and convergence result. ◮ Brenier-Grenier ’96: Characterization of the limit in terms of a

suitable scalar conservation law, uniqueness.

◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08: further

• refinements. Gangbo-Nguyen-Tudorascu ’09: Euler-Poisson.

Basic assumptions: ̺n

0 → ̺0 in the L2-Wasserstein distance,

vn

0 = v0 is given by a continuous function with (at most) linear growth.

In particular the result cover the case when ̺n

0 , ̺0 have a common

compact support and ̺n

0 → ̺0 weakly in the sense of distribution (or,

equivalently, in the duality with continuous functions).

• Pioneering ideas which lies (more or less explicitly) at the core of the

papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by

◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘

ı ’07 in a formal way.

• Different approaches and models:

◮ Bouchut-James ’95, Poupaud-Rascle ’97 ◮ Sobolevski˘

ı ’97, Boudin ’00: viscous regularization.

◮ Wolansky ’07: particles with finite size. 11

Sticky particles Cumulative distribution Lagrangian representation

Main contributions in the case f[̺] ≡ 0

• Existence and convergence

◮ Grenier ’95, E-Rykov-Sinai ’96: first existence and convergence result. ◮ Brenier-Grenier ’96: Characterization of the limit in terms of a

suitable scalar conservation law, uniqueness.

◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08: further

• refinements. Gangbo-Nguyen-Tudorascu ’09: Euler-Poisson.

Basic assumptions: ̺n

0 → ̺0 in the L2-Wasserstein distance,

vn

0 = v0 is given by a continuous function with (at most) linear growth.

In particular the result cover the case when ̺n

0 , ̺0 have a common

compact support and ̺n

0 → ̺0 weakly in the sense of distribution (or,

equivalently, in the duality with continuous functions).

• Pioneering ideas which lies (more or less explicitly) at the core of the

papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by

◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘

ı ’07 in a formal way.

• Different approaches and models:

◮ Bouchut-James ’95, Poupaud-Rascle ’97 ◮ Sobolevski˘

ı ’97, Boudin ’00: viscous regularization.

◮ Wolansky ’07: particles with finite size. 11

Sticky particles Cumulative distribution Lagrangian representation

The cumulative distribution function

For every probability measure ̺ ∈ P(R) x x1 m1 x2 m2 x3 m3 x4 m4 A discrete measure ρ =

4

• j=1

mjδxj

12

Sticky particles Cumulative distribution Lagrangian representation

The cumulative distribution function

For every probability measure ̺ ∈ P(R) x x1 m1 x2 m2 x3 m3 x4 m4 A discrete measure ρ =

4

• j=1

mjδxj we introduce the distribution function Mρ(x) := ρ

• (−∞, x]
• ,

x x1 x2 x3 x4 Its distribution function Mρ Total mass = 1

12

Sticky particles Cumulative distribution Lagrangian representation

The Brenier-Grenier formulation in the absence of force

M̺(x) := ̺

• (−∞, x]
• ,

x ∈ R, so that ̺ = ∂xM̺ in D′(R). Main idea: study the evolution of Mt := M̺t, where ̺t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂tM + ∂xA(M) = 0 in R × (0, +∞) where A : [0, 1] → R is a continuous flux function depending only on the initial data ̺0 and v0. It is characterized by A′(M0(x)) = v0(x).

13

Sticky particles Cumulative distribution Lagrangian representation

The Brenier-Grenier formulation in the absence of force

M̺(x) := ̺

• (−∞, x]
• ,

x ∈ R, so that ̺ = ∂xM̺ in D′(R). Main idea: study the evolution of Mt := M̺t, where ̺t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂tM + ∂xA(M) = 0 in R × (0, +∞) where A : [0, 1] → R is a continuous flux function depending only on the initial data ̺0 and v0. It is characterized by A′(M0(x)) = v0(x).

13

Sticky particles Cumulative distribution Lagrangian representation

The Brenier-Grenier formulation in the absence of force

M̺(x) := ̺

• (−∞, x]
• ,

x ∈ R, so that ̺ = ∂xM̺ in D′(R). Main idea: study the evolution of Mt := M̺t, where ̺t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂tM + ∂xA(M) = 0 in R × (0, +∞) where A : [0, 1] → R is a continuous flux function depending only on the initial data ̺0 and v0. It is characterized by A′(M0(x)) = v0(x).

13

Sticky particles Cumulative distribution Lagrangian representation

The Brenier-Grenier formulation in the absence of force

M̺(x) := ̺

• (−∞, x]
• ,

x ∈ R, so that ̺ = ∂xM̺ in D′(R). Main idea: study the evolution of Mt := M̺t, where ̺t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂tM + ∂xA(M) = 0 in R × (0, +∞) where A : [0, 1] → R is a continuous flux function depending only on the initial data ̺0 and v0. It is characterized by A′(M0(x)) = v0(x).

13

Sticky particles Cumulative distribution Lagrangian representation

Outline

1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative

distribution function

3 Monotone rearrangement and Lagrangian representation

14

Sticky particles Cumulative distribution Lagrangian representation

Monotone rearrangement

x x1 m1 x2 m2 x3 m3 x4 m4 ρ

15

Sticky particles Cumulative distribution Lagrangian representation

Monotone rearrangement

x x1 m1 x2 m2 x3 m3 x4 m4 ρ x x1 x2 x3x4 The distribution function Mρ Mass = 1 w1 w2 w3

15

Sticky particles Cumulative distribution Lagrangian representation

Monotone rearrangement

x x1 m1 x2 m2 x3 m3 x4 m4 ρ x x1 x2 x3x4 The distribution function Mρ Mass = 1 w1 w2 w3 The monotone rearrangement Xρ = Mρ

−1

x1 w1 w2 w3 x2 x3 x4 1 w1 = m1 w2 = m1 + m2 w3 = m1 + m2 + m3

15

Sticky particles Cumulative distribution Lagrangian representation

Optimal transport and monotone rearrangement

Point of view of 1-dimensional optimal transport: instead of using the cumulative distribution function M̺(x) = ̺

• (−∞, x]
• , we

represent each probability measure ̺ by its monotone rearrangement X̺ : (0, 1) → R: X̺(w) is the position x such that ̺((−∞, x]) = w. X̺(w) := inf

• x ∈ R : M̺(x) > w
• w ∈ (0, 1)

which is the so-called pseudo-inverse of M̺. The map X̺ is nondecreasing and right-continuous. It pushes the Lebesgue measure λ := L 1|(0,1) on (0, 1) onto ̺, i.e. (X̺)#L 1 |(0,1) = ̺, L 1(X−1

̺ (B)) = ̺(B)

for every Borel set B ⊂ R It satisfies the change of variable formula

• R

φ(x) d̺(x) = 1 φ(X̺(w)) dw for every nonnegative/bounded Borel function φ : R → R. In particular, m2(̺) :=

• R

|x|2 d̺(x) = 1

• X̺(w)
• 2 dw =
• X̺
• 2

L2(0,1)

16

Sticky particles Cumulative distribution Lagrangian representation

Optimal transport and monotone rearrangement

Point of view of 1-dimensional optimal transport: instead of using the cumulative distribution function M̺(x) = ̺

• (−∞, x]
• , we

represent each probability measure ̺ by its monotone rearrangement X̺ : (0, 1) → R: X̺(w) is the position x such that ̺((−∞, x]) = w. X̺(w) := inf

• x ∈ R : M̺(x) > w
• w ∈ (0, 1)

which is the so-called pseudo-inverse of M̺. The map X̺ is nondecreasing and right-continuous. It pushes the Lebesgue measure λ := L 1|(0,1) on (0, 1) onto ̺, i.e. (X̺)#L 1 |(0,1) = ̺, L 1(X−1

̺ (B)) = ̺(B)

for every Borel set B ⊂ R It satisfies the change of variable formula

• R

φ(x) d̺(x) = 1 φ(X̺(w)) dw for every nonnegative/bounded Borel function φ : R → R. In particular, m2(̺) :=

• R

|x|2 d̺(x) = 1

• X̺(w)
• 2 dw =
• X̺
• 2

L2(0,1)

16

Sticky particles Cumulative distribution Lagrangian representation

Optimal transport and monotone rearrangement

Point of view of 1-dimensional optimal transport: instead of using the cumulative distribution function M̺(x) = ̺

• (−∞, x]
• , we

represent each probability measure ̺ by its monotone rearrangement X̺ : (0, 1) → R: X̺(w) is the position x such that ̺((−∞, x]) = w. X̺(w) := inf

• x ∈ R : M̺(x) > w
• w ∈ (0, 1)

which is the so-called pseudo-inverse of M̺. The map X̺ is nondecreasing and right-continuous. It pushes the Lebesgue measure λ := L 1|(0,1) on (0, 1) onto ̺, i.e. (X̺)#L 1 |(0,1) = ̺, L 1(X−1

̺ (B)) = ̺(B)

for every Borel set B ⊂ R It satisfies the change of variable formula

• R

φ(x) d̺(x) = 1 φ(X̺(w)) dw for every nonnegative/bounded Borel function φ : R → R. In particular, m2(̺) :=

• R

|x|2 d̺(x) = 1

• X̺(w)
• 2 dw =
• X̺
• 2

L2(0,1)

16

Sticky particles Cumulative distribution Lagrangian representation

Optimal transport and monotone rearrangement

Point of view of 1-dimensional optimal transport: instead of using the cumulative distribution function M̺(x) = ̺

• (−∞, x]
• , we

represent each probability measure ̺ by its monotone rearrangement X̺ : (0, 1) → R: X̺(w) is the position x such that ̺((−∞, x]) = w. X̺(w) := inf

• x ∈ R : M̺(x) > w
• w ∈ (0, 1)

which is the so-called pseudo-inverse of M̺. The map X̺ is nondecreasing and right-continuous. It pushes the Lebesgue measure λ := L 1|(0,1) on (0, 1) onto ̺, i.e. (X̺)#L 1 |(0,1) = ̺, L 1(X−1

̺ (B)) = ̺(B)

for every Borel set B ⊂ R It satisfies the change of variable formula

• R

φ(x) d̺(x) = 1 φ(X̺(w)) dw for every nonnegative/bounded Borel function φ : R → R. In particular, m2(̺) :=

• R

|x|2 d̺(x) = 1

• X̺(w)
• 2 dw =
• X̺
• 2

L2(0,1)

16

Sticky particles Cumulative distribution Lagrangian representation

Optimal transport and monotone rearrangement

Point of view of 1-dimensional optimal transport: instead of using the cumulative distribution function M̺(x) = ̺

• (−∞, x]
• , we

represent each probability measure ̺ by its monotone rearrangement X̺ : (0, 1) → R: X̺(w) is the position x such that ̺((−∞, x]) = w. X̺(w) := inf

• x ∈ R : M̺(x) > w
• w ∈ (0, 1)

which is the so-called pseudo-inverse of M̺. The map X̺ is nondecreasing and right-continuous. It pushes the Lebesgue measure λ := L 1|(0,1) on (0, 1) onto ̺, i.e. (X̺)#L 1 |(0,1) = ̺, L 1(X−1

̺ (B)) = ̺(B)

for every Borel set B ⊂ R It satisfies the change of variable formula

• R

φ(x) d̺(x) = 1 φ(X̺(w)) dw for every nonnegative/bounded Borel function φ : R → R. In particular, m2(̺) :=

• R

|x|2 d̺(x) = 1

• X̺(w)
• 2 dw =
• X̺
• 2

L2(0,1)

16

Sticky particles Cumulative distribution Lagrangian representation

Wasserstein distance and the L2 isometry

The map ̺ → X̺ is a one-to-one correspondence between the space P2(R) of probability measures with finite quadratic moment m2(̺) =

• R |x|2 d̺(x) < +∞

and the closed convex cone K of all the nondecreasing function in L2(0, 1) (among which we can always choose the right-continuous representative). L2-Wasserstein distance W2(̺1, ̺2) between ̺1, ̺2 ∈ P2(R): W 2

2 (̺1, ̺2) :=

1

• X̺1(w) − X̺2(w)
• 2 dw =
• X̺1 − X̺2
• 2

L2(0,1)

In this way ̺ ↔ X̺ is an isometry between (P2(R), W2) and (K, · L2(0,1)).

17

Sticky particles Cumulative distribution Lagrangian representation

Wasserstein distance and the L2 isometry

The map ̺ → X̺ is a one-to-one correspondence between the space P2(R) of probability measures with finite quadratic moment m2(̺) =

• R |x|2 d̺(x) < +∞

and the closed convex cone K of all the nondecreasing function in L2(0, 1) (among which we can always choose the right-continuous representative). L2-Wasserstein distance W2(̺1, ̺2) between ̺1, ̺2 ∈ P2(R): W 2

2 (̺1, ̺2) :=

1

• X̺1(w) − X̺2(w)
• 2 dw =
• X̺1 − X̺2
• 2

L2(0,1)

In this way ̺ ↔ X̺ is an isometry between (P2(R), W2) and (K, · L2(0,1)).

17

Sticky particles Cumulative distribution Lagrangian representation

Wasserstein distance and the L2 isometry

The map ̺ → X̺ is a one-to-one correspondence between the space P2(R) of probability measures with finite quadratic moment m2(̺) =

• R |x|2 d̺(x) < +∞

and the closed convex cone K of all the nondecreasing function in L2(0, 1) (among which we can always choose the right-continuous representative). L2-Wasserstein distance W2(̺1, ̺2) between ̺1, ̺2 ∈ P2(R): W 2

2 (̺1, ̺2) :=

1

• X̺1(w) − X̺2(w)
• 2 dw =
• X̺1 − X̺2
• 2

L2(0,1)

In this way ̺ ↔ X̺ is an isometry between (P2(R), W2) and (K, · L2(0,1)).

17

Sticky particles Cumulative distribution Lagrangian representation

Wasserstein distance and the L2 isometry

The map ̺ → X̺ is a one-to-one correspondence between the space P2(R) of probability measures with finite quadratic moment m2(̺) =

• R |x|2 d̺(x) < +∞

and the closed convex cone K of all the nondecreasing function in L2(0, 1) (among which we can always choose the right-continuous representative). L2-Wasserstein distance W2(̺1, ̺2) between ̺1, ̺2 ∈ P2(R): W 2

2 (̺1, ̺2) :=

1

• X̺1(w) − X̺2(w)
• 2 dw =
• X̺1 − X̺2
• 2

L2(0,1)

In this way ̺ ↔ X̺ is an isometry between (P2(R), W2) and (K, · L2(0,1)).

17

Sticky particles Cumulative distribution Lagrangian representation

Lagrangian parametrizations

To the (discrete) data µt = (̺t, ̺tvt) we associate the functions (Xt, Vt) ∈ K × L2(0, 1) by Xt := X̺t, Vt := vt ◦ Xt. Notice that the second component of (Xt, Vt) do not span the whole space L2(R) in general, but it is contained in the closed subspace HXt :=

• V = v ◦ Xt for some Borel map v ∈ L2

̺t(R).

• The

reduced cone

• f

the discrete particle system with masses m = (m1, m2, · · · , mN) Km :=

• X ∈ K : X|[wi−1,wi) ≡ xi

is constant

• 18

Sticky particles Cumulative distribution Lagrangian representation

Lagrangian parametrizations

To the (discrete) data µt = (̺t, ̺tvt) we associate the functions (Xt, Vt) ∈ K × L2(0, 1) by Xt := X̺t, Vt := vt ◦ Xt. Notice that the second component of (Xt, Vt) do not span the whole space L2(R) in general, but it is contained in the closed subspace HXt :=

• V = v ◦ Xt for some Borel map v ∈ L2

̺t(R).

• The

reduced cone

• f

the discrete particle system with masses m = (m1, m2, · · · , mN) Km :=

• X ∈ K : X|[wi−1,wi) ≡ xi

is constant

• 18

Sticky particles Cumulative distribution Lagrangian representation

Lagrangian parametrizations

To the (discrete) data µt = (̺t, ̺tvt) we associate the functions (Xt, Vt) ∈ K × L2(0, 1) by Xt := X̺t, Vt := vt ◦ Xt. Notice that the second component of (Xt, Vt) do not span the whole space L2(R) in general, but it is contained in the closed subspace HXt :=

• V = v ◦ Xt for some Borel map v ∈ L2

̺t(R).

• The

reduced cone

• f

the discrete particle system with masses m = (m1, m2, · · · , mN) Km :=

• X ∈ K : X|[wi−1,wi) ≡ xi

is constant

• The monotone rearrangement

Xρ = Mρ

−1

x1 w1 w2 w3 x2 x3 x4 1 w1 = m1, w2 = m1 + m2, · · ·

18

Sticky particles Cumulative distribution Lagrangian representation

The normal cone NXK

Ξ ∈ NXK if and only if X ∈ K and one of the following equivalent conditions holds (Ξ|Z − X) ≤ 0 for every Z ∈ K, P

K(X + Ξ) = X,

P

K is the L2 projection on K

The normal cone NXK coincides with the subdifferential ∂IK(X) of the indi- cator (convex, lower semicontinuous) function of K in L2(0, 1) IK(X) =

• if X ∈ K,

+∞

• therwise

19

Sticky particles Cumulative distribution Lagrangian representation

The normal cone NXK

Ξ ∈ NXK if and only if X ∈ K and one of the following equivalent conditions holds (Ξ|Z − X) ≤ 0 for every Z ∈ K, P

K(X + Ξ) = X,

P

K is the L2 projection on K

The normal cone NXK coincides with the subdifferential ∂IK(X) of the indi- cator (convex, lower semicontinuous) function of K in L2(0, 1) IK(X) =

• if X ∈ K,

+∞

• therwise

19

Sticky particles Cumulative distribution Lagrangian representation

The normal cone NXK

Ξ ∈ NXK if and only if X ∈ K and one of the following equivalent conditions holds (Ξ|Z − X) ≤ 0 for every Z ∈ K, P

K(X + Ξ) = X,

P

K is the L2 projection on K

The normal cone NXK coincides with the subdifferential ∂IK(X) of the indi- cator (convex, lower semicontinuous) function of K in L2(0, 1) IK(X) =

• if X ∈ K,

+∞

• therwise

19

Sticky particles Cumulative distribution Lagrangian representation

The normal cone NXK

Ξ ∈ NXK if and only if X ∈ K and one of the following equivalent conditions holds (Ξ|Z − X) ≤ 0 for every Z ∈ K, P

K(X + Ξ) = X,

P

K is the L2 projection on K

The normal cone NXK coincides with the subdifferential ∂IK(X) of the indi- cator (convex, lower semicontinuous) function of K in L2(0, 1) IK(X) =

• if X ∈ K,

+∞

• therwise

K Tx1K Nx1K x1 Tx2K Nx2K x2

Figure: Normal and tangent cones.

19

Sticky particles Cumulative distribution Lagrangian representation

Second order differential equations and jump conditions

Force field: f[̺] F[X̺](w) such that

• R

ζ(x) df[̺] = 1 ζ(X̺)F dw for every test function ζ ∈ C0

b (R).

Differential equation for the collision free motion in Lagrangian coordinates: d2 dt2 Xt = F[Xt], X0 = X̺0, X′

0 = V0 = v0 ◦ X0.

20

Sticky particles Cumulative distribution Lagrangian representation

Second order differential equations and jump conditions

Force field: f[̺] F[X̺](w) such that

• R

ζ(x) df[̺] = 1 ζ(X̺)F dw for every test function ζ ∈ C0

b (R).

Differential equation for the collision free motion in Lagrangian coordinates: d2 dt2 Xt = F[Xt], X0 = X̺0, X′

0 = V0 = v0 ◦ X0.

20

Sticky particles Cumulative distribution Lagrangian representation

Second order differential equations and jump conditions

Force field: f[̺] F[X̺](w) such that

• R

ζ(x) df[̺] = 1 ζ(X̺)F dw for every test function ζ ∈ C0

b (R).

Differential equation for the collision free motion in Lagrangian coordinates: d2 dt2 Xt = F[Xt], X0 = X̺0, X′

0 = V0 = v0 ◦ X0.

After a collision at a time ¯ t: V¯

t+ = P TX(¯

t)K(V¯

t−)

i.e. V¯

t+ + NX(¯ t)K ∋ V¯ t−

K particle velocity after collision particle trajectory Figure: Projection of velocities onto the tangent cone.

20

Sticky particles Cumulative distribution Lagrangian representation

Second order differential inclusions: the role of the sticky condition

“Formal” differential inclusion for the SPS: d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt] (⋆) The discrete case still satisfies (⋆) if F satisfies the “non splitting” condition F[X] ∈ HX i.e. F[X] depends on X

• r, more generally,

F[X] − P

HX F[X] ∈ NXK

as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion (Schatzman ’78, Moreau ’83) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: s < t ⇒ Xt ∈ HXs i.e. Xt “depends on” Xs. By the monotonicity property of ∂IK we have s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt)

21

Sticky particles Cumulative distribution Lagrangian representation

Second order differential inclusions: the role of the sticky condition

“Formal” differential inclusion for the SPS: d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt] (⋆) The discrete case still satisfies (⋆) if F satisfies the “non splitting” condition F[X] ∈ HX i.e. F[X] depends on X

• r, more generally,

F[X] − P

HX F[X] ∈ NXK

as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion (Schatzman ’78, Moreau ’83) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: s < t ⇒ Xt ∈ HXs i.e. Xt “depends on” Xs. By the monotonicity property of ∂IK we have s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt)

21

Sticky particles Cumulative distribution Lagrangian representation

Second order differential inclusions: the role of the sticky condition

“Formal” differential inclusion for the SPS: d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt] (⋆) The discrete case still satisfies (⋆) if F satisfies the “non splitting” condition F[X] ∈ HX i.e. F[X] depends on X

• r, more generally,

F[X] − P

HX F[X] ∈ NXK

as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion (Schatzman ’78, Moreau ’83) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: s < t ⇒ Xt ∈ HXs i.e. Xt “depends on” Xs. By the monotonicity property of ∂IK we have s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt)

21

Sticky particles Cumulative distribution Lagrangian representation

Second order differential inclusions: the role of the sticky condition

“Formal” differential inclusion for the SPS: d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt] (⋆) The discrete case still satisfies (⋆) if F satisfies the “non splitting” condition F[X] ∈ HX i.e. F[X] depends on X

• r, more generally,

F[X] − P

HX F[X] ∈ NXK

as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion (Schatzman ’78, Moreau ’83) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: s < t ⇒ Xt ∈ HXs i.e. Xt “depends on” Xs. By the monotonicity property of ∂IK we have s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt)

21

Sticky particles Cumulative distribution Lagrangian representation

Second order differential inclusions: the role of the sticky condition

“Formal” differential inclusion for the SPS: d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt] (⋆) The discrete case still satisfies (⋆) if F satisfies the “non splitting” condition F[X] ∈ HX i.e. F[X] depends on X

• r, more generally,

F[X] − P

HX F[X] ∈ NXK

as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion (Schatzman ’78, Moreau ’83) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: s < t ⇒ Xt ∈ HXs i.e. Xt “depends on” Xs. By the monotonicity property of ∂IK we have s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt)

21

Sticky particles Cumulative distribution Lagrangian representation

Second order differential inclusions: the role of the sticky condition

“Formal” differential inclusion for the SPS: d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt] (⋆) The discrete case still satisfies (⋆) if F satisfies the “non splitting” condition F[X] ∈ HX i.e. F[X] depends on X

• r, more generally,

F[X] − P

HX F[X] ∈ NXK

as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion (Schatzman ’78, Moreau ’83) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: s < t ⇒ Xt ∈ HXs i.e. Xt “depends on” Xs. By the monotonicity property of ∂IK we have s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt)

21

Sticky particles Cumulative distribution Lagrangian representation

Reduction to first-order differential inclusions

d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt], s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt) (⋆) We can then integrate (⋆) with respect to time from 0 to a final time t: since t ∂IK(Xs) ds ∈ ∂IK(Xt). d dtXt − V0 + ∂IK(Xt) ∋ t F[Xs] ds (⋆⋆) Integrating again we get Xt − (X0 + tV0) + ∂IK(Xt) ∋ t (t − s)F[Xs] ds i.e. Xt = P

K

• X0 + tV0 +

t (t − s)F[Xs] ds

• (⋆ ⋆ ⋆)

22

Sticky particles Cumulative distribution Lagrangian representation

Reduction to first-order differential inclusions

d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt], s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt) (⋆) We can then integrate (⋆) with respect to time from 0 to a final time t: since t ∂IK(Xs) ds ∈ ∂IK(Xt). d dtXt − V0 + ∂IK(Xt) ∋ t F[Xs] ds (⋆⋆) Integrating again we get Xt − (X0 + tV0) + ∂IK(Xt) ∋ t (t − s)F[Xs] ds i.e. Xt = P

K

• X0 + tV0 +

t (t − s)F[Xs] ds

• (⋆ ⋆ ⋆)

22

Sticky particles Cumulative distribution Lagrangian representation

Reduction to first-order differential inclusions

d2 dt2 Xt + ∂IK(Xt) ∋ F[Xt], s < t ⇒ ∂IK(Xs) ⊂ ∂IK(Xt) (⋆) We can then integrate (⋆) with respect to time from 0 to a final time t: since t ∂IK(Xs) ds ∈ ∂IK(Xt). d dtXt − V0 + ∂IK(Xt) ∋ t F[Xs] ds (⋆⋆) Integrating again we get Xt − (X0 + tV0) + ∂IK(Xt) ∋ t (t − s)F[Xs] ds i.e. Xt = P

K

• X0 + tV0 +

t (t − s)F[Xs] ds

• (⋆ ⋆ ⋆)

22

Sticky particles Cumulative distribution Lagrangian representation

A description of the evolution by a differential inclusion

Recall that to the (discrete) data µt = (̺t, ̺tvt) we associate the functions (Xt, Vt) ∈ K × L2(0, 1) by Xt := X̺t, Vt := vt ◦ Xt. Theorem (Lagrangian representation) A family µt = (̺t, ̺tvt) is a solution of the (discrete) SPS if and only if X is the unique strong solution of the differential inclusion d dtXt + ∂IK(Xt) ∋ V0 + t F[Xs] ds, lim

t↓0 Xt = X0.

Equivalently, setting Yt := V0 + t

0 F[Xs] ds we get the well posed system

     d dtXt + ∂IK(Xt) ∋ Yt d dtYt = F[Xt]

23

Sticky particles Cumulative distribution Lagrangian representation

A description of the evolution by a differential inclusion

Recall that to the (discrete) data µt = (̺t, ̺tvt) we associate the functions (Xt, Vt) ∈ K × L2(0, 1) by Xt := X̺t, Vt := vt ◦ Xt. Theorem (Lagrangian representation) A family µt = (̺t, ̺tvt) is a solution of the (discrete) SPS if and only if X is the unique strong solution of the differential inclusion d dtXt + ∂IK(Xt) ∋ V0 + t F[Xs] ds, lim

t↓0 Xt = X0.

Equivalently, setting Yt := V0 + t

0 F[Xs] ds we get the well posed system

     d dtXt + ∂IK(Xt) ∋ Yt d dtYt = F[Xt]

23

Sticky particles Cumulative distribution Lagrangian representation

A description of the evolution by a differential inclusion

Recall that to the (discrete) data µt = (̺t, ̺tvt) we associate the functions (Xt, Vt) ∈ K × L2(0, 1) by Xt := X̺t, Vt := vt ◦ Xt. Theorem (Lagrangian representation) A family µt = (̺t, ̺tvt) is a solution of the (discrete) SPS if and only if X is the unique strong solution of the differential inclusion d dtXt + ∂IK(Xt) ∋ V0 + t F[Xs] ds, lim

t↓0 Xt = X0.

Equivalently, setting Yt := V0 + t

0 F[Xs] ds we get the well posed system

     d dtXt + ∂IK(Xt) ∋ Yt d dtYt = F[Xt]

23

Sticky particles Cumulative distribution Lagrangian representation

Stability properties

Suppose that F is Lipschitz in L2(0, 1). By general results on solution of differential inclusion of the type Z′

t + ∂φ(Zt) ∋ Gt, φ convex

Theorem If X1

t , X2 t are the Lagrangian representation of two (discrete) solutions

̺1

t, ̺2 t of the SPS we have

sup

t∈[0,T ]

X1

t − X2 t L2 ≤ CT

• X1

0 − X2 0L2 + V 1 0 − V 2 0 L2

• .

X is right-differentiable in each point and the velocity field vt can be recovered by the formula Vt = d+ dt Xt = vt ◦ Xt ∈ HXt. One gets [S. ’96] the following integral estimate for the velocity component: T V 1

r − V 2 r 2 L2 dr ≤ CT ℓ

Xℓ

0 + V ℓ

• X1

0 − X2 0 + V 1 0 − V 2

• .

24

Sticky particles Cumulative distribution Lagrangian representation

Stability properties

Suppose that F is Lipschitz in L2(0, 1). By general results on solution of differential inclusion of the type Z′

t + ∂φ(Zt) ∋ Gt, φ convex

Theorem If X1

t , X2 t are the Lagrangian representation of two (discrete) solutions

̺1

t, ̺2 t of the SPS we have

sup

t∈[0,T ]

X1

t − X2 t L2 ≤ CT

• X1

0 − X2 0L2 + V 1 0 − V 2 0 L2

• .

X is right-differentiable in each point and the velocity field vt can be recovered by the formula Vt = d+ dt Xt = vt ◦ Xt ∈ HXt. One gets [S. ’96] the following integral estimate for the velocity component: T V 1

r − V 2 r 2 L2 dr ≤ CT ℓ

Xℓ

0 + V ℓ

• X1

0 − X2 0 + V 1 0 − V 2

• .

24

Sticky particles Cumulative distribution Lagrangian representation

Stability properties

Suppose that F is Lipschitz in L2(0, 1). By general results on solution of differential inclusion of the type Z′

t + ∂φ(Zt) ∋ Gt, φ convex

Theorem If X1

t , X2 t are the Lagrangian representation of two (discrete) solutions

̺1

t, ̺2 t of the SPS we have

sup

t∈[0,T ]

X1

t − X2 t L2 ≤ CT

• X1

0 − X2 0L2 + V 1 0 − V 2 0 L2

• .

X is right-differentiable in each point and the velocity field vt can be recovered by the formula Vt = d+ dt Xt = vt ◦ Xt ∈ HXt. One gets [S. ’96] the following integral estimate for the velocity component: T V 1

r − V 2 r 2 L2 dr ≤ CT ℓ

Xℓ

0 + V ℓ

• X1

0 − X2 0 + V 1 0 − V 2

• .

24

Sticky particles Cumulative distribution Lagrangian representation

Convergence of discrete solutions

Theorem Let µn := (̺n, ̺nvn) be a sequence of discrete sticky-particle solutions with ̺n

0 ⇀ ̺0 in P2(R)

and ̺n

0 vn 0 ⇀ ̺0v0 weakly in the space of signed measures with

• |vn

0 |2 dρn 0 →

• |v0|2 d̺0

Then ̺n

t ⇀ ̺t in P2(R) for every t ≥ 0

and ̺n

t vn t ⇀ ̺tvt

weakly in the space of signed measures with

• |vn

t |2 dρn t →

• |vt|2 d̺t

and (ρt, ρtvt) is a solution of

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], ̺|t=0 = ̺0, v|t=0 = v0,

25

Sticky particles Cumulative distribution Lagrangian representation

Convergence of discrete solutions

Theorem Let µn := (̺n, ̺nvn) be a sequence of discrete sticky-particle solutions with ̺n

0 ⇀ ̺0 in P2(R)

and ̺n

0 vn 0 ⇀ ̺0v0 weakly in the space of signed measures with

• |vn

0 |2 dρn 0 →

• |v0|2 d̺0

Then ̺n

t ⇀ ̺t in P2(R) for every t ≥ 0

and ̺n

t vn t ⇀ ̺tvt

weakly in the space of signed measures with

• |vn

t |2 dρn t →

• |vt|2 d̺t

and (ρt, ρtvt) is a solution of

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], ̺|t=0 = ̺0, v|t=0 = v0,

25

Sticky particles Cumulative distribution Lagrangian representation

Convergence of discrete solutions

Theorem Let µn := (̺n, ̺nvn) be a sequence of discrete sticky-particle solutions with ̺n

0 ⇀ ̺0 in P2(R)

and ̺n

0 vn 0 ⇀ ̺0v0 weakly in the space of signed measures with

• |vn

0 |2 dρn 0 →

• |v0|2 d̺0

Then ̺n

t ⇀ ̺t in P2(R) for every t ≥ 0

and ̺n

t vn t ⇀ ̺tvt

weakly in the space of signed measures with

• |vn

t |2 dρn t →

• |vt|2 d̺t

and (ρt, ρtvt) is a solution of

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], ̺|t=0 = ̺0, v|t=0 = v0,

25

Sticky particles Cumulative distribution Lagrangian representation

Convergence of discrete solutions

Theorem Let µn := (̺n, ̺nvn) be a sequence of discrete sticky-particle solutions with ̺n

0 ⇀ ̺0 in P2(R)

and ̺n

0 vn 0 ⇀ ̺0v0 weakly in the space of signed measures with

• |vn

0 |2 dρn 0 →

• |v0|2 d̺0

Then ̺n

t ⇀ ̺t in P2(R) for every t ≥ 0

and ̺n

t vn t ⇀ ̺tvt

weakly in the space of signed measures with

• |vn

t |2 dρn t →

• |vt|2 d̺t

and (ρt, ρtvt) is a solution of

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], ̺|t=0 = ̺0, v|t=0 = v0,

25

Sticky particles Cumulative distribution Lagrangian representation

Convergence of discrete solutions

Theorem Let µn := (̺n, ̺nvn) be a sequence of discrete sticky-particle solutions with ̺n

0 ⇀ ̺0 in P2(R)

and ̺n

0 vn 0 ⇀ ̺0v0 weakly in the space of signed measures with

• |vn

0 |2 dρn 0 →

• |v0|2 d̺0

Then ̺n

t ⇀ ̺t in P2(R) for every t ≥ 0

and ̺n

t vn t ⇀ ̺tvt

weakly in the space of signed measures with

• |vn

t |2 dρn t →

• |vt|2 d̺t

and (ρt, ρtvt) is a solution of

• ∂t̺ + ∂x(̺ v) = 0,

∂t(̺ v) + ∂x(̺ v2) = f[̺], ̺|t=0 = ̺0, v|t=0 = v0,

25

Sticky particles Cumulative distribution Lagrangian representation

An explicit representation formula for the Euler-Poisson system

[Tadmor-Wei] f[̺] = −̺ ∂xq̺ with q̺ solution of − ∂2

xxq̺ = ̺

F[X] = 1 − 2w Xt = P

K

• X0 + tV0 + (1 − 2w)t2

.

26

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

27

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

27

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

27

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

27

Sticky particles Cumulative distribution Lagrangian representation

The L2-projection on K

Theorem If X ∈ L2(0, 1) and X (w) = w

0 X(s) ds is its primitive then

P

K(X) =

d dw X ∗∗ where X ∗∗ is the convex envelope of X .

28

Sticky particles Cumulative distribution Lagrangian representation

The L2-projection on K

Theorem If X ∈ L2(0, 1) and X (w) = w

0 X(s) ds is its primitive then

P

K(X) =

d dw X ∗∗ where X ∗∗ is the convex envelope of X . X 1

28

Sticky particles Cumulative distribution Lagrangian representation

The L2-projection on K

Theorem If X ∈ L2(0, 1) and X (w) = w

0 X(s) ds is its primitive then

P

K(X) =

d dw X ∗∗ where X ∗∗ is the convex envelope of X . X 1 X (w) = w

0 X(s) ds

1

28

Sticky particles Cumulative distribution Lagrangian representation

The L2-projection on K

Theorem If X ∈ L2(0, 1) and X (w) = w

0 X(s) ds is its primitive then

P

K(X) =

d dw X ∗∗ where X ∗∗ is the convex envelope of X . X 1 X (w) = w

0 X(s) ds

1 X ∗∗ 1

28

Sticky particles Cumulative distribution Lagrangian representation

The L2-projection on K

Theorem If X ∈ L2(0, 1) and X (w) = w

0 X(s) ds is its primitive then

P

K(X) =

d dw X ∗∗ where X ∗∗ is the convex envelope of X . X 1 X (w) = w

0 X(s) ds

1 PK(X) 1 X ∗∗ 1

28

Sticky particles Cumulative distribution Lagrangian representation

The subdifferential of IK

If X ∈ K we consider the open set ΩX where X is (essentially) constant: ΩX :=

• w ∈ (0, 1) : X is essentially constant in a neighborhood of w
• ,

and the cone NX :=

• Y ∈ C0([0, 1]) : Y ≥ 0,

Y = 0 in [0, 1] \ ΩX

• .

Theorem Let X ∈ K and Y ∈ L2(0, 1) with Y (w) := w

0 Y (s) ds. Then

Y ∈ ∂IK(X) ⇔ Y ∈ NX. Notice that if Z = f(X) ∈ HX depends on X then ΩX ⊂ ΩZ, NX ⊂ NZ Corollary (Monotonicity property of ∂IK) If Z = f(X) ∈ HX depends on X then ∂IK(X) ⊂ ∂IK(Z).

29

Sticky particles Cumulative distribution Lagrangian representation

The subdifferential of IK

If X ∈ K we consider the open set ΩX where X is (essentially) constant: ΩX :=

• w ∈ (0, 1) : X is essentially constant in a neighborhood of w
• ,

and the cone NX :=

• Y ∈ C0([0, 1]) : Y ≥ 0,

Y = 0 in [0, 1] \ ΩX

• .

Theorem Let X ∈ K and Y ∈ L2(0, 1) with Y (w) := w

0 Y (s) ds. Then

Y ∈ ∂IK(X) ⇔ Y ∈ NX. Notice that if Z = f(X) ∈ HX depends on X then ΩX ⊂ ΩZ, NX ⊂ NZ Corollary (Monotonicity property of ∂IK) If Z = f(X) ∈ HX depends on X then ∂IK(X) ⊂ ∂IK(Z).

29

Sticky particles Cumulative distribution Lagrangian representation

The subdifferential of IK

If X ∈ K we consider the open set ΩX where X is (essentially) constant: ΩX :=

• w ∈ (0, 1) : X is essentially constant in a neighborhood of w
• ,

and the cone NX :=

• Y ∈ C0([0, 1]) : Y ≥ 0,

Y = 0 in [0, 1] \ ΩX

• .

Theorem Let X ∈ K and Y ∈ L2(0, 1) with Y (w) := w

0 Y (s) ds. Then

Y ∈ ∂IK(X) ⇔ Y ∈ NX. Notice that if Z = f(X) ∈ HX depends on X then ΩX ⊂ ΩZ, NX ⊂ NZ Corollary (Monotonicity property of ∂IK) If Z = f(X) ∈ HX depends on X then ∂IK(X) ⊂ ∂IK(Z).

29

Sticky particles Cumulative distribution Lagrangian representation

The subdifferential of IK

If X ∈ K we consider the open set ΩX where X is (essentially) constant: ΩX :=

• w ∈ (0, 1) : X is essentially constant in a neighborhood of w
• ,

and the cone NX :=

• Y ∈ C0([0, 1]) : Y ≥ 0,

Y = 0 in [0, 1] \ ΩX

• .

Theorem Let X ∈ K and Y ∈ L2(0, 1) with Y (w) := w

0 Y (s) ds. Then

Y ∈ ∂IK(X) ⇔ Y ∈ NX. Notice that if Z = f(X) ∈ HX depends on X then ΩX ⊂ ΩZ, NX ⊂ NZ Corollary (Monotonicity property of ∂IK) If Z = f(X) ∈ HX depends on X then ∂IK(X) ⊂ ∂IK(Z).

29

Sticky particles Cumulative distribution Lagrangian representation

The subdifferential of IK

If X ∈ K we consider the open set ΩX where X is (essentially) constant: ΩX :=

• w ∈ (0, 1) : X is essentially constant in a neighborhood of w
• ,

and the cone NX :=

• Y ∈ C0([0, 1]) : Y ≥ 0,

Y = 0 in [0, 1] \ ΩX

• .

Theorem Let X ∈ K and Y ∈ L2(0, 1) with Y (w) := w

0 Y (s) ds. Then

Y ∈ ∂IK(X) ⇔ Y ∈ NX. Notice that if Z = f(X) ∈ HX depends on X then ΩX ⊂ ΩZ, NX ⊂ NZ Corollary (Monotonicity property of ∂IK) If Z = f(X) ∈ HX depends on X then ∂IK(X) ⊂ ∂IK(Z).

29

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

30

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

30

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

30

Sticky particles Cumulative distribution Lagrangian representation

Extensions and open problems

Extensions:

◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime:

W(x) = −|x|. Open problems:

◮ Adding a pressure term.

30