a partial-integro differential equation (pide) for sandpile

A Partial-Integro Differential Equation (PIDE)for Sandpile

Noureddine Igbida

LAMFA CNRS-UMR 6140,Universite de Picardie Jules Verne, 80000 Amiens

Poitiers, 9 February 2010ICPDE - 60th Birthday of M. Chipot

Noureddine Igbida A Partial-Integro Differential Equation (PIDE) for Sandpile

Sandpile with small components

Growing sandpile (numerical simulation with S. Dumont)


Small components : sand surface evolution PDE ([Prigozhin, 1996],[Aronson-Evans-Wu,1999])

x

h(x,t)

f(x,t)

q(x,t)

The flow of the material is confined in a thin boundary layer moving down theslopes of growing piles.

Surface flow is directed by the steepest descent

Angle of stability α : the steepest angle that the surface in bulk made with theground

No pouring over the parts of the pile surface inclined less than α

⇓ ⇓ (ht = ∆ph + f , p → ∞) ⇓ ⇓

• Conservation of mass ht = −∇ · q + f• Phenomenological equations ∃ m = m(t, x) > 0 : q = −m∇h• Gradient constraint |∇h| ≤ γ := tan(α)• No pouring for surface inclined less than α m (|∇h| − γ) = 0


Small components : sand surface evolution PDE

(PPr)

∂h

∂t−∇ ·

(

m∇h)

= f in Q

m > 0, |∇h| ≤ 1, m (1− |∇h|) = 0 in Q

u = 0 on Σ

u(0) = u0 in Ω

Interests, results, refs .... :

Existence and uniqueness of a solution : Aronson, Evans, Wu, Barret, Prigozhin, Igbida

...

Connection with Stochastic model : Evans and Rezakhanlou.

Connection with Monge-Kantorovich problem : Evans and Gangbo

Uniqueness of the flux : Ambrosio ...

Numerical analysis : Barret, Prigozhin, Dumont, Igbida ....

Large time behavior, as t → ∞ : Cardaliaguet, Cannarsa, Igbida.

Stationnary equation : Ambrosio, Evans, Gangbo, Bouchitte, Buttazzo, De Pascale,

Pratelli ....

Regularity of the flux (stationnary equation) : Evans, De Pascale, Pratelli ....


Non small components

The dynamic is generated essentially by the fact that the components (grains, blocks,...) of the structure moves both by jumping if they are not supported by others

⇓ ⇓ ⇓ ⇓

” .... a pure jump process ....”


Non small components

Growing pile of cubes (numerical simulation with F. Karami)


Non small components : Evans-Rezakhanlou Stochastic model [EvReza-98]

X(t5)

0 1 0 0

0

0

0

0

0

f(t1) f(t2) f(t3) f(t4)

f(t5)

f(t5)0

X(0) X(t1) X(t2) X(t3)

X(t4)

X(t4)

X(t5)

X(t5)

Adding a new cube on an existing pile, we have the following possibilities

the new cube remains in place

the new cube moves by falling in one of the forth directions (up, down, left or right).

there are several downhill ”staircases” along which the cube can move, and the cube willrandomly select among the allowable downhill paths.

cubes distribution : mu

i ∼ j ⇐⇒ |i − j| ≤ 1 3D 2D

New cubes will move (or not) in order to get a stable configuartion, which means that the heightsof any two adjacent columns of cubes can differ by at most one :

|η(i) − η(j)| ≤ 1 if i ∼ j,

where η(i) denotes the height of the columns situated at the position i ∈ Z2.

• A stochastic model for growing sandpiles and its continuum limit, L. C. Evans, F. Rezakhanlou, Comm. Math. Phys., 197 (1998),no2, 325-345.


For a given(

f (t))

t>0sequence of l2(Z2),

(

η(t))

t>0is a Markov process describing

the random formatin of piles of cubes.

When the cubes tends to be very small :

Assume that f ∈ BV (0,T ; L2(R2)

f (t, i) =1

Nf

(

t

N,i

N

)

(η(t), t > 0) the associated Markov processus

Theoreme

As N → ∞, we have

IE

[

supx∈R2

∣

∣

∣u(t, x)−

1

Nη(Nt, [N x ])

∣

∣

∣

]

→ 0,

where u is the unique solution of

(PPr)

∂h

∂t(t, x)−∇ · (m(t, x)∇h(t, x)) = f in Q

m > 0, |∇h| ≤ 1, m (1− |∇h|) = 0 in Qu(0) = u0 in Ω

• A stochastic model for growing sandpile and its continuum limit, L. C. Evans, F. Rezakhanlou, Comm. Math. Phys., 197 (1998),no2, 325-345.

• Back on Stochastic Model for Sandpile, N. Igbida, Series in Contemporary Applied Mathematics, 2008.


Jump process : random dynamical description(compound Poisson process)

P a set equipped with a probability IP (for instance a set of particles)(

X (t))

t>0be a R

N valued random variable defined in P describing the positions

of the particles

(

X (t))

t>0only changes by means of jumps, the jumps occur at random times

(separated by exponential intervals) and the heights of the successive jumps is asequence of independent and identically distributed (i.i.d. for short) randomvariables :

⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓

X (t) = X (0) + Y1 + Y2 + ...+ YN(t)

(N(t))t>0 is a poisson process (the quantity N(t) compute the number of jumps in

[0,T ]) with a rate λ (the intensity of jumps which is equal to the average number of jumps

by unit of time)(

Yi

)

i=1,2,....are i.i.d. random variables with probability µ, independent of

(N(t))t>0 , modeling the jumps with a probability measure µ


Jump process : deterministic spatial description (PIDE)

Let u be the density of(

X (t))

t>0; i.e.

IP[

X (t) ∈ A]

=

∫

Au(t, x) dx .

The function u satisfies the following PIDE

∂

∂tu(t, x) + λ

∫

RN(u(x + z)− u(x)) µ(dz) = 0 (1)

⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓

Instead of the description of the dynamical phenomena by random

variables(

X (t))

t>0, the PIDE (1) gives the

description of the spatial distribution by a deterministic equation.

Reference : D. Applebaum, Levy Processes and Stochastic Calculus, 2004.


Example : Nonlocal diffusion equation

(Exple)∂

∂tu(t, x) + λ

∫

RNJ(x − y) (u(x) − u(y)) dy = 0

where, J : R2 → R, J > 0, radial, continuous and

∫

R2J(z) dz = 1.

u(t, x) is a density at the point x at time t

J(x − y) is related to the probability distribution of jumping from location y tolocation x∫

R2J(y − x) u(t, y) dy = (J ∗ u)(t, x) is the rate at which individuals are arriving

at position x from all other places.

−u(t, x) = −

∫

R2J(y − x) u(t, x) dy is the rate at which they are leaving

location x to travel to all other sites.

f (t, x) is the external (or internal) source.

Equation (Exple) is called nonlocal diffusion equation since the diffusion of u at apoint x and time t depends on all the values of u in a neighborhood of x through theconvolution term J ∗ u.


Sandpile with non small components : formal derivation of the PIDE

(

X (t))

t>0random variable describing the position in the plane R

2 of the blocks

moving on the surface.

Assume(

X (t))

t>0has a density u : R

+ ×R2 → R

+

⇓ ⇓ Jump Process ⇓ ⇓

There exists a measure of occurring jumps µ such that u satisfies

∂

∂tu(t, x) + λ

∫

R2(u(x + z)− u(x)) µ(dz) = f (t, x),

where f is an external or an internal source and λ is the intensity of jumps.

But :

µ is unknown

The problem needs to be closed : describe µ ! !


The model is

(Pδε )

∂u(t, x)

∂t+

∫

R2

(

u(t, x)− u(t, y))

µ(t, x , dy) = f (t, x) for (t, x) ∈ (0,T )×R2

u(t) ∈ Kδǫ , µ(t) > 0, Support(µ(t)) ⊆

[

|u(t, x)− u(t, y)| = δ and |x − y | ≤ ǫ]

u(0) = u0

In (Pδε ) :

λ

∫

R2u(t, y) µ(t, x , dy) records blocks arriving to the position u(t, x) from all

other places.

λ

∫

R2µ(t, x , y)u(t, x) µ(t, x , dy) records blocks leaving location u(t, x) to travel

to all other sites.


Solving the PIDE : Euler implicit time discretization

σ > 0, t0 = 0 < t1 < ... < tn−1 < T = tn, ti − ti−1 ≤ ε, f1, ...fn ∈ L2(R2), suchthat

n∑

i=1

∫ ti

ti−1

‖f (t)− fi‖L2(R2) ≤ σ.

uσ is a σ−approximate solution, if

uσ(t) =

u0 for t ∈ ]0, t1],

ui for t ∈ ]ti−1, ti ], i = 1, ...n(2)

and ui solves the Euler implicit time discretization :

ui (x) + σ

∫

R2(ui (x) − ui (y)) µi (x , dy) = σ fi + ui−1(x) in R

2,

ui ∈ Kδǫ , µi > 0, Support(µi ) ⊆ Rδ

ε(ui ).


Solving the PIDE : stationary problem

(PSδε )

u(x) +

∫

R2(u(x) − u(y)) µ(x, dy) = f (x) in R

2

u ∈ Kδǫ , µ > 0, Support(µ) ⊆ Rδ

ε(u).

Let

Msb(R

2 × R2)+ =

µ ∈ Mb(R2 × R

2)+ ;

∫ ∫

ξ(x, y) µ(dx, dy) =

∫ ∫

ξ(y, x) µ(dx, dy)

for any ξ ∈ Cc (R2 × R

2)

.

Theoreme (Ig,2009)

For any f ∈ Cc (R2) ∩ L2(R2), (PSδ

ε ) has a unique solution u in the following sense :

u ∈ Kδǫ ∩ Cc (R

2), µ ∈ Msb(R

2 × R2)+, Support(µ) ⊆ Rδ

ε(u)

∫

R2u(x)ξ(x) dx +

∫

R2

∫

R2(u(x) − u(y)) ξ(x) dµ(x, y) =

∫

R2f (x) ξ(x) dx

for any ξ ∈ Cc (R2). Moreover, u is a solution of (PSδ

ε ) if and only if

u = IPKδǫ(f ), (3)

where IPKδǫ

is the projection with respect to the L2(R2) norm on the convex Kδǫ .


Solving the PIDE : the main operator

u(x) +

∫

R2(u(x) − u(y)) µ(x, dy) = f (x) in R

2

u ∈ Kδǫ , µ > 0, Support(µ) ⊆ Rδ

ε(u)

⇐⇒ u = IPKδε(f )

Corollaire

Let f ∈ Cc (R2) and u ∈ Cc (R

2) ∩ K

δε . Then,

f ∈ ∂IIKδε(u) ⇔

∃ µ ∈ Msb(R

2 × R2)+, Support(µ) ⊆ Rδ

ε(u)

∫

R2

∫

R2(u(x) − u(y)) ξ(x) µ(dx, dy) =

∫

R2f (x) ξ(x) dx

∀ξ ∈ Cc (R2)

Remark

The operator ∂IIKδε

was introduced first by [Andreu-Mazon-Rossi-Toledo,2009] as the

nonlocal ∞−Laplacian and suggest ∂IIKδε

as a governing operator for a nonlocal model of

Sandpile.

Here, we re-write ∂IIKδε

in terms of ”Partial Integro-Differential Equation” (PIDE).


Ideas of the proof

A solution of our PIDE is the projection :

I :=

∫

R2

∫

R2(u(x) − u(y)) (u(x) − z(x)) µ(dx, dy)

then

2 I =

∫

R2

∫

R2(u(x) − u(y))2 µ(dx, dy) +

∫

R2

∫

R2(u(x) − u(y)) (z(y) − z(x)) µ(dx, dy)

>

∫

R2

∫

R2(u(x) − u(y))2

(

1 −|z(y) − z(x)|

δ

)

µ(dx, dy) > 0.

The projection is the solution of our PIDE :

Consider the PIDE p−Laplacian equation ([AMRT,2009]) :

up(x) +

∫

R2J(x − y)

∣

∣

∣

∣

up(x) − up(y)

δ

∣

∣

∣

∣

p−2

(up(x) − up(y)) dy = f (x) for x ∈ R2,

where J ∈ C(R2)+ is radial, compactly supported in B(0, ε), J(0) > 0 and∫

R2J(x)dx = 1.

We prove that

up → u in C(R2)

µp := J(x − y)

∣

∣

∣

∣

up(x) − up(y)

δ

∣

∣

∣

∣

p−2

→ µ in Mb(R2 × R

2) − weak∗

and (u, µ) satisfies the PIDE.


Remarks

In general the solution u of (PSδε is not continuous. So, how to define the

solution ?

Let f ∈ L2(R2), then the f.a.a.e ([Igbida-Mazon-Rossi-Toledo,2009]) :i- u = IP

Kδε(f )

ii- u ∈ Kδε and there exists F ∈ Mas

b (R2 × R2), such that F

[

|x − y| ≤ ε]

,

∫

R2u(x)ξ(x) dx +

∫

R2

∫

R2ξ(x) dF(x, y) =

∫

R2f (x) ξ(x) dx, ∀ξ ∈ Cc (R

2)

and

|F|(R2× R

2) =

2

δ

∫

R2(f (x) − u(x)) u(x) dx.

iii- u ∈ Kδε and there exists F ∈ Mas

b (R2 × R2), such that F

[

|x − y| ≤ ε]

,

∫

R2u(x)ξ(x) dx +

∫ ∫

R2×R2ξ(x) dF(x, y) =

∫


2)

and|F|(R2 × R

2) ≤ |φ|(R2 × R2),

for any φ ∈ Masb (R

2× R

2)+, such that φ

[

|x − y| ≤ ε]

and

∫

R2u(x)ξ(x) dx +

∫ ∫

R2×R2ξ(x) dφ(x, y) =

∫


2).


Solving the PIDE : existence and uniqueness of a solution

If f ∈ BV (0,T ; L2(R2)) and u0 ∈ Kδε , then ([AMRT,09])

(Pδε ) has a unique variational solution ; i.e.

u ∈ W 1,∞([0,T ); L2(R2)), u(0) = u0, u(t) ∈ Kδε

∫

R2u(t) (f (t) − ∂tu(t)) = max

ξ∈Kδε

∫

(f (t) − ∂tu(t)) ξ.

⇐⇒

ut + ∂IIKδε(u) ∋ f ,

u(0) = u0.

If uσ is the σ−approximate solution, then

uσ → u in C([0,T ), L2(R2)), as σ → 0.

Theoreme (Ig,09)

If f ∈ BV (0,T ; L2(R2)) ∩ L∞(0,T ; Cc (R2)) and u0 ∈ Kδ

ε ∩ Cc (R2), then u (the variational

solution) is the unique solution of (Pδε ) in the sense that :

u ∈ W 1,∞(0,T ; L2(Ω))∩ L∞(0,T ; Cc (R2)), u(0) = u0, for any t ∈ (0,T ), u(t) ∈ Kδ

ε and there

exists µt ∈ Mb(R2 × R

2) such that, µt is symmetric,

Support(µt ) ⊆ Rδε(u(t))

and setting F(t, x, y) := (u(t, x) − u(t, y)) µt(x, y), we have

F ∈ L∞(0,T ;w∗ − Mb(R2 × R

2)) and∫

R2

∫

R2ξ(x) dF(t) =

∫

R2

(

f (t, x) −∂u(t, x)

∂t

)

ξ(x) dx a.e. t ∈ (0,T ),

for any ξ ∈ Cc (R2).


Connection : stochastic model ←→ PIDE

Subdivide the plane into unit squares of side length 1/N, with N ∈ N∗.

Assume f ∈ BV (0,T ; L2(R2) with bounded support and

f (t, x) = f

(

t,[Nx ]

N

)

, for any (t, x) ∈ (0,∞)×R2. (4)

Set

f (t, i) = f

(

t,i

N

)

for any (t, i) ∈ [0,∞)×Z2.

Theoreme (Ig,2009)

Let u is a solution of (Pδε ), with ε = δ =

1

N, and let

(

η(., t), t > 0)

be the Markov

process generated by f , then

IE

[∫

R2|u(t, x)− δ η (t, [N x ]) |2

]

≤ δ

∫ t

0

∫

R2|f (s, x)| dxdt,

where u is the solution of (Pδε ).


Connections : stochastic model ←→ PIDE ←→ sand surface evolution PDE

Assume that f ∈ BV (0,T ; L2(R2)

f (t, i) =1

Nf

(

t

N,i

N

)

and (η(t), t > 0) the associated Markov processus.

δ = ε =1

N

uN the solution of (Pδε )

u the solution de (PPr)

Theoreme

As N → ∞, we have

∫

R2|u − uN |+ IE

[∫

R2|u(t, x)−

1

Nη (Nt, [N x ]) |2

]

+IE

[∫

R2|uN (t, x)−

1

Nη (Nt, [N x ]) |2

]

→ 0.


Prospects

Collapsing model : stochatsic collapsing sandpile

Moving sand dunes : Barchanes

Use the PIDE for optimal mass transportation (Ig-Mazon-Rossi-Toledo).

Given two measures µ+ and µ− in RN , such that µ+(RN ) = µ

−(RN) and

spt(µ+) 6= spt(µ

−)

Mint∈A

∫

RNdε(x, t(x)) dµ

+(x)

where

A =

t : spt(µ+) → spt(µ−) ; µ+#t = µ

−i.e. µ−(B) = µ+(t−1(B))

and

dε(x, y) =

0 if x = y

ε

([[

|x − y|

ε

]]

+ 1

)

if x 6= y


Some papers on the subject

http ://www.mathinfo.u-picardie.fr/igbida/

N. Igbida, Nonlocal Equation in Granular Matter, submitted.

N. Igbida, J. Mazon, J. Rossi et J. Toledo, A Nonlocal Monge-Kantorovich Problem,submitted.

S. Dumont and N. Igbida, Back on a Dual Formulation for the Growing Sandpile Problem,European Journal of Applied Mathematics, vol. 20, pp. 169-185.

N. Igbida and F. Karami, Numerical Analysis of Nonlocal Equations for Sandpile andApproximation of Evans-Rezakhanlou Stochastic model, in perparation.

S. Dumont and N. Igbida, Collapsing Sandpile Problem, under revision, Communication onApplied Analysis and Applications.

N. Igbida, Evolution Monge-Kantorovich Equation,submitted.

N. Igbida, Generalized Collapsing Sandpile Problem, to appear in Archiv Der Mathematik,2010.

N. Igbida, Back on Evans-Rezakhanlou Stochastic Model for sandpile, Series inContemporary Applied Mathematics, 2008.

N. Igbida, Equivalent Formulation for Monge-Kantorovich Equation, Nonlinear AnalysisTMA, 2009.


a partial-integro differential equation (pide) for sandpile

Documents