Sur l’´ equation de Monge-Kantorovich Noureddine Igbida LAMFA CNRS-UMR 6140, Universit´ e de Picardie Jules Verne, 80000 Amiens Le Havre, 16 Avril 2009 Noureddine Igbida (LAMFA CNRS-UMR 6140, Universit´ e de Picardie Jules Verne, 80000 Amiens) Sur l’´ equation de Monge-Kantorovich Le Havre, 16 Avril 2009 1 / 34
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Sur l’equation de Monge-Kantorovich
Noureddine Igbida
LAMFA CNRS-UMR 6140,Universite de Picardie Jules Verne, 80000 Amiens
Le Havre, 16 Avril 2009
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 1 / 34
Main of the talk
Monge-Kantorovich evolution equation
Let Ω ⊂ RN be a bounded regular domain, N > 1, T > 0, Q = Ω × (0, T ) and Σ = ∂Ω × (0, T )
(EMK)
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:
∂h
∂t(x , t) −∇ · m(x , t)∇h(x , t) = µ in Q
m(x , t) > 0, |∇h(x , t)| ≤ 1 in Q
m (1 − |∇h(x , t)|) = 0 in Q
u = 0 on Σ
u(0) = u0 in Ω
Main interest :
1 Existence and uniqueness of a solution : µ a Radon measure
2 Numerical analysis : representation of t ∈ [0,T ) → (u(t, x), m(t, x) ∇u(t, x)).
3 Large time behavior, as t → ∞.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 2 / 34
Motivation
Motivation
Optimal mass transportation (Monge-Kantorovich problem) :- L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass
transfer problem. Mem. Am. Math. Soc., 137 (1999), no. 653.
Sandpile model :- L. Prigozhin, Variational model of sandpile growth. Euro. J. Appl. Math. , 7 (1996), 225-236.- G. Aronson, L. C. Evans and Y. Wu, Fast/Slow diffusion and growing sandpiles. J. Differential
Equations, 131 :304-335, 1996.
Cellular Automaton :- L. C. Evans and F. Rezakhanlou. A stochastic model for sandpiles and its continum limit. Comm.
Math. Phys., 197 (1998), no. 2, 325-345.
Mass optimization :- G. Bouchitte and G. Buttazzo, Characterization of optimal shapes and masses through Monge-
Is µ supported in a graph ? i.e. ∃ ? t ∈ A, µ(E) = µ+(x ∈ IR
N; (x, t(x)) ∈ E)
Dual variational principle : Shipper (principe du convoyeur) µ is optimal if and only if
J (µ) = maxξ∈Lip1(RN )
n
Z
IRNξ dµ+ +
Z
IRNξ dµ−
o
=
Z
IRNu dµ+ +
Z
IRNu dµ− .
• Memoire sur la theorie de Deblais et des remblais, Gaspard Monge, Histoire de l’academie des Sciences de Paris, 1781.• On the transfer of massesm, L.V. Kantorovich, Dokl. Acad. Nauk. SSSR 37, 227-229 (1942).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 4 / 34
Motivation
Motivation I : optimal mass transportation
Monge optimal mass transfer [Monge-1780] : Given two measures µ+ and µ− in RN , such that
µ+(RN ) = µ−(RN ) and spt(µ+) 6= spt(µ−)
Mint∈A
Z
RN|x − t(x)| dµ+(x)
A =
t : spt(µ+) → spt(µ−) ; µ+#t = µ−
i.e. µ−(B) = µ+(t−1(B))
ff
˛
˛
˛
˛
˛
˛
˛
˛
˛
˛
x
t(x)mu^+
mu^−
t
Evans and Gangbo [1999] : How to fashion an optimal transport t ?
Monge-Kantorovich equation (Stationary equation of (EMK))
(MK)
−∇ · (m ∇u) = µ+ − µ−
m > 0, |∇u| ≤ 1, m(|∇u| − 1) = 0
ff
=⇒ m ∇u : the flux transportation
m gives the transport density−∇u gives the direction of the optimal transportation
Remark : If (u, m) solves (MK) then u is the Kantorovich potential ; i.e.
maxξ∈Lip1(RN )
Z
IRNξ dµ =
Z
IRNu dµ
• Memoire sur la theorie de Deblais et des remblais, Gaspard Monge, Histoire de l’academie des Sciences de Paris, 1781.Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 5 / 34
Motivation
Motivation II : growing sandpile model
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 6 / 34
Motivation
Motivation II : growing sandpile model
x
h(x,t)
f(x,t)
q(x,t)
Conservation of mass :The flow of the granular material is confined in a thin boundary layer moving down the slopes of agrowing pileThe density of the material is constant
∂h
∂t(x , t) = −∇ · q(x , t) + µ(x , t)
Surface flow directed by the steepest descent :
∃ m = m(x , t) > 0 such that q(x , t) = −m(x , t) ∇h(x , t)
No pouring over the parts of the pile surface inclined less than α :
|∇h(x , t)| < γ =⇒ m(x , t) = 0
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 7 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
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)
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
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 will randomly selectamong the allowable downhill paths.
New cubes will move (or not) in order to get a stable configuartion, which means that the heights of any twoadjacent 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.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 8 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Let p(i , j , η) : probability that a cube placed at the position i will end up at jSo, for any i , j ∈ Z2 we have
c(j , η, τ) : (highly nonlocal factor) records the rate, at time τ, new cubes come to rest at j
c(j , η, t) =X
i∈Z2
p(i , j , η(t)) f (i , t), for any (j , t) ∈ Z2 × [0,∞).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 9 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Stochastic Equation (integral form) :0
B
B
@
∀ F : S × (0,∞) → R Lipchitz continuous in t and F (η(., 0), 0) = 0
IEh
F (η(., t), t) −
Z t
0
„
∂F
∂s+ Ls F
«
(η(., s))i
= 0,
where(Lt F )(ξ) :=
X
j∈ZN
c(j , ξ, t)“
F (Tj (ξ)) − F (ξ)”
.
with
Tj : ξ ∈ S → Tj (ξ) ∈ S with Tj (ξ) =
ξ(i) + 1 if i ∼ jξ(i) otherwise
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 10 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Assume that f ∈ BV (0, T ;L2(R2)
f (t, i) =1
Nf
„
t
N,
i
N
«
(η(t), t > 0) the associated Markov processusu the solution de (EMK)
Theorem
As N → ∞, we have
IE
"
supx∈R2
˛
˛
˛u(t, x) −1
Nη(t, [N x ])
˛
˛
˛
#
→ 0.
• A stochastic model for growing sandpiles 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 (soumis).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 11 / 34
Motivation
Boundary condition : Dirichlet
If µ+(IRN ) = µ−(IRN ) and µ+, µ− are supported in bounded domain : by taking Ω largeenough we can assume that u = 0 on ∂Ω;
If µ+(IRN ) 6= µ−(IRN ) and µ+, µ− are supported in bounded domain, then one needs :
There exists Γ such that HN−1(Γ) = 0 and replace the euclidean c(x, y) = |x − y| by thesemi-geodesic distance taking into account the boundary ; i.e.
cΓ(x, y) = min“
|x − y|, dist(x, Γ) + dist(y, Γ)”
.
Dirichlet Monge-Kantorovich problem (Bouchite, Buttazzo, De Pascale ...)
infn
Z Z
cΓ(x, y) dν(x, y) ; ν ∈ M(Ω × Ω) s.t. π1#ν − π
2#ν = µ1 − µ2
o
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 12 / 34
Main interest
Monge-Kantorovich evolution equation with Dirichlet boundarycondition
(EMK)
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:
∂h
∂t−∇ · (m ∇h) = µ in Q
m > 0, |∇h| ≤ 1 in Q
m (1 − |∇h|) = 0 in Q
u = 0 on Σ
u(0) = u0 in Ω
Main interest :
1 Existence and uniqueness of a solution : µ a Radon measure
2 Numerical analysis : representation of t ∈ [0,T ) → (u(t, x), m(t, x) ∇u(t, x))..
3 Large time behavior, as t → ∞.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 13 / 34
References
References de l’expose
http ://www.mathinfo.u-picardie.fr/igbida/
S. Dumont and N. Igbida,Back on a Dual Formulation for the growing Sandpile Problem, European Journal of AppliedMathematics, vol. 20, pp. 169-185, 2008.
N. Igbida,Evolution Monge-Kantorovich Equation,en revision dans Calculus of Variations and PartialDifferential Equations.
N. Igbida,On Monge-Kantorovich Equation, sous presse dans Nonlinear Analysis TMA.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 14 / 34
Difficulties and definitions of a solution
Main Difficulties
If m is regular then
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:
∂tu −∇ · (m ∇u) = µ
|∇u| ≤ 1, m > 0, m (|∇u| − 1) = 0
u = 0 on ∂Ω
⇒
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:
ut + ∂IIK (u) ∋ µ,
K =n
z ∈ H10 (Ω) ; |z | ≤ 1 in Ω
o
i.e. ∀ t ∈ [0,T ), u(t) ∈ K and
Z
Ωu(t)
“
µ(t) − ∂tu(t)”
= maxξ∈K
Z
“
µ(t) − ∂tu(t)”
ξ.
Nonlinear semi-group theory ⇒ existence and uniqueness of ”variational” solution u