Long time average of mean field games

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Long time average of the mean field games system

Alessio Porretta

Universita’ di Roma Tor Vergata

BCAM, 04/11/2011

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Outlines of the talk

Briefly on the Mean Field Games model and the equilibrium system The long time behavior: ergodic problem and expected behavior. Structural estimates of the Mean Field Games system:

(i) the case of nonlocal coupling (ii) the case of local coupling.

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On the Mean Field Games theory

The Mean Field Games model was introduced by J.-M. Lasry and P.-L. Lions [CRAS ’06, Japn.J.Math. ’07, Cours Coll`ege de France since 2006]. Main goal: describe games with large numbers (a continuum) of players who take their decisions in terms of the distribution density of the other players.

Typical features of the model:

- players act according to the same principles (they are indistinguishable and have the same optimization criteria).

- their strategy takes into account the mass of co-players.

Roughly: players are particles but they take decisions and take into account co-players(more sophisticated than particle physics or similar economics models); however, they only consider the statistical state of the mass of co-players (less sophisticated than agents of N-players games)

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On the Mean Field Games theory

The Mean Field Games model was introduced by J.-M. Lasry and P.-L. Lions [CRAS ’06, Japn.J.Math. ’07, Cours Coll`ege de France since 2006]. Main goal: describe games with large numbers (a continuum) of players who take their decisions in terms of the distribution density of the other players.

Typical features of the model:

- players act according to the same principles (they are indistinguishable and have the same optimization criteria).

- their strategy takes into account the mass of co-players.

Roughly: players are particles but they take decisions and take into account co-players(more sophisticated than particle physics or similar economics models); however, they only consider the statistical state of the mass of co-players (less sophisticated than agents of N-players games)

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Rough description of the model: you have

•A group of agents(indistinguishable).

Each agent (infinitesimal) controls the same stochastic dynamics dXt= αtdt +

√ 2dBt,

where Bt is a standard Brownian motion, in order to minimize, among

controls αt, some cost:

inf α J(α) :=E �� T 0 [L(Xs, αs) + F (Xs, m(s))]ds + G (XT, m(T )) �

where m(t) is the probability density of players at time t.

The associated Hamilton-Jacobi-Bellman equation for the value function

is _

 

−ut− ∆u + H(x, ∇u) = F (x, m(t))

u(x, T ) = G (x, m(T ))

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Then, the optimal control is given by the feedback law α∗_t =−Hp(Xt,∇u(t, Xt)) , Hp:= ∂H ∂p and inf α J(α) = � u(x, 0)dm0(x)

wherem0is the probability distribution of X0.

• In turn,the dynamics of the density function m is governed by the

optimal decisions of the agents.

Recall: given a process dXt= b(Xt)dt +

√

2dBt, the probability measure

m(t) distribution law of Xt satisfies mt− ∆m + div (bm) = 0

Hence m satisfies the Kolmogorov equation 

 

mt− ∆m − div (m Hp(x,∇u)) = 0

m(0) = m0 (initial distribution of the variable X0)

and moreover (since m is a probability measure) �

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Then, the optimal control is given by the feedback law α∗_t =−Hp(Xt,∇u(t, Xt)) , Hp:= ∂H ∂p and inf α J(α) = � u(x, 0)dm0(x)

wherem0is the probability distribution of X0.

• In turn,the dynamics of the density function m is governed by the

optimal decisions of the agents.

Recall: given a process dXt= b(Xt)dt +

√

2dBt, the probability measure

m(t) distribution law of Xt satisfies mt− ∆m + div (bm) = 0

Hence m satisfies the Kolmogorov equation 

 

mt− ∆m − div (m Hp(x,∇u)) = 0

m(0) = m0 (initial distribution of the variable X0)

and moreover (since m is a probability measure) �

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This is the Mean Field Games system (with horizon T ): �

−ut− ∆u + H(x, ∇u) = F (x, m(t))

u(x, T ) = G (x, m(T )) �

mt− ∆m − div (m Hp(x,∇u)) = 0

m(0) = m0

The main novelty with respect to previous models of continuum of players is the interaction in the strategy process: the coupling

F (x, m) (which could possibly also aﬀect the Hamiltonian H).

The existence of a solution is related toeconomic equilibrium with rational anticipations(in Lasry and Lions ’ words). Namely, players take their decisions in terms of an (anticipated) expected

distribution of players. Consistently, the distribution density of players evolves according to the decisions taken.

Mathematically, this is expressed in thebackward-forward coupling, and is related to a successful fixed point argument (m_{�→ u �→ m})

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This is the Mean Field Games system (with horizon T ): �

−ut− ∆u + H(x, ∇u) = F (x, m(t))

u(x, T ) = G (x, m(T )) �

mt− ∆m − div (m Hp(x,∇u)) = 0

m(0) = m0

The main novelty with respect to previous models of continuum of players is the interaction in the strategy process: the coupling

F (x, m) (which could possibly also aﬀect the Hamiltonian H).

The existence of a solution is related toeconomic equilibrium with rational anticipations(in Lasry and Lions ’ words). Namely, players take their decisions in terms of an (anticipated) expected

distribution of players. Consistently, the distribution density of players evolves according to the decisions taken.

Mathematically, this is expressed in thebackward-forward coupling, and is related to a successful fixed point argument (m_{�→ u �→ m})

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Many applications as well as motivations are being developed (see also [O. Gueant- J.M Lasry-P.L. Lions, Paris-Princeton Lectures 2011 ] ).

the system is (formally) obtained as limit of Nash equilibria of N-players games, as N _{→ ∞.}

Moreover, the solution of the mean field equation can provide ε-Nash equilibria for games with finite number of players N, provided N is large. Ex, if the payoﬀ of the i-th player is given by Ji(α) =E � �T 0 [L(Xsi, αis) + F (Xsi,N−11 � j�=i δ_Xj s)]ds + G (X i T,N−11 � j�=i δ_Xj T) � where α = (α1

t, . . . , αNt) (αit being the control of the i-th player).

An ε-Nash equilibrium is given by (αi

t=−Hp(∇u(Xti, t))).

economical models, e.g. producers of an exhaustible resource (petroleum companies etc..)

social (popoulation) models where individuals wishes to resemble to (or to distinguish from) the others (aggregation Vs repulsion models)

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Mathematics on the Mean Field Games system. �

−ut− ∆u + H(x, ∇u) = F (x, m(t)) , u(x, T ) = G (x, m(T ))

mt− ∆m − div (m Hp(x,∇u)) = 0 , m(0) = m0

H(x, p) is C1_{and convex in p.} _{Model case H(x,}_{∇u) =} 1 2|∇u|2.

Periodic setting. We assume that

x �→ F (x, m) , x �→ G (x, m) are ZN_–periodic

andu, m areZN_–periodic_.

Two coupling regimes can be considered:

(i) Nonlocal case: F , G : _RN_{× P}

1→ Rarenonlocal and smoothing

on the space of probability measures_P1 overRN\ ZN.

Typical example: F (x, m) = Φ(k � m), for some smooth Lipschitz function Φ.

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Known (mostly, very recent) results:

Diﬀerent type of existence results(generically: smooth solutions in the nonlocal case, weak solutions in the local case)

Uniqueness conditions: themonotonicity of the coupling functions

�

(F (x, m1)− F (x, m2))d(m1− m2)(x)≥ 0

�

(G (x, m1)− G (x, m2))d(m1− m2)(x)≥ 0

for m1, m2∈ P1, together withconvexity of H(x,·), imply

uniqueness.

Numerical approximations ([Achdou-Capuzzo Dolcetta]). Discrete time, finite state analysis (including long time behavior) in [Gomes-Mohr-Souza].

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Interpretation in terms of optimal control problems (the primal-dual state system arising from optimality conditions), see also

[Lachapelle-Salomon-Turinici]. Ex: Optimize in terms of the field α

inf α � T 0 [ � Ω 1 2m|α| 2₊ F(m(s))]ds + G(m(T )) where _F�_{(m) = F (m) and}_G�_{(m) = G (m) and where}

mt= ∆m + div (α m) , m(0) = m0

Optimality gives αopt =∇u(t, x) solution of the adjoint equation.

(Note: monotonicity of the coupling functions F (m) and G (m) gives convexity for the optimization problem)

Planning problem: prescribed initial and final conditions on m, no conditions on u at time t = 0 or t = T .

(it is a transport optimality problem; bring the density of the agents from the initial configuration m0to a target density mT)

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Long time behavior

Pb: What is the behavior of the agents when the horizon T is large ?

To fix the ideas, consider the model caseH =1 2|∇u|2.

Recall the case of a single equation (with no coupling): 

   

−ut− ∆u +1₂|∇u|2= F (x)

u(x, T ) = G (x) periodic b.c. Then one has:

(i) u(0)_T converges uniformly to a constant ¯λ∈ R.

(ii) u(0)− ¯λT converges uniformly to ¯u, periodic solution of the ergodic problem

¯

λ_{− ∆¯u +}1 2|∇¯u|

2_{= F (x) .}

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Typical arguments:

- comparison principle (¯u− λ(T − t) ± L used as super-subsolution of the evolution equation)

- gradient estimates

- half-relaxed limits, strong maximum principle.

(Note: most of such arguments are not suitable for our system)

As far as m is concerned, if one proves that∇u → ∇¯u, then m(t)

converges to the unique invariant probability measure ¯m associated to the process

dXt=−∇¯u(Xt)dt +

√ 2dBt

Anyway,the convergence of Du is essential to prove convergence of m. (But for the coupled system F = F (x, m), and the convergence of m will be needed for u as well....)

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Typical arguments:

- comparison principle (¯u− λ(T − t) ± L used as super-subsolution of the evolution equation)

- gradient estimates

- half-relaxed limits, strong maximum principle.

(Note: most of such arguments are not suitable for our system)

As far as m is concerned, if one proves that∇u → ∇¯u, then m(t)

converges to the unique invariant probability measure ¯m associated to the process

dXt=−∇¯u(Xt)dt +

√ 2dBt

Anyway,the convergence of Du is essential to prove convergence of m. (But for the coupled system F = F (x, m), and the convergence of m will be needed for u as well....)

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Ergodic problem for the Mean Field Games system.

It was proved by Lasry and Lions thatthere exists a unique couple (¯u, ¯m)

and a unique constant ¯λwhich solve

   ¯ λ− ∆¯u +1 2|∇¯u|2= F (x, ¯m) , � Ω¯u = 0 −∆ ¯m_{− div ( ¯}m_{∇¯u) = 0 ,} �_Ωm = 1¯ where Ω = [0, 1]N _{and the solutions are}_ZN_–periodic.

Moreover,¯u, ¯m are smooth, ¯m > 0(note indeedm = e¯ −¯u_/�� Ωe−¯udx

�

). Rmk: Existence of ¯u, ¯m, ¯λ is proved from limits of Nash equilibria.

As suggested in the non coupled case, one would expect that u/T _{→ ¯λ} and m(t)_{→ ¯}m in the long time behavior.

Main diﬃculties:

- there are no comparison arguments (and no simple estimates)

-forward-backward conditions: some boundary layer could appear at

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Ergodic problem for the Mean Field Games system.

It was proved by Lasry and Lions thatthere exists a unique couple (¯u, ¯m)

and a unique constant ¯λwhich solve

   ¯ λ− ∆¯u +1 2|∇¯u|2= F (x, ¯m) , � Ω¯u = 0 −∆ ¯m_{− div ( ¯}m_{∇¯u) = 0 ,} �_Ωm = 1¯ where Ω = [0, 1]N _{and the solutions are}_ZN_–periodic.

Moreover,¯u, ¯m are smooth, ¯m > 0(note indeedm = e¯ −¯u_/�� Ωe−¯udx

�

). Rmk: Existence of ¯u, ¯m, ¯λ is proved from limits of Nash equilibria. As suggested in the non coupled case, one would expect that u/T _{→ ¯λ} and m(t)_{→ ¯}m in the long time behavior.

Main diﬃculties:

- there are no comparison arguments (and no simple estimates)

-forward-backward conditions: some boundary layer could appear at

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To this purpose, we rescale the problem setting

υT(t, x) := u(tT , x) ; µT(t, x) := m(tT , x) (t, x)_{∈ [0, 1] × R}N. Note that (υT_{, µ}T_{) solve}

� −T1υtT− ∆υT+12|∇υT|2= F (x, µT) t ∈ (0, 1) υT(x, 1) = G (x, µ(1)) �₁ TµTt − ∆µT− div (µT∇υT) = 0 t∈ (0, 1) µT_{(0) = m} 0

The typical result we prove is the following: υT

T converges to ¯λ(1− t)for every t∈ [0, 1] �

υT₋�

ΩυT(y )dy converges to ¯u

DυT _{converges to D ¯}_u (e.g. in L2((0, 1)× Ω)).

µT _{converges to ¯}_m

The type and/or rate of convergence may depend on the case of nonlocal coupling or local coupling.

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To this purpose, we rescale the problem setting

υT(t, x) := u(tT , x) ; µT(t, x) := m(tT , x) (t, x)_{∈ [0, 1] × R}N. Note that (υT_{, µ}T_{) solve}

� −T1υtT− ∆υT+12|∇υT|2= F (x, µT) t ∈ (0, 1) υT(x, 1) = G (x, µ(1)) �₁ TµTt − ∆µT− div (µT∇υT) = 0 t∈ (0, 1) µT_{(0) = m} 0

The typical result we prove is the following: υT

T converges to ¯λ(1− t)for every t∈ [0, 1] �

υT₋�

ΩυT(y )dy converges to ¯u

DυT _{converges to D ¯}_u (e.g. in L2((0, 1)× Ω)).

µT _{converges to ¯}_m

The type and/or rate of convergence may depend on the case of nonlocal coupling or local coupling.

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Main ingredients. In order to prove this result, we use several structural properties of the system.

Instead of comparison arguments, one has to exploitenergy

estimates and the uniqueness principle of the coupled structure.

We always assume that the coupling F (x,_{·) and G (x, ·) are} monotone.

In order to get estimates, we use diﬀerent features according to local or nonlocal coupling.

(i) Nonlocal smoothing coupling F (x, m): we use a semiconcavity

estimate ⇒ Lipschitz bound for u.

(ii) Local coupling F (x, m): we use the property thatthe system

has an Hamiltonian structure_{⇒ there exists an invariant (constant} in time).

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Main ingredients. In order to prove this result, we use several structural properties of the system.

Instead of comparison arguments, one has to exploitenergy

estimates and the uniqueness principle of the coupled structure.

We always assume that the coupling F (x,_{·) and G (x, ·) are} monotone.

In order to get estimates, we use diﬀerent features according to local or nonlocal coupling.

(i) Nonlocal smoothing coupling F (x, m): we use asemiconcavity

estimate ⇒ Lipschitz bound for u.

(ii) Local coupling F (x, m): we use the property thatthe system

has an Hamiltonian structure_{⇒ there exists an invariant (constant} in time).

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Main energy equality: ([Lasry-Lions]_{→ uniqueness).} Any couple of solutions (u1, m1) and (u2, m2) satisfy

−_dtd � Ω (u1− u2)(m1− m2)dx = � Ω (m1+ m2) 2 |Du1− Du2| 2_{+ (F (x, m} 1)− F (x, m2))(m1− m2) dx

Note in particular that

F (x,_{·) nondecreasing ⇒} uniqueness

Rmk: Since m1− m2 has zero mean, you can always keep the equality by

adding a constant to u1or u2(ex.: the ergodic stationary solution ¯u is a solution up to a constant).

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Main energy equality: ([Lasry-Lions]_{→ uniqueness).} Any couple of solutions (u1, m1) and (u2, m2) satisfy

−_dtd � Ω (u1− u2)(m1− m2)dx = � Ω (m1+ m2) 2 |Du1− Du2| 2_{+ (F (x, m} 1)− F (x, m2))(m1− m2) dx

Note in particular that

F (x,_{·) nondecreasing ⇒} uniqueness

Rmk: Since m1− m2 has zero mean, you can always keep the equality by

adding a constant to u1or u2(ex.: the ergodic stationary solution ¯u is a solution up to a constant).

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Apply the energy equality to (u, m) and (¯u, ¯m) between 0 and T: � T 0 � Ω (m + ¯m) 2 |Du − D¯u| 2_{+ (F (x, m)} − F (x, ¯m))(m_{− ¯}m) dx =− �� Ω (u− ¯u)(m − ¯m)dx �T 0 ? ≤ C

Recall thatm, ¯m > 0, and that F (x,_{·) is non decreasing}.

Then,controlling the RHS, one controls the energy of the diﬀerence. Indeed, if the RHS is bounded, one deduce

1 T � T 0 � Ω|Du − D¯u| 2_{+ (F (x, m)} − F (x, ¯m))(m_{− ¯}m) dx _≤ C T

Rescaling the functions this means � 1 0 � Ω|Dυ T_{− D¯u|}2_{+ (F (x, µ}T₎_{− F (x, ¯}_m))(µT_{− ¯}_{m) dx} _≤ C T → 0 and in particular DυT _{→ D¯u in L}2_{((0, 1)}_{× Ω).}

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Apply the energy equality to (u, m) and (¯u, ¯m) between 0 and T: � T 0 � Ω (m + ¯m) 2 |Du − D¯u| 2_{+ (F (x, m)} − F (x, ¯m))(m_{− ¯}m) dx =− �� Ω (u− ¯u)(m − ¯m)dx �T 0 ? ≤ C

Recall thatm, ¯m > 0, and that F (x,_{·) is non decreasing}.

Then,controlling the RHS, one controls the energy of the diﬀerence. Indeed, if the RHS is bounded, one deduce

1 T � T 0 � Ω|Du − D¯u| 2_{+ (F (x, m)} − F (x, ¯m))(m_{− ¯}m) dx _≤ C T

Rescaling the functions this means � 1 0 � Ω|Dυ T_{− D¯u|}2_{+ (F (x, µ}T₎_{− F (x, ¯}_m))(µT_{− ¯}_{m) dx} _≤ C T → 0 and in particular DυT _{→ D¯u in L}2_{((0, 1)}_{× Ω).}

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In order to bound the RHS − �� Ω (u_{− ¯u)(m − ¯}m)dx �T 0

the term at time T is handled by monotonicity of the coupling u(T ) = G (x, m(T )). Hence one is only concerned with the term

�

Ω

(u(0)_{− ¯u)(m}0− ¯m)dx

Rmk: Typically, u(0)_{∼ CT}.

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The Nonlocal case

Using the smoothing eﬀect of F (x,_{·) and G (x, ·), we prove that u is} semiconcave, i.e.

D2u_{≤ K} (uniformly with respect to time T )

hence (since u is periodic) we deduce a Lipschitz bound�Du�∞≤ K.

Then _�

Ω(u(x, 0)− ¯u)(m0− ¯m)dx

=�_Ω(u(x, 0)− u(x0, 0)− ¯u)(m0− ¯m)dx ≤ C

and we obtain the bound for the energy, and the convergence for the rescaled functions: DυT_{→ D¯u (in L}2_{, but actually in L}p _{for every p...).}

Then we can work with µT_{, since}

1 Tµ

T

t − ∆µT− div (µT∇υT) = 0 t∈ (0, 1)

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The Nonlocal case

Using the smoothing eﬀect of F (x,_{·) and G (x, ·), we prove that u is} semiconcave, i.e.

D2u_{≤ K} (uniformly with respect to time T )

hence (since u is periodic) we deduce a Lipschitz bound�Du�∞≤ K.

Then _�

Ω(u(x, 0)− ¯u)(m0− ¯m)dx

=�_Ω(u(x, 0)− u(x0, 0)− ¯u)(m0− ¯m)dx ≤ C

and we obtain the bound for the energy, and the convergence for the rescaled functions: DυT_{→ D¯u (in L}2_{, but actually in L}p _{for every p...).}

Then we can work with µT_{, since}

1 Tµ

T

t − ∆µT− div (µT∇υT) = 0 t∈ (0, 1)

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Main result in the nonlocal case. Assume F , G :_RN_{× P}

1→ R are Lipschitz continuous andregularizing: F (_{·, m) and G (·, m) are bounded in C}2_{uniformly in m}_.

Moreover, x _{→ F (x, m) and x → G (x, m) are Z}N_−periodic. F , G satisfy the monotonicity condition

�

Q1

(F (x, m)_{− F (x, m}�))d(m_{− m}�)(x)_{≥ 0} _{∀m, m}�_{∈ P}1

Ex: F (x, m) = (k � m) � k(x).

m0 is smooth,ZN−periodic, with m0> 0 and�_Q₁m0= 1.

Theorem As T → +∞, υT_{(x, t) = u}T_{(x, tT ) and µ}T_{(x, t) = m}T_{(x, tT ), satisfy:} υT T → ¯λ(1 − t) uniformly in [0, 1]× RN � υT−�ΩυT(y )dy→ ¯u

DυT _{→ D¯u} in Lp((0, 1)× Ω) for all p ≥ 1.

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The local case

One can derive estimates exploiting the fact thatthe system has an Hamiltonian structure. Set_{F(x, m) =}�₀mF (x, r )dr and E(u, m) = � Ω [1 2m|∇u| 2₊ ∇u · ∇m − F(x, m)]dx Lemma

(u, m) is a (classical) solution of (MFG) if and only if (u, m) satisfies the Hamiltonian system              ∂u ∂t = ∂_E ∂m ∂m ∂t =− ∂_E ∂u m(0) = m0, u(x, T ) = G (x, m(T )) .

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Consequence: it is enough to bound_{E at some point}, e.g. if u(T ) is bounded in C2_{, then}_{|E(u(T ), m(T ))| ≤ C , and so as well}

|E(u(0), m(0))| ≤ C If m0is Lipschitz, m0> 0, then

E(u(0), m(0)) =�Ω[12m0|∇u(0)|2+∇u(0) · ∇m0− F(x, m0)]dx

≥ c0�_Ω|∇u(0)|2dx− C

andone concludes a bound on_{|∇u(0)| in L}2_.

Then, as in the previous case, one bounds the energy, obtaining again the average convergence 1 T � T 0 � Ω|Du − D¯u| 2_dxdt ≤ C_T → 0

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Consequence: it is enough to bound_{E at some point}, e.g. if u(T ) is bounded in C2_{, then}_{|E(u(T ), m(T ))| ≤ C , and so as well}

|E(u(0), m(0))| ≤ C If m0is Lipschitz, m0> 0, then

E(u(0), m(0)) =�Ω[12m0|∇u(0)|2+∇u(0) · ∇m0− F(x, m0)]dx

≥ c0�_Ω|∇u(0)|2dx− C

andone concludes a bound on_{|∇u(0)| in L}2_.

Then, as in the previous case, one bounds the energy, obtaining again the average convergence 1 T � T 0 � Ω|Du − D¯u| 2_dxdt ≤ C_T → 0

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We obtain then a similar result as in the nonlocal case, assuming now F , G :RN_{× R → R are Lipschitz continuous, Z}N_{−periodic in x.}

Moreover, G is regularizing, i.e. G (x, m) is bounded in_C2_uniformly

with respect to_�m�L1.

F , G are monotone.

m0 is Lipschitz,ZN−periodic, with m0> 0 and�_Q₁m0= 1.

Theorem As T → +∞, υT_{(x, t) = u}T_{(x, tT ) and µ}T_{(x, t) = m}T_{(x, tT ), satisfy:} υT T → ¯λ(1 − t) in L1(Ω), uniformly for t∈ [0, 1]. � υT₋� ΩυT(y )dy→ ¯u DυT _{→ D¯u} in L2((0, 1)× Ω). µT _{→ ¯}_m _{in L}p_{((0, 1)}_{× Ω) for some p > 1.}

Note thatconvergences are a bit weaker since we lack here of the

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Convergence at exponential rate

We obtain animproved result in the local caseif we strengthen the monotonicity condition

(F (x, m1)− F (x, m2))(m1− m2)≥ γ(m1− m2)2 (1)

for some γ > 0. Then we prove anexponential convergence.

Theorem

Under assumption (1), there is some κ > 0 (independent of T ) such that (we denote ˜u = u₋�_Ωudy )

�˜u(t) − ¯u�L1 ≤ C T _{− t} � e−κ(T −t)+ e−κt� _{∀t ∈ (0, T ) ,} �m(t) − ¯m�L1≤ C t � e−κ(T −t)+ e−κt� ∀t ∈ (0, T )

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Convergence at exponential rate

We obtain animproved result in the local caseif we strengthen the monotonicity condition

(F (x, m1)− F (x, m2))(m1− m2)≥ γ(m1− m2)2 (1)

for some γ > 0. Then we prove anexponential convergence.

Theorem

Under assumption (1), there is some κ > 0 (independent of T ) such that (we denote ˜u = u₋�_Ωudy )

�˜u(t) − ¯u�L1 ≤ C T − t � e−κ(T −t)+ e−κt� _{∀t ∈ (0, T ) ,} �m(t) − ¯m�L1≤ C t � e−κ(T −t)+ e−κt� ∀t ∈ (0, T )

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In particular, the above estimates say thatthe convergence stated before of the rescaled functions υT _{= u(tT ) and µ}T _{= m(tT ) holds with an}

exponential rate(on compact subsets of (0, 1)). �υT_(t)₋� Ω υT(t) dy−¯u�L1 ≤ C T � e−κT (1−t)+ e−κtT 1_{− t} � ∀t ∈ (0, 1) and �µT_(t)_{− ¯}_m_� L1 ≤ C T �_e_{−κT (1−t)}_{+ e}_−κtT t � ∀t ∈ (0, 1) .

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Main tool: back to theenergy estimate. Recall that −d dt � Ω(˜u− ¯u)(m − ¯m)dx = � Ω (m+ ¯m) 2 |Du − D¯u|2+ (F (x, m)− F (x, ¯m))(m− ¯m) dx

where˜u = u₋�_Ωu(y )dy.

Using the strong monotonicity of F and Poincar´e-Wirtinger inequality �

��_Ω(˜u_{− ¯u)(m − ¯}m)dx�� ≤ 1

2�˜u − ¯u�2L2+1₂�m − ¯m�2_L2 ≤ C�Ω|Du − D¯u|2+ (F (x, m)− F (x, ¯m))(m− ¯m) dx

Therefore we have, for some σ > 0

ϕ(t) := � Ω (˜u_{− ¯u)(m − ¯}m)dx satisfies ϕ�(t)_{≤ −σ|ϕ(t)|} hence ϕ(T ) e−σ(T −t)≤ ϕ(t) ≤ ϕ(0) e−σt

Again,bounds only at T and at 0 allow to conclude the exponential decayof ϕ, then, by integration, of the energy.

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��_Ω(˜u_{− ¯u)(m − ¯}m)dx�� ≤ 1

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��_Ω(˜u_{− ¯u)(m − ¯}m)dx�� ≤ 1

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Conclusions

We have shown, as T → ∞

(i) the convergence of u(t)_T to ¯λ(T_{− t)}

(ii) the convergence of u(t)₋�_Ωu(t, y )dy to ¯u (iii) the convergence of m(t) to ¯m

expressed in diﬀerent norms or scales.

The behavior of u(t)− λ(T − t) is not well established yet (comparing with the case of a single equation).

We expect exponential convergence in the nonlocal case as well.

The diﬀerent results obtained point out many delicate issues concerning estimates on the coupled system. Diﬀerent structural properties concern the nonlocal and the local case. Extensions to more general Hamiltonians or coupling is not always easy.

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Conclusions

We have shown, as T → ∞

(i) the convergence of u(t)_T to ¯λ(T_{− t)}

(ii) the convergence of u(t)₋�_Ωu(t, y )dy to ¯u (iii) the convergence of m(t) to ¯m

expressed in diﬀerent norms or scales.

The behavior of u(t)− λ(T − t) is not well established yet (comparing with the case of a single equation).

We expect exponential convergence in the nonlocal case as well.

The diﬀerent results obtained point out many delicate issues concerning estimates on the coupled system. Diﬀerent structural properties concern the nonlocal and the local case. Extensions to more general Hamiltonians or coupling is not always easy.

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