Weak solutions of mean field games systems
Alessio Porretta
Universita’ di Roma Tor Vergata
HJ 2016: “Hamilton-Jacobi equations, new trends and applications”
Rennes, 31/05/2016
A. Porretta
Weak solutions of mean field games systems
Outlines of the talk
Mean Field Games systems lead naturally to work with unbounded
solutions to Hamilton-Jacobi equations.
- Even for viscous HJ, this leads to non trivial uniqueness problems.
- The role of the (adjoint) Fokker-Planck equation is crucial in this
study
Second order case
→ Fokker-Planck with L2 -drift & weak solutions to MFG systems
Vanishing viscosity and first order case
→ new estimates for HJ with data in Lebesgue spaces
A. Porretta
Weak solutions of mean field games systems
Mean field games systems
Mean field games theory ([Lasry-Lions], [Huang-Caines-Malhamé]) leads
to coupled systems of PDEs. Simplest case (in a time horizon T ):
(
(1)
−ut − ∆u + H(t, x, Du) = F (t, x, m) in (0, T ) × Ω
(2)
mt − ∆m − div(m Hp (t, x, Du)) = 0 in (0, T ) × Ω ,
where Hp stands for
∂H(t,x,p)
.
∂p
A. Porretta
Weak solutions of mean field games systems
Mean field games systems
Mean field games theory ([Lasry-Lions], [Huang-Caines-Malhamé]) leads
to coupled systems of PDEs. Simplest case (in a time horizon T ):
(
(1)
−ut − ∆u + H(t, x, Du) = F (t, x, m) in (0, T ) × Ω
(2)
mt − ∆m − div(m Hp (t, x, Du)) = 0 in (0, T ) × Ω ,
where Hp stands for
∂H(t,x,p)
.
∂p
(1) is the Bellman equation for the agents’ value function u.
(2) is the Kolmogorov-Fokker-Planck equation for the distribution of
agents. m(t) is the probability density of the state of players at
time t.
Typically: p 7→ H(t, x, p) is convex.
Model ex: H ' γ(t, x)|Du|q .
A. Porretta
Weak solutions of mean field games systems
Macroscopic derivation
1. Individual optimization: each agent controls a N-d Brownian motion
√
dXs = βs ds + 2 dBs , Xt = x
in order to minimize, among controls βs , the cost
(Z
)
T
Jt,x (β) := Et,x
[L(Xs , βs ) + F (Xs , m(s))]ds + G (XT , m(T ))
t
where m(t) is a probability measure (the supposed law of Xt ).
A. Porretta
Weak solutions of mean field games systems
Macroscopic derivation
1. Individual optimization: each agent controls a N-d Brownian motion
√
dXs = βs ds + 2 dBs , Xt = x
in order to minimize, among controls βs , the cost
(Z
)
T
Jt,x (β) := Et,x
[L(Xs , βs ) + F (Xs , m(s))]ds + G (XT , m(T ))
t
where m(t) is a probability measure (the supposed law of Xt ).
DPP: The value function u(t, x) := inf Jt,x (β) solves the Bellman eq:
β
−ut − ∆u + H(x, Du) = F (x, m(t))
where H is the Legendre transform H = supβ [−β · p − L(x, β)]. This gives
R
• the best value inf β J(β) = u(x, 0)dm0 (x),
m0 = L(X0 ).
•
the optimal control through the feedback law:
βt∗ = −Hp (Xt , Du(t, Xt )) ,
A. Porretta
Hp :=
∂H(x, p)
∂p
Weak solutions of mean field games systems
2. Evolution of the collective state: Given a drift-diffusion process
√
dXt = b(t, Xt )dt + 2dBt
the probability measure m(t) (distribution law of Xt ) satisfies
mt − ∆m + div (bm) = 0
A. Porretta
Weak solutions of mean field games systems
2. Evolution of the collective state: Given a drift-diffusion process
√
dXt = b(t, Xt )dt + 2dBt
the probability measure m(t) (distribution law of Xt ) satisfies
mt − ∆m + div (bm) = 0
The system is at equilibrium if the optimal feedback used by the agents
bt∗ = −Hp (·, Du(·)) produces the distribution measure m(t) which was
assumed in the optimization process.
(
−ut − ∆u + H(x, Du) = F (x, m)
→
mt − ∆m − div(m Hp (x, Du)) = 0 ,
A. Porretta
Weak solutions of mean field games systems
This is the Mean Field Games system (with horizon T ):
(
(1)
−ut − ∆u + H(x, Du) = F (x, m) in (0, T ) × Ω
(2)
mt − ∆m − div(m Hp (x, Du)) = 0 in (0, T ) × Ω ,
usually complemented with initial-terminal conditions:
-m(0) = m0 (initial distribution of the agents)
-u(T ) = G (x, m(T )) (final pay-off)
+ boundary conditions (here for simplicity assume periodic b.c.)
Main novelties are:
the backward-forward structure.
the interaction in the strategy process: the coupling F (x, m)
Rmk 1: This is not the most general structure.
Cost criterion L(Xt , αt , m(t)) → H(x, m, Du).
Rmk 2: The special structure H = H(Du) − F (m) gives to the system a
variational structure → optimality system (so-called mean field control
systems)
A. Porretta
Weak solutions of mean field games systems
Key assumption [Lasry-Lions]:
m 7→ F (x, m) increasing ⇒ uniqueness of smooth solutions
A. Porretta
Weak solutions of mean field games systems
Key assumption [Lasry-Lions]:
m 7→ F (x, m) increasing ⇒ uniqueness of smooth solutions
However, the existence of smooth solutions is mostly unknown:
i
hR
T
Minimization of the cost E 0 F (m(t))dt < ∞
RR
⇒
Estimates of the system only give
F (m)m ≤ C
Typical example: F (m) ' mγ :
→ mγ+1 ∈ L1
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⇒
F (m) ∈ L
γ+1
γ
Weak solutions of mean field games systems
Key assumption [Lasry-Lions]:
m 7→ F (x, m) increasing ⇒ uniqueness of smooth solutions
However, the existence of smooth solutions is mostly unknown:
i
hR
T
Minimization of the cost E 0 F (m(t))dt < ∞
RR
⇒
Estimates of the system only give
F (m)m ≤ C
Typical example: F (m) ' mγ :
→ mγ+1 ∈ L1
⇒
F (m) ∈ L
γ+1
γ
Partial results:
the existence of smooth solutions is known in few cases:
(i) if m 7→ F (x, m) or p 7→ H(x, p) have a mild growth
([Lasry-Lions], [Gomes-Pimentel-Sanchez Morgado])
(iii) in the homogeneous quadratic case H(x, p) = |p|2 , solutions are
smooth for every F (x, m) ≥ 0 ([Cardaliaguet-Lasry-Lions-P.])
A. Porretta
Weak solutions of mean field games systems
Pb: build a complete theory of weak solutions (existence, uniqueness,
stability...).
Motivations & applications:
Convergence of numerical schemes
Existence and uniqueness for the planning problem (prescribed
initial and final densities m(0) and m(T )):


−ut − ∆u + H(x, ∇u) = F (x, m) ,
mt − ∆m − div (m Hp (x, ∇u)) = 0 ,


m(0) = m0 , m(T ) = m1
Here, no condition is assumed on u at time T .
This is an optimal transport model for the distribution law m of the
stochastic flow.
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Weak solutions of mean field games systems
Weak theory for MFG systems
Main problems for a weak theory:
1. The typical setting for well-posedness of the Fokker-Planck
(FP)
mt −∆m+ div (m b) = 0
(t, x) ∈ (0, T )×Ω ,
Ω ⊂ IRN
requires a high integrability for the drift b:
b ∈ L∞ (0, T ; LN (Ω)) ,
o
b ∈ LN+2 ((0, T ) × Ω)
(or any interpolation between the two conditions, cfr. [Aronson-Serrin],
[Ladysenskaya-Solonnikov-Uraltseva])
A. Porretta
Weak solutions of mean field games systems
Weak theory for MFG systems
Main problems for a weak theory:
1. The typical setting for well-posedness of the Fokker-Planck
(FP)
mt −∆m+ div (m b) = 0
(t, x) ∈ (0, T )×Ω ,
Ω ⊂ IRN
requires a high integrability for the drift b:
b ∈ L∞ (0, T ; LN (Ω)) ,
o
b ∈ LN+2 ((0, T ) × Ω)
(or any interpolation between the two conditions, cfr. [Aronson-Serrin],
[Ladysenskaya-Solonnikov-Uraltseva])
Pb. MFGames: b = Hp (x, Du) ' |Du|q−1 is in the right class only if q is
small or Du highly integrable
A. Porretta
Weak solutions of mean field games systems
2. Uniqueness may fail for unbounded solutions of HJB:
(
ut − ∆u + |Du|2 = 0
∃ u ∈ L2 (0, T ; H01 ) , u 6= 0 sol. of
u(0) = 0
Counterexamples are constructed as u = log(1 + v ), v solutions to
(
vt − ∆v = χ
v (0) = 0
provided χ is a concentrated measure (ex. χ = δx0 )
A. Porretta
Weak solutions of mean field games systems
Back to MFG system:
(
−ut − ∆u + H(x, Du) = F (x, m) ,
mt − ∆m − div (m Hp (x, Du)) = 0 ,
Summary:
(i) HJB has no uniqueness of weak solutions
(ii) The drift in FP might not have the right summability
Desperate situation ?......
A. Porretta
Weak solutions of mean field games systems
Back to MFG system:
(
−ut − ∆u + H(x, Du) = F (x, m) ,
mt − ∆m − div (m Hp (x, Du)) = 0 ,
Summary:
(i) HJB has no uniqueness of weak solutions
(ii) The drift in FP might not have the right summability
Desperate situation ?...... But MFG solutions satisfy an extra estimate:
m L(x, Hp (x, Du)) ∈ L1 (QT )
(1)
which comes from optimization:
Z
T
"Z
Z
L(Xt , Hp (Xt , Du(t, Xt )))dt < ∞
L(x, Hp (x, Du)) m ' E
0
Ω
#
T
0
Ex: model case: L, H with quadratic growth
(1) ⇒ m |Du|2 ∈ L1 ,
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↔ m |Hp (x, Du)|2 ∈ L1
Weak solutions of mean field games systems
A new approach to Fokker-Planck equation
Key point: we can consider solutions of Fokker-Planck
mt − ∆m − div (bm) = 0
such that m ≥ 0,
m|b|2 ∈ L1
In this framework, we can prove:
1
Weak (=distributional) solutions of (FP) are unique in this class
2
Weak solutions are renormalized solutions;
(in the sense of [Di Perna-Lions], extended to second order, cfr.
[Lions-Murat])
Moreover, those solutions can be regularized and obtained as limit of
smooth solutions.
Rmk: The importance of the class {m : b ∈ L2 (m)} was also stressed in
[Bogachev-Da Prato-Röckner ’11], [Bogachev-Krylov-Röckner]
A. Porretta
Weak solutions of mean field games systems
The typical statement is the following (adapted to Dirichlet, Neumann,
or to entire space IRN under suitable modifications)
Theorem (P., ARMA 2015)
Let b ∈ L2 (QT )N and m0 ∈ L1 . Then the problem


mt − ∆m − div(m b) = 0 , in (0, T ) × Ω,
m(0) = m0
in Ω.


+ BC
(2)
admits at most one weak sol. m ∈ L1 (QT )+ : m|b|2 ∈ L1 (QT ).
Moreover, in this case any weak solution is a renormalized solution,
belongs to C 0 ([0, T ]; L1 ) and satisfies (for a suitable truncation Tk (·)):
(Tk (m))t − ∆Tk (m) − div(Tk0 (m)m b) = ωk ,
in QT
k→∞
where ωk ∈ L1 (QT ), and ωk → 0 in L1 (QT ).
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Weak solutions of mean field games systems
(3)
Main idea: a nonlinear look at a linear equation
If m|b|2 ∈ L1 , then
m = limε mε ,

√
mtε − ∆mε − div( mε B ε ) = 0 , in (0, T ) × Ω,
mε (0) = m , + BC
0
L2 √
where B ε → m b.
A. Porretta
Weak solutions of mean field games systems
Main idea: a nonlinear look at a linear equation
If m|b|2 ∈ L1 , then
m = limε mε ,

√
mtε − ∆mε − div( mε B ε ) = 0 , in (0, T ) × Ω,
mε (0) = m , + BC
0
L2 √
where B ε → m b.
for general convection-diffusion problems (possibly nonlinear)
(
mtε + Amε = div (φ(t, x, mε )) in QT
mε (0) = m0ε , +BC
we have that if
|φ(t, x, m)| ≤ c(1 +
√
m) k(t, x) ,
k ∈ L2 (QT )
then
mε → m
in C 0 ([0, T ]; L1 )
and m is renormalized solution relative to m0 .
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Weak solutions of mean field games systems
(4)
One can apply this idea even in the Di Perna-Lions approach,
regularizing m by convolution:
mt − ∆m − div(m b) = 0
⇒ mε := m ? ρε
mtε
? ρε
solves
− ∆mε − div((mb) ? ρε ) = 0
where Schwartz’s inequality + m ≥ 0 imply
1
1
|(mb) ? ρε | ≤ (m ? ρε ) 2 (m|b|2 ) ? ρε 2
| {z } |
{z
}
√
mε
Bε
with B ε converging in L2 (QT ).
→
for purely second order operators, no need of commutators !
A. Porretta
Weak solutions of mean field games systems
Summary on FP:
the class of weak solutions m such that m|b|2 ∈ L1 gives:
uniqueness, renormalized formulation, solutions obtained by
regularization, estimates. Ex:
m0 > 0 , log m0 ∈ L1loc (Ω) ⇒ log m(t) ∈ L1loc (Ω),
hence m(t) > 0 a.e.
the class m|b|2 ∈ L1 is consistent with the stochastic flow:
we are considering only trajectories Xt along which the drift is
L2 -integrable:
"Z
#
T
2
E
|b(Xt )| dt < ∞
0
Work in progress: prove that uniqueness in law holds for (SDE)
under this condition and establish rigorously the connection with a
stochastic flow
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Weak solutions of mean field games systems
Weak solutions to Mean Field Games systems


−ut − ∆u + H(x, ∇u) = F (x, m) ,
mt − ∆m − div (m Hp (x, ∇u)) = 0 ,


u(T ) = G (x, m(T )) , m(0) = m0
• F , G ∈ C 0 (Ω × R)
• p 7→ H(x, p) is convex and satisfies structure conditions
Ex: H ' γ(t, x)|∇u|q , q ≤ 2.
Def. of weak solutions:
- u, m ∈ C 0 ([0, T ]; L1 ), m |Du|q ∈ L1
-G (x, m(T )) ∈ L1 , H(x, Du) ∈ L1 , F (x, m) ∈ L1 ,
- the equations hold in the sense of distributions.
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Weak solutions of mean field games systems
Theorem (P., ARMA ’15)
Assume that m 7→ G (x, m) is nondecreasing, and let m0 ∈ L∞
+.
(i) If F , G are bounded below, then there exists a weak solution.
(ii) If in addition m 7→ F (x, m) is nondecreasing, p 7→ H(x, p) is strictly
convex (at infinity), and log m0 ∈ L1loc (Ω), then there is a unique weak
solution.
Rmk: The coupling functions F , G have no growth restriction from above
• The case F = F (x) is included !! New viewpoint for
(
ut − ∆u + H(x, Du) = F (x)
u∂Ω = 0 , u(0) = u0
Uniqueness ⇐⇒ mt − ∆m − div (Hp (x, Du) m) = 0 admits a sol. m
with Hp (Du) ∈ L2 (m).
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Weak solutions of mean field games systems
Convergence of numerical schemes
[Achdou-P., SINUM 2016]
We use finite differences approximations of the mean field games system
as in [Achdou-Capuzzo Dolcetta], [Achdou-Camilli-Capuzzo Dolcetta]:
 k+1 k
u
−u
k+1

 i,j ∆t i,j − (∆h u k )i,j + g (xi,j , ∇h u k ) = F (mi,j
),
i,j
k+1
k

 mi,j −mi,j − (∆ mk+1 ) + T (u k , mk+1 ) = 0,
h
i,j
i,j
∆t
where g is a monotone approximation of the Hamiltonian H as in upwind
schemes:
Ex (1-d): g = g ui+1h−ui , ui −uh i−1 with g (p1 , p2 ) increasing in p2 and
decreasing in p1 , g (q, q) = H(q).
while T is the discrete adjoint of the associated linearized transport:
T (v , m) · w = m gp ([∇h v ]) · [∇h w ]
This structure allows us to have discrete estimates and compactness as in
the continuous model.
A. Porretta
Weak solutions of mean field games systems
For simplicity, assume dimension N = 2.
Let uh,∆t and mh,∆t be the piecewise constant functions respectively
n+1
n
in
taking the values ui,j
and mi,j
(tn , tn+1 ) × (ih − h/2, ih + h/2) × (jh − h/2, jh + h/2).
Theorem (Achdou-P. 2015)
Up to the extraction of a subsequence of h and ∆t, mh,∆t → m̃ and
uh,∆t → ũ in Lβ ((0, T ) × Ω) for all β ∈ [1, 2), where
ũ, m̃ ∈ C 0 ([0, T ]; L1 ) is a weak solution of the system.
Rmks:
Provides a new proof for the existence of weak solutions.
Uses discrete energy estimates for the Fokker-Planck equation +
compactness through discrete Aubin-Simon lemma
Uses new tools for discrete approximation of viscous HJ with
L1 -data (monotonicity of the discrete Hamiltonian is crucial)
A. Porretta
Weak solutions of mean field games systems
Extensions, work in progress
Similar results exist for the case of RN . If m0 ∈ L1 ∩ L∞ , existence
and uniqueness of solutions (u, m) such that m|Du|q ∈ L1 and
u ∈ L∞ + L∞ ((0, T ); L1 ) (global integrability).
A. Porretta
Weak solutions of mean field games systems
Extensions, work in progress
Similar results exist for the case of RN . If m0 ∈ L1 ∩ L∞ , existence
and uniqueness of solutions (u, m) such that m|Du|q ∈ L1 and
u ∈ L∞ + L∞ ((0, T ); L1 ) (global integrability).
[Achdou-P.]
Extension to congestion models in mean field games:
H=
|Du|q
,
(c + mγ )
c ≥ 0.
This corresponds to a cost which penalizes the areas at high density:
0
L(m, α) ' mγ/q−1 |α|q .
We have existence and uniqueness of weak solutions for
0 < γ ≤ 4 q−1
q (optimal range).
Important: the structure of optimality system is missing and optimal
control approach cannot be used. Yet the weak theory works...
A. Porretta
Weak solutions of mean field games systems
Extensions, work in progress
Similar results exist for the case of RN . If m0 ∈ L1 ∩ L∞ , existence
and uniqueness of solutions (u, m) such that m|Du|q ∈ L1 and
u ∈ L∞ + L∞ ((0, T ); L1 ) (global integrability).
[Achdou-P.]
Extension to congestion models in mean field games:
H=
|Du|q
,
(c + mγ )
c ≥ 0.
This corresponds to a cost which penalizes the areas at high density:
0
L(m, α) ' mγ/q−1 |α|q .
We have existence and uniqueness of weak solutions for
0 < γ ≤ 4 q−1
q (optimal range).
Important: the structure of optimality system is missing and optimal
control approach cannot be used. Yet the weak theory works...
(in progress) Verification theorems for weak solutions
→ build trajectories of the SDE associated to weak solutions:
√
dXt = b(Xt )dt + 2dBt
and prove uniqueness in law for martingale solutions
A. Porretta
Weak solutions of mean field games systems
Vanishing viscosity and first order case
[joint work with Cardaliaguet, Graber & Tonon]


−∂t u + H(t, x, Du) = F (t, x, m)
∂t m − div(m Hp (t, x, Du)) = 0


u(T ) = uT , m(0) = m0
in (0, T ) × Ω
in (0, T ) × Ω ,
where uT , m0 are regular.
Structure conditions:
c0 (mγ − 1) ≤ F (t, x, m) ≤ C0 (mγ + 1)
γ>0
c1 (|ξ|q − 1) ≤ H(t, x, ξ) ≤ C1 (|ξ|q + 1)
p>1
A. Porretta
Weak solutions of mean field games systems
Vanishing viscosity and first order case
[joint work with Cardaliaguet, Graber & Tonon]


−∂t u + H(t, x, Du) = F (t, x, m)
∂t m − div(m Hp (t, x, Du)) = 0


u(T ) = uT , m(0) = m0
in (0, T ) × Ω
in (0, T ) × Ω ,
where uT , m0 are regular.
Structure conditions:
c0 (mγ − 1) ≤ F (t, x, m) ≤ C0 (mγ + 1)
γ>0
c1 (|ξ|q − 1) ≤ H(t, x, ξ) ≤ C1 (|ξ|q + 1)
p>1
Energy estimates ⇒
m|Du|q ∈ L1 , F (m)m ∈ L1
A. Porretta
Weak solutions of mean field games systems
Vanishing viscosity and first order case
[joint work with Cardaliaguet, Graber & Tonon]


−∂t u + H(t, x, Du) = F (t, x, m)
∂t m − div(m Hp (t, x, Du)) = 0


u(T ) = uT , m(0) = m0
in (0, T ) × Ω
in (0, T ) × Ω ,
where uT , m0 are regular.
Structure conditions:
c0 (mγ − 1) ≤ F (t, x, m) ≤ C0 (mγ + 1)
γ>0
c1 (|ξ|q − 1) ≤ H(t, x, ξ) ≤ C1 (|ξ|q + 1)
p>1
Energy estimates ⇒
→ F (m) ∈ L
γ+1
γ
m|Du|q ∈ L1 , F (m)m ∈ L1 ' m ∈ Lγ+1
.
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Weak solutions of mean field games systems
Weak solutions of the first order system are defined as:
u ∈ W 1,q , m ∈ Lγ+1 , m|Du|q ∈ L1 and
(i) u is a distributional subsolution:
−ut + H(x, ∇u) ≤ F (x, m)
(ii) m is a distributional solution for the continuity equation
mt − div (m Hp (x, ∇u)) = 0
(iii) the energy equality holds
Z TZ
Z TZ
m F (x, m)dxdt +
0
Ω
Z0
=
m {Hp (x, Du)Du − H(x, Du)} dxdt
Z
m0 u(0) −
uT m(T )
Ω
Ω
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Ω
Weak solutions of mean field games systems
Theorem (CGPT ’15)
Assume that p 7→ H(x, p) is strictly convex and m 7→ F (x, m) is
increasing. Then the first order system


−ut + H(x, ∇u) = F (x, m) ,
mt − div (m Hξ (x, ∇u)) = 0 ,


m(0) = m0 , u(T ) = uT
admits a unique weak solution (u, m) in the sense that m is unique and u
is unique in {m > 0}.
N
Something more is known if γ+1
γ > 1 + p [Cardaliaguet-Graber]
u is Hölder continuous and is also globally unique (not only in
{m > 0}) if one requires
− ∂t u + H(x, Du) ≥ 0
in QT
(5)
in the viscosity sense.
The result extends to second order degenerate case −Tr(A(x)D 2 (·))
provided A is Lipschitz and q ≥ γ+1
γ .
A. Porretta
Weak solutions of mean field games systems
Sketch of proof: vanishing viscosity limit
(
−∂t u − ε∆u + H(t, x, Du) = F (t, x, m)
∂t m − ε∆m − div(m Hp (t, x, Du)) = 0
in (0, T ) × Ω
in (0, T ) × Ω
Energy estimates give F (mε ) bounded in Lr , r =
γ+1
γ .
Integral bounds for subsolutions of HJ → u ε bounded in W 1,p ∩ Ls
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Weak solutions of mean field games systems
Sketch of proof: vanishing viscosity limit
(
−∂t u − ε∆u + H(t, x, Du) = F (t, x, m)
∂t m − ε∆m − div(m Hp (t, x, Du)) = 0
in (0, T ) × Ω
in (0, T ) × Ω
Energy estimates give F (mε ) bounded in Lr , r =
γ+1
γ .
Integral bounds for subsolutions of HJ → u ε bounded in W 1,p ∩ Ls
We have
uε * u
ε
F (m ) * α
in W 1,p ,
in Lp ,
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mε * m
ε
in Lγ+1
ε
m Hξ (Du ) * w
in L1 .
Weak solutions of mean field games systems
Sketch of proof: vanishing viscosity limit
(
−∂t u − ε∆u + H(t, x, Du) = F (t, x, m)
∂t m − ε∆m − div(m Hp (t, x, Du)) = 0
in (0, T ) × Ω
in (0, T ) × Ω
Energy estimates give F (mε ) bounded in Lr , r =
γ+1
γ .
Integral bounds for subsolutions of HJ → u ε bounded in W 1,p ∩ Ls
We have
uε * u
ε
F (m ) * α
in W 1,p ,
in Lp ,
mε * m
ε
in Lγ+1
ε
m Hξ (Du ) * w
in L1 .
In the limit we get that (u, m) satisfy in distributional sense
−∂t u + H(t, x, Du) ≤ α ,
A. Porretta
∂t m − div(w ) = 0
Weak solutions of mean field games systems
Sketch of proof: vanishing viscosity limit
(
−∂t u − ε∆u + H(t, x, Du) = F (t, x, m)
∂t m − ε∆m − div(m Hp (t, x, Du)) = 0
in (0, T ) × Ω
in (0, T ) × Ω
Energy estimates give F (mε ) bounded in Lr , r =
γ+1
γ .
Integral bounds for subsolutions of HJ → u ε bounded in W 1,p ∩ Ls
We have
uε * u
ε
F (m ) * α
in W 1,p ,
in Lp ,
mε * m
ε
in Lγ+1
ε
m Hξ (Du ) * w
in L1 .
In the limit we get that (u, m) satisfy in distributional sense
−∂t u + H(t, x, Du) ≤ α ,
∂t m − div(w ) = 0
but the energy equality also holds:
Z TZ
Z TZ
Z
Z
m α+
{w · Du − m H(x, Du)} =
m0 u(0)− uT m(T )
0
Ω
0
Ω
Ω
Ω
By Minty’s argument we identify w = mHp (Du), α = F (m).
A. Porretta
Weak solutions of mean field games systems
Integral estimates for HJ
Assume H coercive & superlinear : H(t, x, Du) ≥ α|Du|q , with q > 1.
This leads to estimates for distributional subsolutions
(
−∂t u + c0 |Du|q ≤ f ∈ Lr (QT )
u(T ) = uT ∈ L∞
Then we have, if N = space dimension:
N
⇒ u ∈ L∞ (QT )
q
N
rq(1 + N)
1<r <1+
⇒ u ∈ Ls (QT ) , s =
q
N − q(r − 1)
r >1+
A. Porretta
Weak solutions of mean field games systems
Integral estimates for HJ
Assume H coercive & superlinear : H(t, x, Du) ≥ α|Du|q , with q > 1.
This leads to estimates for distributional subsolutions
(
−∂t u + c0 |Du|q ≤ f ∈ Lr (QT )
u(T ) = uT ∈ L∞
Then we have, if N = space dimension:
N
⇒ u ∈ L∞ (QT )
q
N
rq(1 + N)
1<r <1+
⇒ u ∈ Ls (QT ) , s =
q
N − q(r − 1)
r >1+
Qn: where does the exponent come from ?
A. Porretta
Weak solutions of mean field games systems
Integral estimates for HJ
Assume H coercive & superlinear : H(t, x, Du) ≥ α|Du|q , with q > 1.
This leads to estimates for distributional subsolutions
(
−∂t u + c0 |Du|q ≤ f ∈ Lr (QT )
u(T ) = uT ∈ L∞
Then we have, if N = space dimension:
N
⇒ u ∈ L∞ (QT )
q
N
rq(1 + N)
1<r <1+
⇒ u ∈ Ls (QT ) , s =
q
N − q(r − 1)
r >1+
Qn: where does the exponent come from ?
If we knew that ∂t u, |Du|q ∈ Lr - same space as f - then apply Sobolev
embedding + space-time interpolation (a la Gagliardo-Nirenberg)...
Qn: maximal regularity for (coercive) first order problems ?
∂t u + |Du|q = f ∈ Lr (QT )
A. Porretta
⇒
|Du|q , ∂t u ∈ Lr
??
Weak solutions of mean field games systems
Case r > 1 +
N
q
(for solutions)

−∂t u + H(t, x, Du) = f ∈ Lr (QT )
α |Du|q − c ≤ H(t, x, Du) ≤ β|Du|q + c
[Cardaliaguet-Silvestre ’12] ⇒ u ∈ C 0,θ (QT )
A. Porretta
Weak solutions of mean field games systems
Case r > 1 +
N
q
(for solutions)

−∂t u + H(t, x, Du) = f ∈ Lr (QT )
α |Du|q − c ≤ H(t, x, Du) ≤ β|Du|q + c
[Cardaliaguet-Silvestre ’12] ⇒ u ∈ C 0,θ (QT )
1,1
[Cardaliaguet-P.-Tonon ’15] ⇒ u ∈ Wloc
(QT ) and
(i) there exists ε > 0 : |Du|q , ut ∈ L1+ε
loc (QT ) and
kut kL1+ε (K ) + k|Du|q kL1+ε (K ) ≤ C (K , kf kLr , kDukq , r , N, q))
(ii) u is a.e. differentiable in QT .
ε depends on α, β, N, q, r and cannot be arbitrarily large
(counterexample !.. in which the conclusion is false for ε ≥ q1 )
This is a Meyers’ type result: measurable coefficients, gain of
integrability
A. Porretta
Weak solutions of mean field games systems
Compare with elliptic-parabolic regularity:
∂t u − div(A(x)Du) = div(G ) ,
G ∈ Lr r > 2
(i) [Meyers] → If A(x) elliptic with bounded measurable entries
then ∃ ε > 0: Du ∈ L2(1+ε)
(also true for nonlinear operators, systems.. cfr. [Kinnunen-Lewis]
A. Porretta
Weak solutions of mean field games systems
Compare with elliptic-parabolic regularity:
∂t u − div(A(x)Du) = div(G ) ,
G ∈ Lr r > 2
(i) [Meyers] → If A(x) elliptic with bounded measurable entries
then ∃ ε > 0: Du ∈ L2(1+ε)
(also true for nonlinear operators, systems.. cfr. [Kinnunen-Lewis]
(ii) (maximal regularity) → If A(x) is elliptic and continuous, then
Du ∈ Lr (also true for nonlinear operators....cfr. [Mingione])
A. Porretta
Weak solutions of mean field games systems
Compare with elliptic-parabolic regularity:
∂t u − div(A(x)Du) = div(G ) ,
G ∈ Lr r > 2
(i) [Meyers] → If A(x) elliptic with bounded measurable entries
then ∃ ε > 0: Du ∈ L2(1+ε)
(also true for nonlinear operators, systems.. cfr. [Kinnunen-Lewis]
(ii) (maximal regularity) → If A(x) is elliptic and continuous, then
Du ∈ Lr (also true for nonlinear operators....cfr. [Mingione])
Is something similar happening for first order equations ?
∂t u + a(t, x)|Du|q = f ,
f ∈ Lr r > 1 +
N
q
(i) [CPT] : a(t, x) coercive and bounded, only measurable
→ Du ∈ Lq(1+ε)
(ii) a(t, x) coercive and continuous (smooth as needed...) ⇒
maximal regularity ?
A. Porretta
Weak solutions of mean field games systems
Thanks for the attention !
A. Porretta
Weak solutions of mean field games systems