Regularity theory for mean-field games
systems
Dr. Edgard A. Pimentel
(Based on joint works with D. Gomes and H. Sánchez-Morgado)
Instituto Superior Técnico - Universidade de Lisboa
Center for Mathematical Analysis, Geometry and Dynamical Systems
June, 2014
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
1 / 50
Outline
1
Introduction
Time dependent mean-field games
Motivation for the MFG system
Main assumptions
Previous results
2
Two results
Subquadratic setting
Superquadratic setting
3
Strategy of the proofs
Subquadratic case
Superquadratic case
Further regularity
4
Follow-up work: forward-forward mean-field games
5
And beyond...
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
2 / 50
Outline
1
Introduction
Time dependent mean-field games
Motivation for the MFG system
Main assumptions
Previous results
2
Two results
Subquadratic setting
Superquadratic setting
3
Strategy of the proofs
Subquadratic case
Superquadratic case
Further regularity
4
Follow-up work: forward-forward mean-field games
5
And beyond...
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
3 / 50
Mean-field games
1
Differential games with a (very) large number of rational,
indistinguishable and intelligent players
2
J-M. Lasry and P-L. Lions as well as M. Huang, P. Caines and R.
Malhamé
3
(Stochastic) Optimal control problem coupled with the transport of
a density through the mean-field coupling and the feedback
optimal control (Nash equilibrium)
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
4 / 50
Mean-field games
1
Differential games with a (very) large number of rational,
indistinguishable and intelligent players
2
J-M. Lasry and P-L. Lions as well as M. Huang, P. Caines and R.
Malhamé
3
(Stochastic) Optimal control problem coupled with the transport of
a density through the mean-field coupling and the feedback
optimal control (Nash equilibrium)
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
4 / 50
Mean-field games
1
Differential games with a (very) large number of rational,
indistinguishable and intelligent players
2
J-M. Lasry and P-L. Lions as well as M. Huang, P. Caines and R.
Malhamé
3
(Stochastic) Optimal control problem coupled with the transport of
a density through the mean-field coupling and the feedback
optimal control (Nash equilibrium)
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
4 / 50
A model problem
(
−ut + H(x, Du) = ∆u + g[m]
mt − div(Dp Hm) = ∆m
on Td × [0, T ]
on Td × [0, T ] ,
equipped with the initial-terminal boundary conditions
(
u(x, T ) = uT (x)
m(x, 0) = m0 (x),
(1)
(2)
where
1
g[m] is the mean-field coupling,
2
Td is the d-dimensional torus, and
3
T > 0 is a fixed terminal instant.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
5 / 50
A model problem
(
−ut + H(x, Du) = ∆u + g[m]
mt − div(Dp Hm) = ∆m
on Td × [0, T ]
on Td × [0, T ] ,
equipped with the initial-terminal boundary conditions
(
u(x, T ) = uT (x)
m(x, 0) = m0 (x),
(1)
(2)
where
1
g[m] is the mean-field coupling,
2
Td is the d-dimensional torus, and
3
T > 0 is a fixed terminal instant.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
5 / 50
A model problem
(
−ut + H(x, Du) = ∆u + g[m]
mt − div(Dp Hm) = ∆m
on Td × [0, T ]
on Td × [0, T ] ,
equipped with the initial-terminal boundary conditions
(
u(x, T ) = uT (x)
m(x, 0) = m0 (x),
(1)
(2)
where
1
g[m] is the mean-field coupling,
2
Td is the d-dimensional torus, and
3
T > 0 is a fixed terminal instant.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
5 / 50
A model problem
(
−ut + H(x, Du) = ∆u + g[m]
mt − div(Dp Hm) = ∆m
on Td × [0, T ]
on Td × [0, T ] ,
equipped with the initial-terminal boundary conditions
(
u(x, T ) = uT (x)
m(x, 0) = m0 (x),
(1)
(2)
where
1
g[m] is the mean-field coupling,
2
Td is the d-dimensional torus, and
3
T > 0 is a fixed terminal instant.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
5 / 50
The optimal control
Consider
(
√
dxt = vdt + 2dWt
x0 = x,
(3)
and
x
"Z
#
T
J(x, v, m) = E
L(xs , vs ) + g[m](x, s)ds + uT (xT ) .
(4)
0
We know that a (viscosity) solution to (1) is the value function of the
optimal control problem described by (3)-(4). Also, the feedback
optimal control is given by
v∗ = −Dp H(x, Du).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
6 / 50
The transport of a density
The population governed by (3) evolves according to
mt + div(vm) = ∆m.
However, every agent knows that the feedback optimal control is
v∗ = −Dp H(x, Du). The former equation becomes then
mt − div(Dp Hm) = ∆m.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
7 / 50
Main assumptions I
We assume that g is a local power-like non-linearity, i.e.,
.
g[m](x, t) = mα (x, t);
(5)
The Hamiltonian H is subquadratic, i.e.,
H(x, p) ≤ C|p|γ + C,
and
|Dp H| ≤ C|p|γ−1 + C,
for 1 < γ < 2.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
8 / 50
Main assumptions I
We assume that g is a local power-like non-linearity, i.e.,
.
g[m](x, t) = mα (x, t);
(5)
The Hamiltonian H is subquadratic, i.e.,
H(x, p) ≤ C|p|γ + C,
and
|Dp H| ≤ C|p|γ−1 + C,
for 1 < γ < 2.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
8 / 50
Main assumptions II
The Hamiltonian H is superquadratic, i.e.,
C1 |p|2+µ + C1 ≤ H(x, p) ≤ C2 |p|2+µ + C2 ,
and
|Dp H| ≤ C|p|µ H + C,
for 0 < µ < 1;
uT and m0 smooth functions on Td and m0 ≥ κ0 , where κ0 > 0.
Additionally,
Z
Td
Edgard A. Pimentel (CAMGSD-IST-UTL)
m0 (x)dx = 1.
Regularity theory for MFG
June, 2014
9 / 50
Main assumptions II
The Hamiltonian H is superquadratic, i.e.,
C1 |p|2+µ + C1 ≤ H(x, p) ≤ C2 |p|2+µ + C2 ,
and
|Dp H| ≤ C|p|µ H + C,
for 0 < µ < 1;
uT and m0 smooth functions on Td and m0 ≥ κ0 , where κ0 > 0.
Additionally,
Z
Td
Edgard A. Pimentel (CAMGSD-IST-UTL)
m0 (x)dx = 1.
Regularity theory for MFG
June, 2014
9 / 50
Examples
1
Subquadratic case:
2
Hs (x, p) = a(x) 1 + |p|
2
γ
2
+ V (x).
Superquadratic case:
2+µ
2
HS (x, p) = a(x) 1 + |p|2
+ V (x).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
10 / 50
Examples
1
Subquadratic case:
2
Hs (x, p) = a(x) 1 + |p|
2
γ
2
+ V (x).
Superquadratic case:
2+µ
2
HS (x, p) = a(x) 1 + |p|2
+ V (x).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
10 / 50
Existence of solutions - stationary case
1
Existence of weak solutions
I
2
J-M. Lasry and P-L. Lions, 2006.
Existence of smooth solutions
I
I
I
D. Gomes, G. Pires and H. Sánchez-Morgado, 2012;
D. Gomes and H. Sánchez-Morgado, 2013;
D. Gomes, S. Patrizi and V. Voskanyan, 2013.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
11 / 50
Existence of solutions - stationary case
1
Existence of weak solutions
I
2
J-M. Lasry and P-L. Lions, 2006.
Existence of smooth solutions
I
I
I
D. Gomes, G. Pires and H. Sánchez-Morgado, 2012;
D. Gomes and H. Sánchez-Morgado, 2013;
D. Gomes, S. Patrizi and V. Voskanyan, 2013.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
11 / 50
Existence of solutions - stationary case
1
Existence of weak solutions
I
2
J-M. Lasry and P-L. Lions, 2006.
Existence of smooth solutions
I
I
I
D. Gomes, G. Pires and H. Sánchez-Morgado, 2012;
D. Gomes and H. Sánchez-Morgado, 2013;
D. Gomes, S. Patrizi and V. Voskanyan, 2013.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
11 / 50
Existence of solutions - stationary case
1
Existence of weak solutions
I
2
J-M. Lasry and P-L. Lions, 2006.
Existence of smooth solutions
I
I
I
D. Gomes, G. Pires and H. Sánchez-Morgado, 2012;
D. Gomes and H. Sánchez-Morgado, 2013;
D. Gomes, S. Patrizi and V. Voskanyan, 2013.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
11 / 50
Existence of solutions - stationary case
1
Existence of weak solutions
I
2
J-M. Lasry and P-L. Lions, 2006.
Existence of smooth solutions
I
I
I
D. Gomes, G. Pires and H. Sánchez-Morgado, 2012;
D. Gomes and H. Sánchez-Morgado, 2013;
D. Gomes, S. Patrizi and V. Voskanyan, 2013.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
11 / 50
Existence of solutions - stationary case
1
Existence of weak solutions
I
2
J-M. Lasry and P-L. Lions, 2006.
Existence of smooth solutions
I
I
I
D. Gomes, G. Pires and H. Sánchez-Morgado, 2012;
D. Gomes and H. Sánchez-Morgado, 2013;
D. Gomes, S. Patrizi and V. Voskanyan, 2013.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
11 / 50
Existence of solutions - time dependent case
1
Existence of weak solutions to (1)-(2)
I
2
I
3
J-M. Lasry and P-L. Lions, 2006.
Existence of weak solutions to the planning problem
A. Porretta, 2013.
Smooth solutions for quadratic Hamiltonians
I
P. Cardaliaguet, J-M. Lasry, P-L. Lions and A. Porreta, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
12 / 50
Existence of solutions - time dependent case
1
Existence of weak solutions to (1)-(2)
I
2
I
3
J-M. Lasry and P-L. Lions, 2006.
Existence of weak solutions to the planning problem
A. Porretta, 2013.
Smooth solutions for quadratic Hamiltonians
I
P. Cardaliaguet, J-M. Lasry, P-L. Lions and A. Porreta, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
12 / 50
Existence of solutions - time dependent case
1
Existence of weak solutions to (1)-(2)
I
2
I
3
J-M. Lasry and P-L. Lions, 2006.
Existence of weak solutions to the planning problem
A. Porretta, 2013.
Smooth solutions for quadratic Hamiltonians
I
P. Cardaliaguet, J-M. Lasry, P-L. Lions and A. Porreta, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
12 / 50
Existence of solutions - time dependent case
1
Existence of weak solutions to (1)-(2)
I
2
I
3
J-M. Lasry and P-L. Lions, 2006.
Existence of weak solutions to the planning problem
A. Porretta, 2013.
Smooth solutions for quadratic Hamiltonians
I
P. Cardaliaguet, J-M. Lasry, P-L. Lions and A. Porreta, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
12 / 50
Existence of solutions - time dependent case
1
Existence of weak solutions to (1)-(2)
I
2
I
3
J-M. Lasry and P-L. Lions, 2006.
Existence of weak solutions to the planning problem
A. Porretta, 2013.
Smooth solutions for quadratic Hamiltonians
I
P. Cardaliaguet, J-M. Lasry, P-L. Lions and A. Porreta, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
12 / 50
Existence of solutions - time dependent case
1
Existence of weak solutions to (1)-(2)
I
2
I
3
J-M. Lasry and P-L. Lions, 2006.
Existence of weak solutions to the planning problem
A. Porretta, 2013.
Smooth solutions for quadratic Hamiltonians
I
P. Cardaliaguet, J-M. Lasry, P-L. Lions and A. Porreta, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
12 / 50
Existence of solutions - time dependent case
Hamiltonians with quadratic or subquadratic growth and
g[m] = mα
I
P-L. Lions, 2012:
1
2
Existence of smooth
solutions for α > 0 provided that
1
γ ∈ 1, 1 + d+1
;
2
Existence of smooth solutions for α < d−2
provided that
1
γ ∈ 1 + d+1 , 2 .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
13 / 50
Existence of solutions - time dependent case
Hamiltonians with quadratic or subquadratic growth and
g[m] = mα
I
P-L. Lions, 2012:
1
2
Existence of smooth
solutions for α > 0 provided that
1
γ ∈ 1, 1 + d+1
;
2
Existence of smooth solutions for α < d−2
provided that
1
γ ∈ 1 + d+1 , 2 .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
13 / 50
Existence of solutions - time dependent case
Hamiltonians with quadratic or subquadratic growth and
g[m] = mα
I
P-L. Lions, 2012:
1
2
Existence of smooth
solutions for α > 0 provided that
1
γ ∈ 1, 1 + d+1
;
2
Existence of smooth solutions for α < d−2
provided that
1
γ ∈ 1 + d+1 , 2 .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
13 / 50
Existence of solutions - time dependent case
Hamiltonians with quadratic or subquadratic growth and
g[m] = mα
I
P-L. Lions, 2012:
1
2
Existence of smooth
solutions for α > 0 provided that
1
γ ∈ 1, 1 + d+1
;
2
Existence of smooth solutions for α < d−2
provided that
1
γ ∈ 1 + d+1 , 2 .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
13 / 50
In face of these results...
(At least) two questions can be posed:
Is the result due to P-L. Lions optimal?
I
Can we improve it by increasing the range in which α varies?
What about the superquadratic case?
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
14 / 50
In face of these results...
(At least) two questions can be posed:
Is the result due to P-L. Lions optimal?
I
Can we improve it by increasing the range in which α varies?
What about the superquadratic case?
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
14 / 50
In face of these results...
(At least) two questions can be posed:
Is the result due to P-L. Lions optimal?
I
Can we improve it by increasing the range in which α varies?
What about the superquadratic case?
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
14 / 50
In face of these results...
(At least) two questions can be posed:
Is the result due to P-L. Lions optimal?
I
Can we improve it by increasing the range in which α varies?
What about the superquadratic case?
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
14 / 50
Outline
1
Introduction
Time dependent mean-field games
Motivation for the MFG system
Main assumptions
Previous results
2
Two results
Subquadratic setting
Superquadratic setting
3
Strategy of the proofs
Subquadratic case
Superquadratic case
Further regularity
4
Follow-up work: forward-forward mean-field games
5
And beyond...
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
15 / 50
Existence of classical solution, subquadratic case
Theorem (Gomes-P.-Sánchez-Morgado, Comm. in PDE’s, 2014)
Let g be given as in (5) and assume that H is subquadratic. Then,
there exists a classical solution (u, m) for (1) with the initial-terminal
boundary conditions (2), provided that
α < αγ,d ,
where
αγ,d >
2
.
d −2
This result improves and extends substantially those obtained in P-L.
Lions (2012).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
16 / 50
Existence of classical solution, superquadratic case
Theorem (Gomes-P.-Sánchez-Morgado)
Let g be given as in (5) and assume that H is superquadratic. Then,
there exists a classical solution (u, m) for (1) with the initial-terminal
boundary conditions (2), provided that
α<
Edgard A. Pimentel (CAMGSD-IST-UTL)
2
.
d(1 + µ) − 2
Regularity theory for MFG
June, 2014
17 / 50
Outline
1
Introduction
Time dependent mean-field games
Motivation for the MFG system
Main assumptions
Previous results
2
Two results
Subquadratic setting
Superquadratic setting
3
Strategy of the proofs
Subquadratic case
Superquadratic case
Further regularity
4
Follow-up work: forward-forward mean-field games
5
And beyond...
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
18 / 50
A regularization argument
To prove these results, a regularization of (1) is considered. It is done
by replacing g[m] by the nonlocal operator
g [m] = η ∗ g[η ∗ m],
where η is a standard mollifying kernel, which in particular is
symmetric. This yields the system:
(
−ut + H(x, Du ) = ∆u + g [m ]
mt − div(Dp Hm ) = ∆m .
(6)
We use the convention g0 = g.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
19 / 50
Subquadratic case
1
Polynomial estimates for the Fokker-Planck equation
2
Upper bounds for the Hamilton-Jacobi equation
3
Gagliardo-Nirenberg inequality
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
20 / 50
Polynomial estimates for the Fokker-Planck equation
Theorem
Let (u , m ) be a solution of (6) and km kL∞ ([0,T ],Lβ0 (Td )) ≤ C, for some
β0 ≥ 1. Suppose further that p >
Z
Td
d
2
and r =
p(d(θ−1)+2)
.
2p−d
rn
(m )βn (τ, x) dx ≤ C + C |Dp H|2 r
Then,
L (0,T ;Lp (Td ))
where
rn = r
Edgard A. Pimentel (CAMGSD-IST-UTL)
,
θn − 1
, θ > 1 and βn = θn β0 .
θ−1
Regularity theory for MFG
June, 2014
21 / 50
Upper bounds for the Hamilton-Jacobi equation
Lemma
Let (u , m ) be a solution of (6) and assume that H is subquadratic.
Let a, b > 1 be such that
d
b(a − 1)
<
.
2
a
Then there exists C > 0 such that
ku kL∞ (0,T ;L∞ (Td )) ≤ C + Ckg (m)kLa (0,T ;Lb (Td )) .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
22 / 50
Gagliardo-Nirenberg inequality
Theorem
Let (u , m ) be a solution of (6) and assume that H is subquadratic.
For 1 < p, r < ∞ there are positive constants c and C such that
kD 2 u kLr (0,T ;Lp (Td )) ≤ ckg (m )kLr (0,T ;Lp (Td ))
γ
+ cku kL2−γ
∞ (0,T ;L∞ (Td )) + C.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
23 / 50
Superquadratic case
1
Polynomial estimates for the Fokker-Planck equation
2
Upper bounds for the Hamilton-Jacobi equation
3
Estimates by the non-linear adjoint method
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
24 / 50
Polynomial estimates for the Fokker-Planck equation
Theorem
Let (u , m ) be a solution of (6). Assume that H is superquadratic.
Assume further that 0 < µ < 1 < β0 , θ, p, r , and 0 ≤ υ ≤ 1 satisfy
αp =
and r =
d(θ−1)+2
.
2
θ n β0
,
θn + υ − θn υ
Then
kg kL∞ (0,T ;Lp (Td )) ≤ C + C kDu k
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
(2+2µ)(θ n −1)r υα
θ n β0 (θ−1)
L∞ (0,T ;L∞ (Td ))
.
June, 2014
25 / 50
Upper bounds for the Hamilton-Jacobi equation
Lemma
Suppose (u , m ) is a solution of (6) and H is superquadratic. Then, if
p > d2 ,
ku kL∞ (Td ×[0,T ]) ≤ C + Ckg (m)kL∞ (0,T ;Lp (Td )) .
Furthermore, if
1 1
1 1
p
d
+ = + = 1 and
> ,
r
s
p q
s
2
ku kL∞ (Td ×[0,T ]) ≤ C + Ckg (m)kLr (0,T ;Lp (Td )) .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
26 / 50
Estimates by the non-linear adjoint method
Theorem
Suppose that H is superquadratic. Let (u , m ) be a solution of (6) and
assume that p > d. Then
1
kDu kL∞ (0,T ;L∞ (Td )) ≤C + Ckg (m)kL1−µ
∞ (0,T ;Lp (Td ))
1
1
1−µ
+ Ckg (m)kL1−µ
∞ (0,T ;Lp (Td )) kukL∞ (0,T ;L∞ (Td )) .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
27 / 50
Bootstrapping regularity and passing to the limit
1
Lipschitz regularity for u in the subquadratic case
2
Bounds for m from below
3
Regularity for (u , m ) in Sobolev spaces
4
Additional estimates allowing one to pass to the limit → 0
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
28 / 50
Bootstrapping regularity and passing to the limit
1
Lipschitz regularity for u in the subquadratic case
2
Bounds for m from below
3
Regularity for (u , m ) in Sobolev spaces
4
Additional estimates allowing one to pass to the limit → 0
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
28 / 50
Bootstrapping regularity and passing to the limit
1
Lipschitz regularity for u in the subquadratic case
2
Bounds for m from below
3
Regularity for (u , m ) in Sobolev spaces
4
Additional estimates allowing one to pass to the limit → 0
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
28 / 50
Bootstrapping regularity and passing to the limit
1
Lipschitz regularity for u in the subquadratic case
2
Bounds for m from below
3
Regularity for (u , m ) in Sobolev spaces
4
Additional estimates allowing one to pass to the limit → 0
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
28 / 50
Regularity for the Fokker-Planck equation in Sobolev
spaces
Corollary
Let (u , m ) be a solution of (6) with initial-terminal conditions (2).
Assume that g[m] = mα and that H is either sub or superquadratic.
Then
2 m , m ∈ L2 (Td × [0, T ]), and D m ∈ L∞ ([0, T ], L2 (Td ));
Dxx
x
t
3 m , D 2 m ∈ L2 (Td × [0, T ]) D 2 m ∈ L∞ ([0, T ], L2 (Td )) and
Dxxx
xx
xt
2 m , m ∈ Lr (Td × [0, T ]) and
there is r > d such that Dx m , Dxx
t
0,1−d/r
d
then m ∈ C
(T × [0, T ]).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
29 / 50
Regularity by the Hopf-Cole transformation
Consider the following Hopf-Cole transformation:
.
w = ln m .
Lemma
Let (u , m ) be a solution of (6) with initial-terminal conditions (2).
Assume that g[m] = mα and that H is either sub or superquadratic.
Then ln m is Lipschitz and, therefore, m is bounded by above and
below.
In particular the Lemma ensures that m is bounded away from zero.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
30 / 50
Regularity for the Hamilton-Jacobi equation
Lemma
Let (u , m ) be a solution of (6) with initial-terminal conditions (2).
Assume that g[m] = mα and that H is either sub or superquadratic.
Then
2 u , u ∈ Lr (Td × [0, T ]), for any r < ∞;
Dxx
t
3 u , D 2 u ∈ L2 (Td × [0, T ]) D u ∈ L∞ ([0, T ], L2 (Td ));
Dxxx
xx
xt
There exists γ ∈ (0, 1) such that u ∈ C 0,γ Td × [0, T ] .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
31 / 50
Passing to the limit
We observe that
1
2
3
There exists u such that u → u in C 0,γ Td × [0, T ] through some
sub-sequence, uniformly in compacts
Compactness implies that we also have Du → Du
There exists m such that m → m in C 0,γ Td × [0, T ] through
some sub-sequence, uniformly in compacts
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
32 / 50
Passing to the limit
We observe that
1
2
3
There exists u such that u → u in C 0,γ Td × [0, T ] through some
sub-sequence, uniformly in compacts
Compactness implies that we also have Du → Du
There exists m such that m → m in C 0,γ Td × [0, T ] through
some sub-sequence, uniformly in compacts
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
32 / 50
Passing to the limit
We observe that
1
2
3
There exists u such that u → u in C 0,γ Td × [0, T ] through some
sub-sequence, uniformly in compacts
Compactness implies that we also have Du → Du
There exists m such that m → m in C 0,γ Td × [0, T ] through
some sub-sequence, uniformly in compacts
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
32 / 50
Outline
1
Introduction
Time dependent mean-field games
Motivation for the MFG system
Main assumptions
Previous results
2
Two results
Subquadratic setting
Superquadratic setting
3
Strategy of the proofs
Subquadratic case
Superquadratic case
Further regularity
4
Follow-up work: forward-forward mean-field games
5
And beyond...
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
33 / 50
Forward-forward mean-field games
Long-time behavior of MFG systems, T → ∞
Convergence to the equilibrium problem - numerical methods
I
I
Reverse time in the Hamilton-Jacobi equation
Prescribe initial-initial conditions for the MFG systems
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
34 / 50
Forward-forward mean-field games
Long-time behavior of MFG systems, T → ∞
Convergence to the equilibrium problem - numerical methods
I
I
Reverse time in the Hamilton-Jacobi equation
Prescribe initial-initial conditions for the MFG systems
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
34 / 50
Forward-forward mean-field games
Long-time behavior of MFG systems, T → ∞
Convergence to the equilibrium problem - numerical methods
I
I
Reverse time in the Hamilton-Jacobi equation
Prescribe initial-initial conditions for the MFG systems
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
34 / 50
Forward-forward mean-field games
Long-time behavior of MFG systems, T → ∞
Convergence to the equilibrium problem - numerical methods
I
I
Reverse time in the Hamilton-Jacobi equation
Prescribe initial-initial conditions for the MFG systems
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
34 / 50
Forward-forward mean-field games
Long-time behavior of MFG systems, T → ∞
Convergence to the equilibrium problem - numerical methods
I
I
Reverse time in the Hamilton-Jacobi equation
Prescribe initial-initial conditions for the MFG systems
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
34 / 50
Forward-forward mean-field games
Consider the following variation of the original MFG system
(
ut + H(x, Du) = ∆u + g[m]
on Td × [0, T ]
mt − div(Dp Hm) = ∆m
on Td × [0, T ] ,
equipped with initial-initial boundary conditions:
(
u(x, 0) = u0 (x)
m(x, 0) = m0 (x).
(7)
(8)
Main difficulty: Fokker-Planck is no longer the (formal) adjoint
equation to the Hamilton-Jacobi
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
35 / 50
Forward-forward mean-field games
Consider the following variation of the original MFG system
(
ut + H(x, Du) = ∆u + g[m]
on Td × [0, T ]
mt − div(Dp Hm) = ∆m
on Td × [0, T ] ,
equipped with initial-initial boundary conditions:
(
u(x, 0) = u0 (x)
m(x, 0) = m0 (x).
(7)
(8)
Main difficulty: Fokker-Planck is no longer the (formal) adjoint
equation to the Hamilton-Jacobi
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
35 / 50
A result on existence of classical solutions
Theorem (Gomes-P.)
Let g be given as in (5) and assume that H is subquadratic. Then,
there exists a classical solution (u, m) for (7) with the initial-initial
boundary conditions (8), provided that
α < αγ,d ,
where
αγ,d >
Edgard A. Pimentel (CAMGSD-IST-UTL)
2
.
d
Regularity theory for MFG
June, 2014
36 / 50
Outline
1
Introduction
Time dependent mean-field games
Motivation for the MFG system
Main assumptions
Previous results
2
Two results
Subquadratic setting
Superquadratic setting
3
Strategy of the proofs
Subquadratic case
Superquadratic case
Further regularity
4
Follow-up work: forward-forward mean-field games
5
And beyond...
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
37 / 50
Building blocks of the theory
The previously discussed techniques address mean-field games
systems with two main features:
Periodic setting - both the value function and the density are supposed
to be periodic functions in space;
Non-negative MF couplings - the non-linearity g is supposed to be
non-negative, e.g.,
g[m](x, t) = mα (x, t).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
38 / 50
Building blocks of the theory
The previously discussed techniques address mean-field games
systems with two main features:
Periodic setting - both the value function and the density are supposed
to be periodic functions in space;
Non-negative MF couplings - the non-linearity g is supposed to be
non-negative, e.g.,
g[m](x, t) = mα (x, t).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
38 / 50
Building blocks of the theory
The previously discussed techniques address mean-field games
systems with two main features:
Periodic setting - both the value function and the density are supposed
to be periodic functions in space;
Non-negative MF couplings - the non-linearity g is supposed to be
non-negative, e.g.,
g[m](x, t) = mα (x, t).
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
38 / 50
MFG systems in the whole space Rd
1. The previous results on the existence of solutions for
MFG were mostly obtained for the periodic setting;
2. The case of bounded domains, A. Porretta, 2013;
3. Whole space: linear quadratic case, Bardi, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
39 / 50
MFG systems in the whole space Rd
1. The previous results on the existence of solutions for
MFG were mostly obtained for the periodic setting;
2. The case of bounded domains, A. Porretta, 2013;
3. Whole space: linear quadratic case, Bardi, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
39 / 50
MFG systems in the whole space Rd
1. The previous results on the existence of solutions for
MFG were mostly obtained for the periodic setting;
2. The case of bounded domains, A. Porretta, 2013;
3. Whole space: linear quadratic case, Bardi, 2012.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
39 / 50
The case of Rd - main difficulties
1. Compactness is lost - Sobolev inequalities require some
integrability to hold true;
2. Some quantities are naturally unbounded, e.g.,
Z
H(x, Du(x))dx,
Rd
for, say, H(x, p) ≥ 1. Key estimates simply does not hold;
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
40 / 50
The case of Rd - main difficulties
1. Compactness is lost - Sobolev inequalities require some
integrability to hold true;
2. Some quantities are naturally unbounded, e.g.,
Z
H(x, Du(x))dx,
Rd
for, say, H(x, p) ≥ 1. Key estimates simply does not hold;
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
40 / 50
The case of Rd - main difficulties
1. Compactness is lost - Sobolev inequalities require some
integrability to hold true;
2. Some quantities are naturally unbounded, e.g.,
Z
H(x, Du(x))dx,
Rd
for, say, H(x, p) ≥ 1. Key estimates simply does not hold;
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
40 / 50
The case of Rd - regularity solutions
Theorem (Gomes-P.)
Assume that the Hamiltonian H satisfies a (sub-)quadratic growth
condition and suppose that g is a power-like non-linearity with
exponent α, where
1
0<α<
.
d −1
Then,
kDu kL∞ (0,T ;Lp (Td )) ≤ C,
uniformly in .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
41 / 50
The case of Rd - strategy of the proof I
1
2
Bounds for u in L∞ (BR );
The nonlinear adjoint method
Key novelty 1: Use of the adjoint method to obtain local Sobolev
regularity;
Key novelty 2: Sobolev regularity for the adjoint variable: entropy
dissipation.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
42 / 50
The case of Rd - strategy of the proof II
Lemma
Let (u , m ) be a solution to (6). Then,
kDu kL∞ (0,T ;Lp (BR )) ≤ C + C kg (m )kθL1c (0,T ;La (Rd ))
+ C kg (m )kθL2c (0,T ;La (Rd )) ,
for some θi > 1, where a, c > 1.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
43 / 50
The case of Rd - strategy of the proof III
Lemma
Let (u , m ) be a solution to (6). Then, there exists a constant C > 0
such that
km k
Lα+1 (0,T ;L
Edgard A. Pimentel (CAMGSD-IST-UTL)
2∗ (α+1)
2
(Rd ))
Regularity theory for MFG
≤ C.
June, 2014
44 / 50
Logarithmic non-linearities
1
Existence of solutions has been studied in the case of
non-linearities bounded by below;
Bounds for u: The optimal control formulation for the HJ equation
yields immediately a lower bound for u.
2
Logarithmic non-linearities entail two main difficulties:
1. Quantitative bounds for L∞ -norms of u;
2. Lebesgue norms of g in terms of Du.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
45 / 50
Logarithmic non-linearities
1
Existence of solutions has been studied in the case of
non-linearities bounded by below;
Bounds for u: The optimal control formulation for the HJ equation
yields immediately a lower bound for u.
2
Logarithmic non-linearities entail two main difficulties:
1. Quantitative bounds for L∞ -norms of u;
2. Lebesgue norms of g in terms of Du.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
45 / 50
Logarithmic non-linearities
1
Existence of solutions has been studied in the case of
non-linearities bounded by below;
Bounds for u: The optimal control formulation for the HJ equation
yields immediately a lower bound for u.
2
Logarithmic non-linearities entail two main difficulties:
1. Quantitative bounds for L∞ -norms of u;
2. Lebesgue norms of g in terms of Du.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
45 / 50
Logarithmic non-linearities
1
Existence of solutions has been studied in the case of
non-linearities bounded by below;
Bounds for u: The optimal control formulation for the HJ equation
yields immediately a lower bound for u.
2
Logarithmic non-linearities entail two main difficulties:
1. Quantitative bounds for L∞ -norms of u;
2. Lebesgue norms of g in terms of Du.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
45 / 50
Logarithmic non-linearities
1
Existence of solutions has been studied in the case of
non-linearities bounded by below;
Bounds for u: The optimal control formulation for the HJ equation
yields immediately a lower bound for u.
2
Logarithmic non-linearities entail two main difficulties:
1. Quantitative bounds for L∞ -norms of u;
2. Lebesgue norms of g in terms of Du.
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
45 / 50
Logarithmic non-linearities - existence of classical
solutions
Theorem (Gomes-P.)
Let the Hamiltonian H be such that
0 ≤ H(x, p) ≤ C|p|γ + C,
and assume that g is given by g[m] = ln m.Then, there exists a
classical solution (u, m) to the MFG system provided d > 2 and
1<γ<
Edgard A. Pimentel (CAMGSD-IST-UTL)
5
.
4
Regularity theory for MFG
June, 2014
46 / 50
Logarithmic non-linearities - strategy of the proof I
1
A regularization argument:
.
g [m](x, t) = ln(m + ),
which generates the approximate MFG system
(
−ut + H(x, Du ) = ∆u + g [m ]
mt − div(Dp Hm ) = ∆m ;
(9)
2
Lebesgue norms of ln(m + ) controlled in terms of Du
3
Regularity for the Hamilton-Jacobi equation by the adjoint method
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
47 / 50
Logarithmic non-linearities - strategy of the proof II
Lemma
Let (u , m ) solve (9). Then
2(γ−1)
kg [m ]kL∞ (0,T ;Lp (Td )) ≤ C + C kDu kL∞ (Td ×[0,T ]) .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
48 / 50
Logarithmic non-linearities - strategy of the proof III
Lemma
Let (u , m ) solve (9). Then
kDu kL∞ (Td ×[0,T ]) ≤ C + C kg kL∞ (0,T ;Lp (Td ))
2(γ−1)
+ C kg kL∞ (0,T ;Lp (Td )) kDu kL∞ (Td ×[0,T ]) .
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
49 / 50
Thank you!!
Edgard A. Pimentel (CAMGSD-IST-UTL)
Regularity theory for MFG
June, 2014
50 / 50