Existence and uniqueness results for the MFG system

Transcription

1 Existence and uniqueness results for the MFG system P. Cardaliaguet Paris-Dauphine Based on Lasry-Lions (2006) and Lions lectures at Collège de France IMA Special Short Course Mean Field Games", November 12-13, 2012 P. Cardaliaguet (Paris-Dauphine) MFG 1 / 86

2 The Mean Field Game system The general MFG system takes the form (with unknown (u, m) : [0, T ] R d R 2 ) : (i) t u σ 2 u + H(x, Du, m) = 0 in [0, T ] R d (MFG) (ii) t m σ 2 m div(m D p H(x, Du, m)) = 0 in [0, T ] R d (iii) m(0) = m 0, u(x, T ) = G(x, m(t )) in R d where σ R, H = H(x, p, m) is a convex Hamiltonian (in p) depending on the density m, G = G(x, m(t )) is a function depending on the position x and the density m(t ) at time T. m 0 is a probability density on R d. P. Cardaliaguet (Paris-Dauphine) MFG 2 / 86

3 System introduced in Lasry-Lions 06, as a finite dimensional reduction of a master equation", Huang-Caines-Malhame, 06 to model large population differential games. P. Cardaliaguet (Paris-Dauphine) MFG 3 / 86

4 Outline 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 4 / 86

5 Outline 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 4 / 86

6 Outline First order MFG equations with nonlocal coupling 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 5 / 86

7 First order MFG equations with nonlocal coupling Discuss existence and uniqueness for the first order, non local system (MFG) where (i) t u Du 2 = F (x, m(t, )) in R d (0, T ) (ii) t m div (mdu) = 0 in R d (0, T ) (iii) m(0) = m 0, u(x, T ) = u f (x) the first equation is backward in time the second one is forward in time the map F is nonlocal and regularizing. P. Cardaliaguet (Paris-Dauphine) MFG 6 / 86

8 First order MFG equations with nonlocal coupling Approach by fixed point We fix a family (m(t)) t [0,T ]. Let u be the solution of the (backward) HJ equation (i) t u + 1 (HJ) 2 Du 2 = F (x, m(t, )) in R d (0, T ) (iii) u(x, T ) = u f (x) and m be the solution of the (forward) Kolmogorov equation (ii) t m div ( mdu) = 0 (K ) in R d (0, T ) (iii) m(0) = m 0 Problem Prove that the map m m has a fixed point. P. Cardaliaguet (Paris-Dauphine) MFG 7 / 86

9 First order MFG equations with nonlocal coupling Difficulties (HJ) has a Lipschitz continuous solution, but not C 1 solutions in general (shocks). (K) is ill-posed for vector fields Du which are only L (multiple solutions). A system arising in geometric optics : The (forward-forward) system (i) t u Du 2 = F(x, m(t, )) in R d (0, T ) (ii) t m div (mdu) = 0 in R d (0, T ) (iii) m(0) = m 0, u(0, x) = u 0 (x) was studied by Gosse and James (2002), Ben Moussa and Kossioris (2003)) and Strömberg (2007). Well-posed of measured-valued solutions. P. Cardaliaguet (Paris-Dauphine) MFG 8 / 86

10 First order MFG equations with nonlocal coupling Difficulties (HJ) has a Lipschitz continuous solution, but not C 1 solutions in general (shocks). (K) is ill-posed for vector fields Du which are only L (multiple solutions). A system arising in geometric optics : The (forward-forward) system (i) t u Du 2 = F(x, m(t, )) in R d (0, T ) (ii) t m div (mdu) = 0 in R d (0, T ) (iii) m(0) = m 0, u(0, x) = u 0 (x) was studied by Gosse and James (2002), Ben Moussa and Kossioris (2003)) and Strömberg (2007). Well-posed of measured-valued solutions. P. Cardaliaguet (Paris-Dauphine) MFG 8 / 86

11 First order MFG equations with nonlocal coupling Existence and stability properties of the (MFG) system rely on : Semi-concavity estimate for HJ equations Well-posedness of the Kolmogorov equation for gradient vector fields of semi-concave functions Further simplifying assumption : m 0 is bounded, with bounded support. P. Cardaliaguet (Paris-Dauphine) MFG 9 / 86

12 Outline First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 10 / 86

13 First order MFG equations with nonlocal coupling Semi-concave functions Semi-concavity properties of HJ equations Definition A map w : R d R is semi-concave if there is some C > 0 such that w(λx + (1 λ)y) λw(x) + (1 λ)w(y) Cλ(1 λ) x y 2 for any x, y R d, λ [0, 1] P. Cardaliaguet (Paris-Dauphine) MFG 11 / 86

14 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Proposition w : R d R is semi-concave iff one of the following conditions is satisfied : 1 the map x w(x) C 2 x 2 is concave in R d, 2 D 2 w C I d in the sense of distributions, 3 p q, x y C x y 2 for any x, y R d, t [0, T ], p D x + w(x) and q D x + w(y), where D x + w denotes the super-differential of w with respect to the x variable, namely { } D x + w(x) = p R d w(y) w(x) p, y x ; lim sup 0 y x y x In particular, a semi-concave function is locally Lipschitz continuous in R d. P. Cardaliaguet (Paris-Dauphine) MFG 12 / 86

15 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Lemma [Stability] Let (w n ) be a sequence of uniformly semi-concave maps on R d which point-wisely converge to a map w : R d R. Then the convergence is locally uniform and w is semi-concave. Moreover, Dw n (x) converge to Dw(x) for a.e. x R d. P. Cardaliaguet (Paris-Dauphine) MFG 13 / 86

16 First order MFG equations with nonlocal coupling Semi-concavity and HJ equations Semi-concavity properties of HJ equations Semi-concavity is the natural framework for HJ equations of the form (HJ) { t u Du 2 = f (t, x) in [0, T ] R d u(t, ) = g(x) for data such that sup f (, t) C 2 C, g C 2 C. t [0,T ] Proposition The unique viscosity solution u of (HJ) is semi-concave in space with modulus C = C( C) and given by the representation formula : u(x, t) = where x(s) = x + s α(τ)dτ. t T 1 inf α L 2 ([t,t ],R d ) t 2 α(s) 2 + f (s, x(s))ds + g(x(t )), P. Cardaliaguet (Paris-Dauphine) MFG 14 / 86

17 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Lemma (Optimal synthesis) Let (x, t) R d [0, T ) and x( ) be an a.c. solution of (grad eq) { x (s) = D x u(s, x(s)) a.e. in [t, T ] x(t) = x Then the control α := x is optimal for the problem T 1 inf α L 2 ([t,t ],R d ) t 2 α(s) 2 + f (s, x(s))ds + g(x(t )) If u(, t) is differentiable at x, then equation (grad eq) has a unique solution, corresponding to the optimal trajectory. In particular, for a.e. x R d, equation (grad eq) has a unique solution. P. Cardaliaguet (Paris-Dauphine) MFG 15 / 86

18 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations We denote by Φ = Φ(x, t, s) the flow of (grad eq) : (grad eq) { s Φ(x, t, s) = D xu(s, Φ(x, t, s)) a.e. in [t, T ] Φ(x, t, t) = x Remark : Φ(x, t, s) is defined for a.e. x R d. Lemma The flow Φ has the semi-group property Φ(x, t, s ) = Φ(Φ(x, t, s), s, s ) t s s T P. Cardaliaguet (Paris-Dauphine) MFG 16 / 86

19 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Lemma (contraction property of the flow) There is a constant C = C( C) such that x y C Φ(x, t, s) Φ(y, t, s) 0 t < s T, x, y R d. In particular the map x Φ(x, s, t) has a Lipschitz continuous inverse on the set Φ(R d, t, s). P. Cardaliaguet (Paris-Dauphine) MFG 17 / 86

20 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Proof : Let x(τ) = Φ(x, t, s τ) and y(τ) = Φ(y, t, s τ). Then x( ) and y( ) solve x (τ) = D x u(x(τ), s τ), y (τ) = D x u(y(τ), s τ) on [0, s t) with initial condition x(0) = Φ(x, t, s) and y(0) = Φ(y, t, s). Hence ( ) d 1 (x y)(τ) 2 = (x y )(τ), (x y)(τ) C (x y)(τ) 2 dτ 2 since u(τ, ) is C semiconcave. Therefore x(0) y(0) e C(s τ) (x y)(τ) τ [0, s t], which proves the claim. P. Cardaliaguet (Paris-Dauphine) MFG 18 / 86

21 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Proof : Let x(τ) = Φ(x, t, s τ) and y(τ) = Φ(y, t, s τ). Then x( ) and y( ) solve x (τ) = D x u(x(τ), s τ), y (τ) = D x u(y(τ), s τ) on [0, s t) with initial condition x(0) = Φ(x, t, s) and y(0) = Φ(y, t, s). Hence ( ) d 1 (x y)(τ) 2 = (x y )(τ), (x y)(τ) C (x y)(τ) 2 dτ 2 since u(τ, ) is C semiconcave. Therefore x(0) y(0) e C(s τ) (x y)(τ) τ [0, s t], which proves the claim. P. Cardaliaguet (Paris-Dauphine) MFG 18 / 86

22 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations Proof : Let x(τ) = Φ(x, t, s τ) and y(τ) = Φ(y, t, s τ). Then x( ) and y( ) solve x (τ) = D x u(x(τ), s τ), y (τ) = D x u(y(τ), s τ) on [0, s t) with initial condition x(0) = Φ(x, t, s) and y(0) = Φ(y, t, s). Hence ( ) d 1 (x y)(τ) 2 = (x y )(τ), (x y)(τ) C (x y)(τ) 2 dτ 2 since u(τ, ) is C semiconcave. Therefore x(0) y(0) e C(s τ) (x y)(τ) τ [0, s t], which proves the claim. P. Cardaliaguet (Paris-Dauphine) MFG 18 / 86

23 Outline First order MFG equations with nonlocal coupling The Kolmogorov equation 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 19 / 86

24 First order MFG equations with nonlocal coupling The Kolmogorov equation Fix u = u(t, x) the solution of (HJ). Recall that D 2 u C and Du L where C = C( C). We study the Kolmogorov equation (K ) { t µ div (µdu) = 0 in R d (0, T ) µ(x, 0) = m 0 (x) in R d Goal : show that (K ) has a unique solution given by the flow Φ. P. Cardaliaguet (Paris-Dauphine) MFG 20 / 86

25 First order MFG equations with nonlocal coupling The Kolmogorov equation Definition Let µ L 1 loc ([0, T ] Rd ). Then µ is a weak solution to (K ) { t µ div (µdu) = 0 in R d (0, T ) µ(x, 0) = m 0 (x) in R d if, for any ϕ C c ([0, T ) R d ), T 0 ( t ϕ + Du, Dϕ ) µ dxdt = ϕ(0, x)m 0 (x)dx R d R d P. Cardaliaguet (Paris-Dauphine) MFG 21 / 86

26 First order MFG equations with nonlocal coupling The Kolmogorov equation Pull-back of a measure : Let µ be a Borel probability measure on R d and Ψ : R d R d be a Borel map. The measure ν := Ψ µ is defined by on of the equivalent statements : f (x)dν(x) = f (Ψ(x))dµ(x) for any f Cb 0(Rd ). R d R d ν(e) = µ(ψ 1 (E)) for any Borel set E R d. The Monge-Kantorovitch distance : Let P 1 (R d ) be the set of Borel probability measures m on R d such that x dm(x) < +. R d The Monge-Kantorovitch distance between measures m, m P 1 (R d ) is given by { } d 1 (m, m ) = sup h d(m m ). Q where the supremum is taken over the maps h : R d R which are 1 Lipschitz continuous. P. Cardaliaguet (Paris-Dauphine) MFG 22 / 86

27 First order MFG equations with nonlocal coupling The Kolmogorov equation Pull-back of a measure : Let µ be a Borel probability measure on R d and Ψ : R d R d be a Borel map. The measure ν := Ψ µ is defined by on of the equivalent statements : f (x)dν(x) = f (Ψ(x))dµ(x) for any f Cb 0(Rd ). R d R d ν(e) = µ(ψ 1 (E)) for any Borel set E R d. The Monge-Kantorovitch distance : Let P 1 (R d ) be the set of Borel probability measures m on R d such that x dm(x) < +. R d The Monge-Kantorovitch distance between measures m, m P 1 (R d ) is given by { } d 1 (m, m ) = sup h d(m m ). Q where the supremum is taken over the maps h : R d R which are 1 Lipschitz continuous. P. Cardaliaguet (Paris-Dauphine) MFG 22 / 86

28 First order MFG equations with nonlocal coupling The Kolmogorov equation Pull-back of a measure : Let µ be a Borel probability measure on R d and Ψ : R d R d be a Borel map. The measure ν := Ψ µ is defined by on of the equivalent statements : f (x)dν(x) = f (Ψ(x))dµ(x) for any f Cb 0(Rd ). R d R d ν(e) = µ(ψ 1 (E)) for any Borel set E R d. The Monge-Kantorovitch distance : Let P 1 (R d ) be the set of Borel probability measures m on R d such that x dm(x) < +. R d The Monge-Kantorovitch distance between measures m, m P 1 (R d ) is given by { } d 1 (m, m ) = sup h d(m m ). Q where the supremum is taken over the maps h : R d R which are 1 Lipschitz continuous. P. Cardaliaguet (Paris-Dauphine) MFG 22 / 86

29 First order MFG equations with nonlocal coupling The Kolmogorov equation Lemma Any weak solution µ of (K ) { t µ div (µdu) = 0 in R d (0, T ) µ(x, 0) = m 0 (x) in R d such that µ(t) P 1 (R d ) for any t, satisfies d 1 (µ(t 1 ), µ(t 2 )) Du t 2 t 1 P. Cardaliaguet (Paris-Dauphine) MFG 23 / 86

30 First order MFG equations with nonlocal coupling The Kolmogorov equation Proof : Fix 0 < t 1 < t 2 T and let h Cc (R d ) be 1 Lipschitz continuous. Let ε > 0 and (t t 1 )h(x)/ε if t [t 1, t 1 + ε] h(x) if t [t ϕ ε (t, x) = 1 + ε, t 2 ε] (t 2 t)h(x)/ε if t [t 2 ε, t 2 ] 0 otherwise As we have T 0 t1 +ε t 1 ( t ϕ ε + Du, Dϕ ε ) µ = ϕ ε (0, x)m 0 (x)dx R d R d R d hµ ε + t2 t 1 R d Du, Dh µ + t2 t 2 ε R d hµ ε 0 P. Cardaliaguet (Paris-Dauphine) MFG 24 / 86

31 First order MFG equations with nonlocal coupling The Kolmogorov equation Letting ε 0 gives for a.e. 0 < t 1 < t 2 < T : R d h(µ(t 1 ) µ(t 2 )) + t2 t 1 Du, Dh µ = 0. R d So R d h(µ(t 1 ) µ(t 2 )) Du Dh t 2 t 1. Take the sup over h : d 1 (µ(t 1 ), µ(t 2 )) Du t 2 t 1. P. Cardaliaguet (Paris-Dauphine) MFG 25 / 86

32 First order MFG equations with nonlocal coupling The Kolmogorov equation Letting ε 0 gives for a.e. 0 < t 1 < t 2 < T : R d h(µ(t 1 ) µ(t 2 )) + t2 t 1 Du, Dh µ = 0. R d So R d h(µ(t 1 ) µ(t 2 )) Du Dh t 2 t 1. Take the sup over h : d 1 (µ(t 1 ), µ(t 2 )) Du t 2 t 1. P. Cardaliaguet (Paris-Dauphine) MFG 25 / 86

33 First order MFG equations with nonlocal coupling The Kolmogorov equation Recall that Φ = Φ(t, x, s) is the flow of (grad eq) : (grad eq) { s Φ(t, x, s) = D xu(s, Φ(t, x, s)) a.e. in [t, T ] Φ(t, x, t) = x Assume that m 0 has compact support contained in B(0, C). Lemma There is C = C( C) such that, for any s [0, T ], the measure µ(s) := Φ(, 0, s) m 0 has a support contained in the ball B(0, C), is absolutely continuous, and its density µ(s, ) satisfies µ(s, ) L C. P. Cardaliaguet (Paris-Dauphine) MFG 26 / 86

34 First order MFG equations with nonlocal coupling The Kolmogorov equation Proof Since D x u C and Spt(m 0 ) B(0, C), the support of (µ(s)) is contained in B(0, R) where R = C + TC. Fix s [0, T ]. From the contraction Lemma, the map x Φ(x, 0, s) has a C Lipschitz continuous inverse Ψ(s, ) on the set Φ(R d, 0, s). If E is a Borel subset of R d, µ(s, E) = m 0 (Φ 1 (, 0, s)(e)) = m 0 (Ψ(s, E)) m 0 L d (Ψ(s, E)) m 0 CL d (E). Therefore µ(s) is a.c. with a density µ(t, ) satisfying µ(t, ) m 0 C t [0, T ]. P. Cardaliaguet (Paris-Dauphine) MFG 27 / 86

35 First order MFG equations with nonlocal coupling The Kolmogorov equation Proof Since D x u C and Spt(m 0 ) B(0, C), the support of (µ(s)) is contained in B(0, R) where R = C + TC. Fix s [0, T ]. From the contraction Lemma, the map x Φ(x, 0, s) has a C Lipschitz continuous inverse Ψ(s, ) on the set Φ(R d, 0, s). If E is a Borel subset of R d, µ(s, E) = m 0 (Φ 1 (, 0, s)(e)) = m 0 (Ψ(s, E)) m 0 L d (Ψ(s, E)) m 0 CL d (E). Therefore µ(s) is a.c. with a density µ(t, ) satisfying µ(t, ) m 0 C t [0, T ]. P. Cardaliaguet (Paris-Dauphine) MFG 27 / 86

36 First order MFG equations with nonlocal coupling The Kolmogorov equation Proof Since D x u C and Spt(m 0 ) B(0, C), the support of (µ(s)) is contained in B(0, R) where R = C + TC. Fix s [0, T ]. From the contraction Lemma, the map x Φ(x, 0, s) has a C Lipschitz continuous inverse Ψ(s, ) on the set Φ(R d, 0, s). If E is a Borel subset of R d, µ(s, E) = m 0 (Φ 1 (, 0, s)(e)) = m 0 (Ψ(s, E)) m 0 L d (Ψ(s, E)) m 0 CL d (E). Therefore µ(s) is a.c. with a density µ(t, ) satisfying µ(t, ) m 0 C t [0, T ]. P. Cardaliaguet (Paris-Dauphine) MFG 27 / 86

37 First order MFG equations with nonlocal coupling The Kolmogorov equation Proof Since D x u C and Spt(m 0 ) B(0, C), the support of (µ(s)) is contained in B(0, R) where R = C + TC. Fix s [0, T ]. From the contraction Lemma, the map x Φ(x, 0, s) has a C Lipschitz continuous inverse Ψ(s, ) on the set Φ(R d, 0, s). If E is a Borel subset of R d, µ(s, E) = m 0 (Φ 1 (, 0, s)(e)) = m 0 (Ψ(s, E)) m 0 L d (Ψ(s, E)) m 0 CL d (E). Therefore µ(s) is a.c. with a density µ(t, ) satisfying µ(t, ) m 0 C t [0, T ]. P. Cardaliaguet (Paris-Dauphine) MFG 27 / 86

38 First order MFG equations with nonlocal coupling The Kolmogorov equation Lemma The map s µ(s) := Φ(, 0, s) m 0 is a weak solution of (K ) { t µ div (µdu) = 0 in R d (0, T ) µ(0, x) = m 0 (x) in R d P. Cardaliaguet (Paris-Dauphine) MFG 28 / 86

39 First order MFG equations with nonlocal coupling The Kolmogorov equation Proof : Let ϕ Cc (R N [0, T )). Then d ϕ(s, x)µ(s, x) ds R d N = d ϕ(s, Φ(x, 0, s))m 0 (x)dx ds R = ( s ϕ(s, Φ(x, 0, s)) + D x ϕ(s, Φ(x, 0, s)), s Φ(x, 0, s) ) m 0 (x)dx R d = ( s ϕ(s, Φ(x, 0, s)) D x ϕ(s, Φ(x, 0, s)), D x u(s, Φ(x, 0, s)) ) m 0 (x)dx R d = ( s ϕ(s, y) D x ϕ(s, y), D x u(s, y) ) µ(s, y) R d We conclude by integrating between 0 and T. P. Cardaliaguet (Paris-Dauphine) MFG 29 / 86

40 First order MFG equations with nonlocal coupling The Kolmogorov equation Theorem Given a solution u to (HJ), the map s µ(s) := Φ(, 0, s) m 0 is the unique weak solution of (K ). Special case of Di Perna-Lions theory (1989). P. Cardaliaguet (Paris-Dauphine) MFG 30 / 86

41 First order MFG equations with nonlocal coupling The Kolmogorov equation Theorem Given a solution u to (HJ), the map s µ(s) := Φ(, 0, s) m 0 is the unique weak solution of (K ). Special case of Di Perna-Lions theory (1989). P. Cardaliaguet (Paris-Dauphine) MFG 30 / 86

42 Outline First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 31 / 86

43 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system We go back to the MFG system : (MFG) (i) t u Du 2 = F (x, m(t, )) in R d (0, T ) (ii) t m div (mdu) = 0 in R d (0, T ) (iii) m(0) = m 0, u(0, x) = G(x, m(t )) P. Cardaliaguet (Paris-Dauphine) MFG 32 / 86

44 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Standing assumptions : 1 F and G are continuous over R d P 1. 2 There is a constant C such that, for any m P 1, F(, m) C 2 C, G(, m) C 2 C m P 1, where C 2 is the space of function with continuous second order derivatives endowed with the norm f C 2 = sup x R d [ f (x) + Df (x) + D 2 f (x) ]. 3 Finally we suppose that m 0 is absolutely continuous, with a density still denoted m 0 which is bounded and has a compact support. P. Cardaliaguet (Paris-Dauphine) MFG 33 / 86

45 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Definition A pair (u, m) is a solution to the (MFG) system (MFG) (i) t u Du 2 = F (x, m(t, )) in R d (0, T ) (ii) t m div (mdu) = 0 in R d (0, T ) (iii) m(0) = m 0, u(0, x) = G(x, m(t )) if u W 1, loc ([0, T ] R d ) and m L 1 loc ([0, T ] Rd ), u is a (backward) viscosity solution to (MFG (i)) with terminal condition u(0, x) = G(x, m(t )). m is a weak solution to (MFG (ii)) with initial condition m(0) = m 0. P. Cardaliaguet (Paris-Dauphine) MFG 34 / 86

46 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system A stability property Let (m n ) be a sequence of C([0, T ], P 1 ) which uniformly converges to m C([0, T ], P 1 (R d )). Let u n be the solution to { t u n Du n 2 = F(x, m n (t)) in R d (0, T ) u n (x, T ) = g(x, m n (T )) in R d and u be the solution to { t u Du 2 = F(x, m(t)) in R d (0, T ) u(x, T ) = g(x, m(t )) in R d Let Φ n (resp. Φ) the flow associated to u n (resp. to u) as above and let us set µ n (s) = Φ n (, 0, s) m 0 and µ(s) = Φ(, 0, s) m 0. P. Cardaliaguet (Paris-Dauphine) MFG 35 / 86

47 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Lemma (Stability) The solution (u n ) locally uniformly converges u in R d [0, T ] while (µ n ) converges to µ in C([0, T ], P 1 (R d )). P. Cardaliaguet (Paris-Dauphine) MFG 36 / 86

48 Proof : First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As the F(x, m n (t)) and the g(x, m n (T )) locally uniformly converge to F (x, m(t)) and g(x, m(t )), local uniform convergence of (u n ) to u is a consequence of the standard stability of viscosity solutions. Since the u n are uniformly semi-concave, Du n converges almost everywhere in R d (0, T ) to Du. The (µ n (t)) have a uniformly bounded support (say in some compact K R d ) and are uniformly Lipschitz continuous : d 1 (µ n (t 1 ), µ n (t 2 )) Du n t 2 t 1 C t 2 t 1. and uniformly bounded in L. By Ascoli and Banach-Alaoglu, a subsequence (still denoted (µ n )) of the (µ n ) converges in C([0, T ], P 1 (K )) and in L weak-* to some m which has a support in K [0, T ], belongs to L (R d [0, T ]) and to C([0, T ], P 1 (K )). One can pass to the limit in the Kolmogorov equation to get that µ is the unique solution associated to u : µ(s) = Φ(, 0, s) m 0. P. Cardaliaguet (Paris-Dauphine) MFG 37 / 86

49 Proof : First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As the F(x, m n (t)) and the g(x, m n (T )) locally uniformly converge to F (x, m(t)) and g(x, m(t )), local uniform convergence of (u n ) to u is a consequence of the standard stability of viscosity solutions. Since the u n are uniformly semi-concave, Du n converges almost everywhere in R d (0, T ) to Du. The (µ n (t)) have a uniformly bounded support (say in some compact K R d ) and are uniformly Lipschitz continuous : d 1 (µ n (t 1 ), µ n (t 2 )) Du n t 2 t 1 C t 2 t 1. and uniformly bounded in L. By Ascoli and Banach-Alaoglu, a subsequence (still denoted (µ n )) of the (µ n ) converges in C([0, T ], P 1 (K )) and in L weak-* to some m which has a support in K [0, T ], belongs to L (R d [0, T ]) and to C([0, T ], P 1 (K )). One can pass to the limit in the Kolmogorov equation to get that µ is the unique solution associated to u : µ(s) = Φ(, 0, s) m 0. P. Cardaliaguet (Paris-Dauphine) MFG 37 / 86

50 Proof : First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As the F(x, m n (t)) and the g(x, m n (T )) locally uniformly converge to F (x, m(t)) and g(x, m(t )), local uniform convergence of (u n ) to u is a consequence of the standard stability of viscosity solutions. Since the u n are uniformly semi-concave, Du n converges almost everywhere in R d (0, T ) to Du. The (µ n (t)) have a uniformly bounded support (say in some compact K R d ) and are uniformly Lipschitz continuous : d 1 (µ n (t 1 ), µ n (t 2 )) Du n t 2 t 1 C t 2 t 1. and uniformly bounded in L. By Ascoli and Banach-Alaoglu, a subsequence (still denoted (µ n )) of the (µ n ) converges in C([0, T ], P 1 (K )) and in L weak-* to some m which has a support in K [0, T ], belongs to L (R d [0, T ]) and to C([0, T ], P 1 (K )). One can pass to the limit in the Kolmogorov equation to get that µ is the unique solution associated to u : µ(s) = Φ(, 0, s) m 0. P. Cardaliaguet (Paris-Dauphine) MFG 37 / 86

51 Proof : First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As the F(x, m n (t)) and the g(x, m n (T )) locally uniformly converge to F (x, m(t)) and g(x, m(t )), local uniform convergence of (u n ) to u is a consequence of the standard stability of viscosity solutions. Since the u n are uniformly semi-concave, Du n converges almost everywhere in R d (0, T ) to Du. The (µ n (t)) have a uniformly bounded support (say in some compact K R d ) and are uniformly Lipschitz continuous : d 1 (µ n (t 1 ), µ n (t 2 )) Du n t 2 t 1 C t 2 t 1. and uniformly bounded in L. By Ascoli and Banach-Alaoglu, a subsequence (still denoted (µ n )) of the (µ n ) converges in C([0, T ], P 1 (K )) and in L weak-* to some m which has a support in K [0, T ], belongs to L (R d [0, T ]) and to C([0, T ], P 1 (K )). One can pass to the limit in the Kolmogorov equation to get that µ is the unique solution associated to u : µ(s) = Φ(, 0, s) m 0. P. Cardaliaguet (Paris-Dauphine) MFG 37 / 86

52 Proof : First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As the F(x, m n (t)) and the g(x, m n (T )) locally uniformly converge to F (x, m(t)) and g(x, m(t )), local uniform convergence of (u n ) to u is a consequence of the standard stability of viscosity solutions. Since the u n are uniformly semi-concave, Du n converges almost everywhere in R d (0, T ) to Du. The (µ n (t)) have a uniformly bounded support (say in some compact K R d ) and are uniformly Lipschitz continuous : d 1 (µ n (t 1 ), µ n (t 2 )) Du n t 2 t 1 C t 2 t 1. and uniformly bounded in L. By Ascoli and Banach-Alaoglu, a subsequence (still denoted (µ n )) of the (µ n ) converges in C([0, T ], P 1 (K )) and in L weak-* to some m which has a support in K [0, T ], belongs to L (R d [0, T ]) and to C([0, T ], P 1 (K )). One can pass to the limit in the Kolmogorov equation to get that µ is the unique solution associated to u : µ(s) = Φ(, 0, s) m 0. P. Cardaliaguet (Paris-Dauphine) MFG 37 / 86

53 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Theorem (Lasry and Lions, 2006) There is a least one solution to the MFG system (MFG) (i) t u Du 2 = F (x, m(t, )) in R d (0, T ) (ii) t m div (mdu) = 0 in R d (0, T ) (iii) m(0) = m 0, u(t, x) = G(x, m(t )) P. Cardaliaguet (Paris-Dauphine) MFG 38 / 86

54 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Proof : Let C be the convex subset of maps m C([0, T ], P 1 ) such that m(0) = m 0. To any m C one associates the unique solution u to { t u Du 2 = F (x, m(t)) in R d (0, T ) u(x, T ) = G(x, m(t )) in R d and to this solution one associates the unique solution to the continuity equation { t m div ( m)du) = 0 in R d (0, T ) m(0) = m 0 Then m C and, from the stability Lemma, the mapping m m is continuous. It is also compact because there is a constant C = C( C) such that, for any s [0, T ], m(s) has a support in B(0, C) and satisfies d 1 ( m(t 1 ), m(t 2)) C t 1 t 2 t 1, t 2 [0, T ]. One completes the proof by Schauder fix point Theorem. P. Cardaliaguet (Paris-Dauphine) MFG 39 / 86

55 Comments First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Semi-concavity properties of value function in optimal control are borrowed from the monograph by Cannarsa and Sinestrari For the analysis of transport equations with discontinuous vector fields, see, in particular, Di Perna-Lions, 1989, for vector fields in Sobolev spaces, Ambrosio, 2004, for vector fields in BV spaces. P. Cardaliaguet (Paris-Dauphine) MFG 40 / 86

56 First order MFG equations with nonlocal coupling Uniqueness Existence/uniqueness of a solution for the MFG system Assume furthermore that F and G satisfy the monotonicity condition : R d (F (x, m 1 ) F(x, m 2 ))d(m 1 m 2 )(x) 0 m 1, m 2 P 1, and R d (G(x, m 1 ) G(x, m 2 ))d(m 1 m 2 )(x) 0 m 1, m 2 P 1. Theorem Under the above conditions, the (MFG) system has a unique solution. P. Cardaliaguet (Paris-Dauphine) MFG 41 / 86

57 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Proof : Let (u 1, m 1 ) and (u 2, m 2 ) two solutions of (MFG). Set ũ = u 1 u 2 and m = m 1 m 2. Then ũ solves (1) t ũ while m solves We compute d ũ(t) m(t) = dt R d Note that m 2 ( Du1 2 Du 2 2) (F(x, m 1 ) F(x, m 2 )) = 0 (2) t m div (m 1 Du 1 m 2 Du 2 ) = 0. = ( t ũ(t)) m(t) + ũ(t)( t m(t)) R d ( 1 ( Du1 2 Du 2 2) ) (F(x, m 1 ) F (x, m 2 )) m R 2 d Dũ, (m 1 Du 1 m 2 Du 2 ) R d ( Du1 2 Du 2 2) Dũ, m 1 Du 1 m 2 Du 2 = (m 1 + m 2 ) Du 1 Du P. Cardaliaguet (Paris-Dauphine) MFG 42 / 86

58 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Proof : Let (u 1, m 1 ) and (u 2, m 2 ) two solutions of (MFG). Set ũ = u 1 u 2 and m = m 1 m 2. Then ũ solves (1) t ũ while m solves We compute d ũ(t) m(t) = dt R d Note that m 2 ( Du1 2 Du 2 2) (F(x, m 1 ) F(x, m 2 )) = 0 (2) t m div (m 1 Du 1 m 2 Du 2 ) = 0. = ( t ũ(t)) m(t) + ũ(t)( t m(t)) R d ( 1 ( Du1 2 Du 2 2) ) (F(x, m 1 ) F (x, m 2 )) m R 2 d Dũ, (m 1 Du 1 m 2 Du 2 ) R d ( Du1 2 Du 2 2) Dũ, m 1 Du 1 m 2 Du 2 = (m 1 + m 2 ) Du 1 Du P. Cardaliaguet (Paris-Dauphine) MFG 42 / 86

59 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As we get Integrate over [0, T ] : As R d R d (F(x, m 1 ) F (x, m 2 )) m 0, d (m 1 + m 2 ) ũ(t) m(t) Du 1 Du 2 2 dt R d R 2 d m(t )ũ(t ) R d T (m 1 + m 2 ) m(0)ũ(0) 0 R 2 d Du 1 Du 2 2 m(t )ũ(t ) = (G(x, m 1 ) G(x, m 2 ))(m 1 m 2 ) 0 R d R d while m(0) = 0, we get T (m 1 + m 2 ) 0 Du 1 Du R 2 d P. Cardaliaguet (Paris-Dauphine) MFG 43 / 86

60 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As we get Integrate over [0, T ] : As R d R d (F(x, m 1 ) F (x, m 2 )) m 0, d (m 1 + m 2 ) ũ(t) m(t) Du 1 Du 2 2 dt R d R 2 d m(t )ũ(t ) R d T (m 1 + m 2 ) m(0)ũ(0) 0 R 2 d Du 1 Du 2 2 m(t )ũ(t ) = (G(x, m 1 ) G(x, m 2 ))(m 1 m 2 ) 0 R d R d while m(0) = 0, we get T (m 1 + m 2 ) 0 Du 1 Du R 2 d P. Cardaliaguet (Paris-Dauphine) MFG 43 / 86

61 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system As we get Integrate over [0, T ] : As R d R d (F(x, m 1 ) F (x, m 2 )) m 0, d (m 1 + m 2 ) ũ(t) m(t) Du 1 Du 2 2 dt R d R 2 d m(t )ũ(t ) R d T (m 1 + m 2 ) m(0)ũ(0) 0 R 2 d Du 1 Du 2 2 m(t )ũ(t ) = (G(x, m 1 ) G(x, m 2 ))(m 1 m 2 ) 0 R d R d while m(0) = 0, we get T (m 1 + m 2 ) 0 Du 1 Du R 2 d P. Cardaliaguet (Paris-Dauphine) MFG 43 / 86

62 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system So Du 1 = Du 2 in {m 1 > 0} {m 2 > 0} Hence m 1 is a solution of { t µ div (µdu 2 ) = 0 in R d (0, T ) µ(x, 0) = m 0 (x) in R d By uniqueness, m 1 = m 2. Then u 1 = u 2 because u 1 and u 2 solve the same (HJ) equation. P. Cardaliaguet (Paris-Dauphine) MFG 44 / 86

63 First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system Conclusion for first order, nonlocal (MFG) systems The existence/uniqueness theory can be easily generalized to general Hamiltonians, Stability of solutions can be proved along the same lines, Other existence proof : by vanishing viscosity, Numerical approximation of the solution : Camilli and Silva (preprint, 2012) P. Cardaliaguet (Paris-Dauphine) MFG 45 / 86

64 Outline First order MFG systems with local coupling 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 46 / 86

65 First order MFG systems with local coupling The Mean Field Game system We now study the MFG system (i) t u + H(x, Du) = f (x, m(x, t)) (MFG) (ii) t m div(md p H(x, Du)) = 0 (iii) m(0) = m 0, u(x, T ) = u T (x) where f : R d [0, + ) [0, + ) is a local coupling term. Problem : Existence/uniqueness of a solutions - smoothness properties. P. Cardaliaguet (Paris-Dauphine) MFG 47 / 86

66 First order MFG systems with local coupling Similar systems Adjoint methods for Hamilton-Jacobi PDE : analysis of the vanishing viscosity limit : t u ε + H(Du ε ) = ɛ u ε, ) t m ε div (DH(Du ε )m ε = ɛ m ε, u ε (0, x) = u 0 (x), m ε (1,.) = m 1. better understand the convergence of the vanishing viscosity method when H is nonconvex. (Evans (2010)) P. Cardaliaguet (Paris-Dauphine) MFG 48 / 86

67 First order MFG systems with local coupling A congestion model : An optimal transport problem related to congestion yields to the following system of PDEs : for α (0, 1), t u + α 4 mα 1 Du 2 = 0, 1 ) t m + div( 2 mα Du = 0, m(0,.) = m 0, m(1,.) = m 1. (Benamou-Brenier formulation of Wasserstein distance : α = 1 - Dolbeault-Nazaret-Savaré, 2009) analysis of the system : Existence and uniqueness of a weak solution (C.-Carlier-Nazaret 2012.) P. Cardaliaguet (Paris-Dauphine) MFG 49 / 86

68 First order MFG systems with local coupling Back to the MFG system (MFG) (i) t u + H(x, Du) = f (x, m(x, t)) (ii) t m div(md p H(x, Du)) = 0 (iii) m(0) = m 0, u(x, T ) = u T (x) under the assumptions that f : R d [0, + ) [0, + ) is a local coupling term, strictly increasing in m with f (x, 0) = 0, H is uniformly convex in p, all the data are Z d periodic in space : denoted by. P. Cardaliaguet (Paris-Dauphine) MFG 50 / 86

69 First order MFG systems with local coupling Difficulties for the MFG with local coupling For nonlocal coupling, existence/regularity of the MFG system come from semi-concavity estimates for HJ equations with C 2 RHS, preservation of the absolute continuity of m 0 by Kolmogorov equation for gradient flows of semi-concave functions. For local equations, the RHS is a priori only measurable (or integrable), the HJ with discontinuous RHS is poorly understood, the Kolmogorov equation with Du L is ill-posed as well. requires a completely different approach. P. Cardaliaguet (Paris-Dauphine) MFG 51 / 86

70 First order MFG systems with local coupling Difficulties for the MFG with local coupling For nonlocal coupling, existence/regularity of the MFG system come from semi-concavity estimates for HJ equations with C 2 RHS, preservation of the absolute continuity of m 0 by Kolmogorov equation for gradient flows of semi-concave functions. For local equations, the RHS is a priori only measurable (or integrable), the HJ with discontinuous RHS is poorly understood, the Kolmogorov equation with Du L is ill-posed as well. requires a completely different approach. P. Cardaliaguet (Paris-Dauphine) MFG 51 / 86

71 First order MFG systems with local coupling Difficulties for the MFG with local coupling For nonlocal coupling, existence/regularity of the MFG system come from semi-concavity estimates for HJ equations with C 2 RHS, preservation of the absolute continuity of m 0 by Kolmogorov equation for gradient flows of semi-concave functions. For local equations, the RHS is a priori only measurable (or integrable), the HJ with discontinuous RHS is poorly understood, the Kolmogorov equation with Du L is ill-posed as well. requires a completely different approach. P. Cardaliaguet (Paris-Dauphine) MFG 51 / 86

72 First order MFG systems with local coupling Ideas for the MFG system Look at the MFG system as a quasilinear elliptic equation for u in time-space. or/and as necessary condition for optimal control problems. This is the case For an optimal control problem for an Hamilton-Jacobi equation For an optimal control problem for a Kolmogorov equation The second one being the dual of the first one. P. Cardaliaguet (Paris-Dauphine) MFG 52 / 86

73 Outline First order MFG systems with local coupling The quasilinear elliptic equation 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 53 / 86

74 First order MFG systems with local coupling The quasilinear elliptic equation For simplicity of presentation, we assume that H = H(p), f = f (m). The MFG system becomes (i) t u + H(Du) = f (m(x, t)) (MFG) (ii) t m div(mh (Du)) = 0 (iii) m(0) = m 0, u(x, T ) = u T (x) So m(x, t) = f 1 ( t u + H(Du)) P. Cardaliaguet (Paris-Dauphine) MFG 54 / 86

75 First order MFG systems with local coupling The quasilinear elliptic equation Therefore 0 = t m div(mh (Du)) = t ( f 1 ( t u + H(Du)) ) div ( (f 1 ( t u + H(Du)) H (Du) ) = (f 1 ) (... ) ( tt u + 2 D t u, H (Du) D 2 uh (Du), H (Du) ) f 1 ( )Tr ( H (Du)D 2 u ) = Tr ( A(D t,x u)d 2 t,xu ) where A(D t,x u) = ( (f 1 ) 1 H ( t u + H(Du)) (Du) T ( H (Du) H )(Du) H (Du) +f ( t u + H(Du)) 0 H 0 (Du) ) (because (f 1 ) > 0 and f 1 ( t u + H(Du)) = m 0) P. Cardaliaguet (Paris-Dauphine) MFG 55 / 86

76 First order MFG systems with local coupling The quasilinear elliptic equation The MFG systems reduces to the quasilinear elliptic equation Tr ( A(D t,x u)dt,xu 2 ) = 0 in (0, 1) T d t u + H(Du) = f (m 0 ) at t = 0 u(t, ) = u T at t = T P. Cardaliaguet (Paris-Dauphine) MFG 56 / 86

77 First order MFG systems with local coupling A priori estimates The quasilinear elliptic equation 1 The map Φ(t, x) := t u + H(Du) is bounded above. Indeed : Φ satisfies an equation of the form Tr(A(D t,x Φ) B. φ 0 in (0, 1) T d So max Φ reached at the boundary, Φ(0, ) = f (m 0 ) is bounded, Φ(T, x) is bounded (barrier argument). 2 Du is bounded (Bernstein method) P. Cardaliaguet (Paris-Dauphine) MFG 57 / 86

78 First order MFG systems with local coupling The quasilinear elliptic equation Theorem (Lasry-Lions, 2009) 1 Existence/uniqueness of solution for the MFG system with u W 1, and m L, 2 If f (m) log(m) at 0, then the solution is smooth. P. Cardaliaguet (Paris-Dauphine) MFG 58 / 86

79 Outline First order MFG systems with local coupling MFG as optimality conditions 1 First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations The Kolmogorov equation Existence/uniqueness of a solution for the MFG system 2 First order MFG systems with local coupling The quasilinear elliptic equation MFG as optimality conditions P. Cardaliaguet (Paris-Dauphine) MFG 59 / 86

80 First order MFG systems with local coupling Some assumptions MFG as optimality conditions We study the MFG system (i) t u + H(x, Du) = f (x, m(x, t)) (MFG) (ii) t m div(md p H(x, Du)) = 0 (iii) m(0) = m 0, u(x, T ) = u T (x) under the following conditions : f : R d [0, + ) R is smooth and increasing w.r. to m with f (x, 0) = 0, with C + 1 C m q 1 f (x, m) C(1 + m q 1 ) (where q > 1). There is r > d(q 1) such that 1 C ξ r C H(x, ξ) C( ξ r + 1) (x, ξ) R d R d. + technical conditions of D x H... P. Cardaliaguet (Paris-Dauphine) MFG 60 / 86

81 First order MFG systems with local coupling MFG as optimality conditions The optimal control of HJ equation { T inf 0 Q } F (x, α(t, x)) dxdt u(0, x)dm 0 (x) Q where u is the solution to the HJ equation { t u + H(x, Du) = α in (0, T ) R d u(t, ) = u T in R d and F (x, a) = sup(am F(x, m)) where m R m f (x, m F (x, m) = )dm if m otherwise Note that F (x, a) = 0 for a 0. P. Cardaliaguet (Paris-Dauphine) MFG 61 / 86

82 First order MFG systems with local coupling MFG as optimality conditions The optimal control of Kolmogorov equation { T inf 0 Q } mh (x, v) + F(x, m) dxdt + u T (x)m(t, x)dx Q where the infimum is taken over the pairs (m, v) such that in the sense of distributions. t m + div(mv) = 0, m(0) = m 0 P. Cardaliaguet (Paris-Dauphine) MFG 62 / 86

83 First order MFG systems with local coupling MFG as optimality conditions Aim : Provide a framework in which both problems are well-posed and in duality, Write the MFG system as optimality conditions. Requires estimates on HJ (HJ) { t u + H(x, Du) = α in (0, T ) R d u(t, x) = u T (x) in R d where α is in some L p. P. Cardaliaguet (Paris-Dauphine) MFG 63 / 86

84 First order MFG systems with local coupling Estimates for HJB equations MFG as optimality conditions Assume that p > 1 + d/r and r > 1 such that 1 C ξ r C H(x, ξ) C( ξ r + 1) (x, ξ) R d R d. and u be a subsolution to t u + H(x, Du) α in (0, T ) R d Lemma (Upper bounds) There is a universal constant C such that u(t 1, x) u(t 2, x) + C(t 2 t 1 ) p 1 d/r p+d/r (α) + p for any 0 t 1 < t 2 T (where p 1 d/r > 0 by assumption). P. Cardaliaguet (Paris-Dauphine) MFG 64 / 86

85 First order MFG systems with local coupling MFG as optimality conditions Theorem (Regularity, C.-Silvestre, 2012) Let u be a bounded viscosity solution of { t u + H(x, Du) = α in (0, T ) R d u(t, x) = u T (x) in R d where α 0, α L p with p > 1 + d/r. Then, for any δ > 0, u is Hölder continuous in [0, T δ] R d : u(t, x) u(s, y) C (t, x) (s, y) γ where γ = γ( u, α p, d, r), C = C( u, α p, d, r, δ). P. Cardaliaguet (Paris-Dauphine) MFG 65 / 86

86 First order MFG systems with local coupling MFG as optimality conditions Related results Capuzzo Dolcetta-Leoni-Porretta (2010), Barles (2010) : stationary equations, bounded RHS, C. (2009), Cannarsa-C. (2010), C. Rainer (2011) : evolution equations, bounded RHS, Dall Aglio-Porretta (preprint) : stationary setting, unbounded RHS, C.-Silvestre (2012) : evolution equations, unbounded RHS. P. Cardaliaguet (Paris-Dauphine) MFG 66 / 86

87 Summary First order MFG systems with local coupling MFG as optimality conditions Let (u, α) solve { t u + H(x, Du) = α in (0, T ) R d u(t, ) = u T in R d with α = α(t, x) 0. Then (Upper bound) for any 0 t 1 < t 2 T. u(t 1, ) u(t 2, ) + C(t 2 t 1 ) p 1 d/r d+d/r α p a.e. (Lower bound) u(t, x) u T (x) C(T t). (Regularity) u is locally Hölder continuous in [0, T ) R d with a modulus depending only on α p and u T. P. Cardaliaguet (Paris-Dauphine) MFG 67 / 86

88 First order MFG systems with local coupling Back to the optimal control of HJB MFG as optimality conditions We study the optimal control of the HJ equation : { T } (HJ Pb) inf F (x, α(t, x)) dxdt u(0, x)dm 0 (x) 0 Q Q where u is the solution to the HJ equation { t u + H(x, Du) = α in (0, T ) R d u(t, ) = u T in R d Proposition If (u n, α n ) is a minimizing sequence, then the (α n ) are bounded in L p. the (u n ) are uniformly continuous in [0, T ] R d. (Integral bounds) Du n is bounded in L r and ( tu n ) is bounded in L 1. P. Cardaliaguet (Paris-Dauphine) MFG 68 / 86

89 First order MFG systems with local coupling The relaxed problem MFG as optimality conditions Let K be the set (u, α) BV ((0, T ) R d ) L p ((0, T ) Rd ) such that Du L r ((0, T ) Rd ) α 0 a.e. u(t, x) = u T (x) and t u + H(x, Du) α in (0, T ) R d. holds in the sense of distribution. Note that K is a convex set. The relaxed problem is (HJ Rpb) inf (u,α) K { T 0 Q } F (x, α(t, x)) dxdt u(0, x)dm 0 (x) Q P. Cardaliaguet (Paris-Dauphine) MFG 69 / 86

90 First order MFG systems with local coupling MFG as optimality conditions Theorem The relaxed problem has (HJ-Rpb) at least one minimum (u, α), where u is continuous and satisfies in the viscosity sense t u + H(x, Du) 0 in (0, T ) R d The value of the relaxed problem (HJ-Rpb) is equal to the value of the optimal control of the HJ equation (HJ-Pb). P. Cardaliaguet (Paris-Dauphine) MFG 70 / 86

91 First order MFG systems with local coupling MFG as optimality conditions The optimal control problem of HJ equation (HJ-Pb) can be rewritten as { T } inf F (x, t u + H(x, Du)) dxdt u(0, x)dm 0 (x) 0 Q Q with constraint u(t, ) = u T. This is a convex problem. Suggests a duality approach. P. Cardaliaguet (Paris-Dauphine) MFG 71 / 86

92 First order MFG systems with local coupling MFG as optimality conditions The Fenchel-Rockafellar duality Theorem Let E and F be two normed spaces, Λ L c (E, F ) and F : E R {+ } and G : F R {+ } be two lsc proper convex maps. Theorem Under the qualification condition : there is x 0 such that F(x 0 ) < + and G continuous at Λ(x 0 ), we have inf {F(x) + G(Λ(x))} = max { F (Λ (y )) G ( y )} x E y F as soon as the LHS is finite. P. Cardaliaguet (Paris-Dauphine) MFG 72 / 86

93 First order MFG systems with local coupling MFG as optimality conditions Remark : If x and ȳ are respectively optimal in inf {F(x) + G(Λ(x))} and sup { F (Λ (y )) G ( y )} x E y F then and i.e., and F( x) + F (Λ (ȳ )) = Λ (ȳ ), x G(Λ( x)) + G ( ȳ ) = ȳ, Λ( x) Λ (ȳ ) F( x) Λ( x) G ( ȳ ) P. Cardaliaguet (Paris-Dauphine) MFG 73 / 86

94 First order MFG systems with local coupling MFG as optimality conditions The optimal control problem of HJ equation (HJ-Pb) { T } inf F (x, t u + H(x, Du)) dxdt u(0, x)dm 0 (x) 0 Q Q (with constraint u(t, ) = u T ) can be rewritten as where E = C 1 ([0, T ] R d ), inf {F(u) + G(Λ(u))} u E F = C 0 ([0, T ] R d, R) C 0 ([0, T ] R d, R d ), F(u) = m 0 (x)u(0, x)dx if u(t, ) = u T (+ otherwise). G(a, b) = Q T 0 Q Λ(u) = ( t u, Du). F (x, a(t, x) + H(x, b(t, x))) dxdt. P. Cardaliaguet (Paris-Dauphine) MFG 74 / 86

95 First order MFG systems with local coupling MFG as optimality conditions Then F is convex and lower semi-continuous on E while G is convex and continuous on F. Moreover Λ is bounded and linear. The qualification condition is satisfied by u(t, x) = u T (x). By Fenchel-Rockafellar duality theorem we have inf {F(u) + G(Λ(u))} = max { F (Λ (m, w)) G ( (m, w))} u E (m,w) F where F is the set of Radon measures (m, w) M ([0, T ] R d, R R d ) and F and G are the convex conjugates of F and G. P. Cardaliaguet (Paris-Dauphine) MFG 75 / 86

96 First order MFG systems with local coupling MFG as optimality conditions Recall that F(u) = m 0 (x)u(0, x)dx if u(t, ) = u T (+ otherwise) Q and G(a, b) = T 0 Q F (x, a(t, x) + H(x, b(t, x))) dxdt. Lemma F (Λ u (m, w)) = T (x)dm(t, x) if t m + div(w) = 0, m(0) = m 0 Q + otherwise and T G F(x, m) mh (m, w) = (x, w 0 Q m ) dtdx if (m, w) L1 + otherwise P. Cardaliaguet (Paris-Dauphine) MFG 76 / 86

97 First order MFG systems with local coupling MFG as optimality conditions Consequence of the Lemma : max { F (Λ (m, w)) G(m, w)} (m,w) F { T = max F(x, m) mh (x, w } m ) dtdx u T (x)m(t, x) dx 0 Q Q where the maximum is taken over the L 1 maps (m, w) such that m 0 a.e. and t m + div(w) = 0, m(0) = m 0 P. Cardaliaguet (Paris-Dauphine) MFG 77 / 86

98 First order MFG systems with local coupling MFG as optimality conditions Idea of proof : F (Λ (m, w)) = sup Λ (m, w), u F(u) u(t )=u T = sup (m, w), Λ(u) + m 0 (x)u(0, x)dx u(t )=u T Q T = sup (m t u + w, Du ) + m 0 (x)u(0, x)dx u(t )=u T 0 Q Q T = sup u( t m + div(w)) u(t )=u T 0 Q + m(t )u T + (m 0 m(0))u(0)dx Q Q u = T (x)dm(t, x) if t m + div(w) = 0, m(0) = m 0 Q + otherwise P. Cardaliaguet (Paris-Dauphine) MFG 78 / 86

99 First order MFG systems with local coupling MFG as optimality conditions The dual of the optimal control of HJ eqs Theorem The dual of the optimal control of HJ (HJ-Pb) equation is given by { T (K Pb) inf mh (x, w } m ) + F (x, m) dxdt + u T (x)m(t, x)dx 0 Q where the infimum is taken over the pairs (m, w) L 1 ((0, T ) Rd ) L 1 ((0, T ) Rd, R d ) such that t m + div(w) = 0, m(0) = m 0 in the sense of distributions. Moreover this dual problem has a unique minimum. Q P. Cardaliaguet (Paris-Dauphine) MFG 79 / 86

100 First order MFG systems with local coupling MFG as optimality conditions (K-Pb) as an optimal control problem for Kolmogorov equation : Set v = w/m. Then (K-Pb) becomes { T } inf mh (x, v) + F(x, m) dxdt + u T (x)m(t, x)dx 0 Q Q where the infimum is taken over the pairs (m, v) such that in the sense of distributions. t m + div(mv) = 0, m(0) = m 0 P. Cardaliaguet (Paris-Dauphine) MFG 80 / 86

101 First order MFG systems with local coupling Back to the (MFG) system MFG as optimality conditions We now define weak solutions of the (MFG) system (i) t u + H(x, Du) = f (x, m(x, t)) (MFG) (ii) t m div(md p H(x, Du)) = 0 (iii) m(0) = m 0, u(x, T ) = u T (x) and explain the relation with the two optimal control problems for the HJ equation (problem (HJ-Pb)) for the Kolmogorov equation (problem (K-Pb)) P. Cardaliaguet (Paris-Dauphine) MFG 81 / 86

102 First order MFG systems with local coupling MFG as optimality conditions Definition A pair (m, u) L 1 ((0, T ) R d ) BV ((0, T ) R d ) (with u continuous in [0, T ] R d, Du L r ((0, T ) R d, m), md p H(x, Du) L 1 ) is a weak solution of (MFG) if (i) t u + H(x, Du) f (x, m) holds in the sense of distribution, with u(t, x) = u T (x) in the sense of trace, (iii) t m div(md p H(x, Du)) = 0 holds in the sense of distribution in (0, T ) R d and m(0) = m 0, T (iv) Equality m ( t u ac Du, D p H(x, Du) ) = m(t )u T m 0 u(0) holds. 0 Q (where t u ac is the a.c. part of the measure t u). Q P. Cardaliaguet (Paris-Dauphine) MFG 82 / 86

103 First order MFG systems with local coupling MFG as optimality conditions Remarks If (i) holds with an equality and if u is in W 1,1, then (ii) implies (iii). Conditions (i) and (iii) imply that t u ac (t, x) + H(x, Du(t, x)) = f (x, m(t, x)) holds a.e. in {m > 0}. P. Cardaliaguet (Paris-Dauphine) MFG 83 / 86

104 First order MFG systems with local coupling MFG as optimality conditions Existence/uniqueness of weak solutions Theorem (Cannarsa, C., in preparation) There is a unique weak solution (m, u) of (MFG) such that u is locally Hölder continuous in [0, T ) R d and which satisfies in the viscosity sense Idea of proof : t u + H(x, Du) 0 in (0, T ) R d. Let (m, w) is a minimizer of (K-Pb) and (u, α) is a minimizer of (HJ-Rpb) such that u is continuous. Then one can show that (m, u) is a solution of mean field game system (MFG) and w = md p H(, Du) while α = f (, m) a.e.. Conversely, any solution of (MFG) such that u is continuous is such that the pair (m, md p H(, Du)) is the minimizer of (K-Pb) while (u, f (, m)) is a minimizer of (HJ-Rpb). P. Cardaliaguet (Paris-Dauphine) MFG 84 / 86

105 First order MFG systems with local coupling MFG as optimality conditions Existence/uniqueness of weak solutions Theorem (Cannarsa, C., in preparation) There is a unique weak solution (m, u) of (MFG) such that u is locally Hölder continuous in [0, T ) R d and which satisfies in the viscosity sense Idea of proof : t u + H(x, Du) 0 in (0, T ) R d. Let (m, w) is a minimizer of (K-Pb) and (u, α) is a minimizer of (HJ-Rpb) such that u is continuous. Then one can show that (m, u) is a solution of mean field game system (MFG) and w = md p H(, Du) while α = f (, m) a.e.. Conversely, any solution of (MFG) such that u is continuous is such that the pair (m, md p H(, Du)) is the minimizer of (K-Pb) while (u, f (, m)) is a minimizer of (HJ-Rpb). P. Cardaliaguet (Paris-Dauphine) MFG 84 / 86