PDE Methods for Mean Field Games with Non-Separable Hamiltonian: Data in Sobolev Spaces (Continued) David Ambrose June 29, 2018

Transcription

1 PDE Methods for Mean Field Games with Non-Separable Hamiltonian: Data in Sobolev Spaces Continued David Ambrose June 29, 218

2 Steps of the energy method Introduce an approximate problem. Prove existence of solutions for the approximate problem. Prove an estimate for these solutions which is uniform in the approximation parameter. Use this estimate to help pass to the limit as the approximation parameters vanish. Prove that the limiting solution solves the original problem. Make similar estimates for uniqueness. David Ambrose Drexel University Some Existence Theory June 29, / 26

3 The mean field games system For reference, we again write down the mean field games system we are considering: u t + εht, x, m, Du = u, 1 m t + εdivmh p t, x, m, Du = m, 2 in the context of the planning problem, i.e., with boundary conditions m, = m, ut, = u T. 3 David Ambrose Drexel University Some Existence Theory June 29, / 26

4 The approximate problems We had set up the approximate problems with two approximating parameters, n and δ. We had let P δ be a mollifier in the x variable; a good choice is truncation of the Fourier series at the level 1/δ. We mollified the data: µ n,δ, = P δ µ, w n,δ T, = P δ w T. We had set up an iterative scheme: µ n+1,δ t w n+1,δ t + εp Θt, x, µ n,δ, Dw n,δ + w n+1,δ =, + εdiv µ n,δ + mθ p t, x, µ n,δ, Dw n,δ µ n+1,δ =. We initialized with w,δ = µ,δ =. David Ambrose Drexel University Some Existence Theory June 29, / 26

5 We skipped the uniform estimate The most important step in the energy method is finding an estimate, uniform in the approximation parameters. We will now establish that sup µ n,δ 2 s 1 + sup w n,δ 2 s + t [,T ] t [,T ] T µ n,δ 2 s + w n,δ 2 s+1 dτ is bounded by some constant independent of n and δ. To do this, we will introduce a smallness constraint. David Ambrose Drexel University Some Existence Theory June 29, / 26

6 The theorem we are proving Remember the theorem I stated last time: Theorem Let T > and ε > be given. Let s d+5 2 and let µ H s 1 T d be such that m + µ is a probability measure. Let u T H s T d be given. Assume that the conditions H1 and H2 are satisfied. Then there exists µ L [, T ]; H s 1 L 2 [, T ]; H s and there exists u L [, T ]; H s L 2 [, T ]; H s+1 such that m + µ is a probability measure for all t [, T ], and such that u, m + µ satisfies 1, 2, 3. Furthermore, for all s [, s, we have µ C[, T ]; H s 1 and u C[, T ]; H s. David Ambrose Drexel University Some Existence Theory June 29, / 26

7 The hypothesis H1 H1 The function H is such that there exists a non-decreasing function F : [, [, such that for all β N 2d+1 with β s + 2, β Θ,, ν, Dy F ν + Dy. This has the immediate consequence that Θ and its derivatives are bounded in terms of the norm of the solution: β Θt,, µ, Dw β 2 CF s 1 µ 2 d Dw µ d+1 2 s 1 + w s s 1. David Ambrose Drexel University Some Existence Theory June 29, / 26

8 Our norms to estimate For all n N, we define M n and N n to be M n = sup t [,T ] Dw n 2 s 1 + sup t [,T ] µ n 2 s 1, N n = 1 α s T α Dw n 2 dτ + α s 1 T α Dµ n 2 dτ. Our solutions actually depend on δ as well, but we will suppress this dependence for now. The N n characterizes the parabolic gain of regularity; we will focus on estimating M n and we will get an estimate for N n as we go. David Ambrose Drexel University Some Existence Theory June 29, / 26

9 We begin with µ n : d 1 α µ n+1 2 dx dt 2 T d = α µ n+1 α µ n+1 dx ε α µ n+1 α Dµ n Θ p, x, µ n, Dw n dx T d T d ε α µ n+1 d α µ n + m Θ xip i, x, µ n, Dw n dx T d i=1 ε α µ n+1 d α µ n + m [Θ qpi, x, µ n, Dw n xi µ n ] dx T d i=1 ε α µ n+1 d d [ Θpip α µ n + m j, x, µ n, Dw n wn] 2 T d xixj dx. i=1 j=1 David Ambrose Drexel University Some Existence Theory June 29, / 26

10 We integrate with respect to time and rearrange: 1 α µ n t, x dx α µ n+1 2 t, x dx + D α µ n+1 2 dxdτ 2 T d 2 T d T d t = ε α µ n+1 α Dµ n Θ pτ, x, µ n, Dw n dxdτ T d t ε α µ n+1 d α µ n + m Θ xi T d p i τ, x, µ n, Dw n dxdτ i=1 t ε α µ n+1 d α µ n [ + m Θ qpi τ, x, µ n, Dw n xi µ n] dxdτ T d i=1 t ε α µ n+1 d d [ α µ n + m Θ pi T d pj τ, x, µn, Dw n ] 2 xi xj wn dxdτ i=1 j=1 = I + II + III + IV. David Ambrose Drexel University Some Existence Theory June 29, / 26

11 Work to do: We need to estimate these terms, I, II, III, and IV in terms of M n and N n. We will actually do some further decomposing: I = I A + I B, III = III A + III B, and IV = IV A + IV B. Each of these further decompositions involves separating out the leading term, as in an example in the last lecture. This is for µ; we have corresponding work to do for w as well. David Ambrose Drexel University Some Existence Theory June 29, / 26

12 The term I A We have the definition t I A = ε α µ n+1 α Dµ n Θ p τ, x, µ n, Dw n dxdτ T d We pull Θ p through the integral: t I A ε Θ p t,, µ n, Dw n We use the assumption H1: T d α µ n+1 α Dµ n dxdτ. T I A εf M n α µ n+1 α Dµ n dxdτ. T d Next we use Young s inequality, with parameter σ 1 >. David Ambrose Drexel University Some Existence Theory June 29, / 26

13 The term I A continued Using Young s inequality yields the following: 1 T I A εf M n α µ n+1 2 dτ + σ 1 2σ T εf M n 2σ 1 We choose σ 1 = 28T εf M n. This implies I A 1 56T T T α Dµ n 2 dτ α µ n+1 2 dτ + σ 1 2 N n α µ n+1 2 dτ + 14ε 2 T F M n 2 N n 1 56 sup α µ n+1 2 t [,T ]. + 14ε 2 T F M n 2 N n. David Ambrose Drexel University Some Existence Theory June 29, / 26

14 The term I B We have the definition t I B = ε α µ n+1 T d [ α Dµ n Θ p τ, x, µ n, Dw n α Dµ n Θ p τ, x, µ n, Dw n ] dxdτ. We use Young s inequality with parameter σ 2 = 28T ε, finding I B ε 2 T T sup α µ n+1 2 t [,T ] α Dµ n Θ p τ,, µ n, Dw n α Dµ n Θ p τ,, µ n, Dw n We then use our lemma about derivatives of products, and Sobolev embedding, and H1, finding I B 1 56 sup α µ n+1 2 t [,T ] 2 + cε 2 T 2 F M n 2 M n 1 + M n s 1. dτ. David Ambrose Drexel University Some Existence Theory June 29, / 26

15 The term II We have the definition II = ε t i=1 T d α µ n+1 α µ n + m d Θ xip i τ, x, µ n, Dw n dxdτ There is no need to extract a leading-order contribution this time. We use Young s inequality, the Sobolev algebra property, and H1, finding II 1 sup α µ n cε 2 T 2 F M n M n s. 56 t [,T ] David Ambrose Drexel University Some Existence Theory June 29, / 26

16 The term III A We have the definition III A = ε t α µ n+1 µ n + m T d d [Θ qpi τ, x, µ n, Dw n α xi µ n ] dxdτ. i=1 We estimate this by putting µ n and Θ qpi in L using Sobolev embedding and H1, and we use N n for α xi µ n. The result is III A 1 56 sup α µ n+1 2 t [,T ] + cε 2 T 1 + M n F M n 2 N n. David Ambrose Drexel University Some Existence Theory June 29, / 26

17 The term III B We have the definition t { III B = ε α µ n+1 d α µ n + m T d i=1 [ Θ qpi τ, x, µ n, Dw n xi µ n] d µ n [ + m Θ qpi τ, x, µ n, Dw n } α xi µ n] dxdτ. i=1 We estimate this similarly to how we estimated I B : III B 1 56 sup α µ n+1 2 t [,T ] + ct 2 ε 2 F M n M n s+1. David Ambrose Drexel University Some Existence Theory June 29, / 26

18 The term IV A We have the definition IV A = ε t α µ n+1 µ n + m T d d d [ Θpi p j τ, x, µ n, Dw n α 2 wn] xixj dxdτ. i=1 j=1 As in III A, we put µ n and Θ pi p j in L, using Sobolev embedding and H1, and we use N n to bound α x 2 i x j w n : IV A 1 sup α µ n t [,T ] + cε 2 T 1 + M n F M n 2 N n. David Ambrose Drexel University Some Existence Theory June 29, / 26

19 The term IV B We have the definition t { IV B = ε α µ n+1 d d [ α µ n + m Θ pi T d pj τ, x, µn, Dw n ] 2 xi xj wn i=1 j=1 d d [ µ n + m Θ pi pj τ, x, µn, Dw n ] } α 2 xi xj wn dxdτ. i=1 j=1 We estimate this similarly to I B and III B : IV B 1 sup α µ n ct 2 ε 2 F M n M n s t [,T ] David Ambrose Drexel University Some Existence Theory June 29, / 26

20 What have we been doing? We need to bound the growth of the approximate solutions. We have shown how to bound µ n+1,δ, mainly in terms of the previous iterates. The structure in the equations that we have used: considering the MFG system as a system for m, Du, other than the linear terms, the equations only have first spatial derivatives. The linear terms are parabolic, giving a gain of regularity of one derivative. We have used this gain of regularity to control the first spatial derivatives present in the nonlinear terms. David Ambrose Drexel University Some Existence Theory June 29, / 26

21 Using our estimate for µ n+1,δ We have shown I + II + III + IV 1 sup α µ n t [,T ] + cε 2 T F M n T 1 + N n 1 + M n. s+1 This has as one consequence 1 2 α µ n+1 t, α µ n+1, sup α µ n t [,T ] + cε 2 T F M n T 1 + N n 1 + M n. s+1 David Ambrose Drexel University Some Existence Theory June 29, / 26

22 Using our estimate for µ n+1,δ continued Another consequence is t D α µ n+1 2 dτ 1 2 α µ n+1, sup α µ n t [,T ] + cε 2 T F M n T 1 + N n 1 + M n. s+1 We take the supremum in time of both of these consequences, and we add, finding 1 T sup α µ n D α µ n+1 2 dτ 4 t [,T ] α µ n+1, 2 +cε 2 T F M n 2 1+T 1+N n 1+M n. s+1 David Ambrose Drexel University Some Existence Theory June 29, / 26

23 We do the same thing for w n+1,δ We go through the same steps for the w equation, and we estimate V, V I A, V I B, V II A, and V II B. We get a corresponding conclusion for w and we add it to the conclusion for µ, finding 1 4 sup t [,T ] T + α µ n sup α xj w n+1 2 t [,T ] D α µ n+1 τ, 2 dτ + T D α xj w n+1 τ, 2 dτ α µ n+1, 2 + α xj w n+1 T, 2 + cε 2 T F M n T 1 + N n 1 + M n s+1. David Ambrose Drexel University Some Existence Theory June 29, / 26

24 Ready for our induction We sum over multi-indices and whatnot, we multiply by 4, and we substitute the boundary conditions: M n+1 + 4N n+1 4 P δ µ 2 s DP δ w T 2 s 1 + cε 2 T F M n T 1 + N n 1 + M n s+1. 4 Let S be a real number such that 4 µ 2 s Dw T 2 s 1 S. We state our smallness assumption: H2 The function F and the constants c, ε, T, and S satisfy cε 2 T F 2S T 1 + 2S s+2 S. David Ambrose Drexel University Some Existence Theory June 29, / 26

25 Induction proof: the end of the estimate Claim: for all n, we have M n + 4N n 2S. Base case: µ = w =, so M + N =. Inductive hypothesis: Assume M n + 4N n 2S. Notice that M n 2S and N n 2S. Also, recall that P δ f s 1 f s 1, for any f. We then have M n+1 + 4N n+1 4 µ 2 s Dw T 2 s 1 + cε 2 T F 2S T 1 + 2S s+2. 5 So, M n+1 + 4N n+1 S + S = 2S. This completes the proof. David Ambrose Drexel University Some Existence Theory June 29, / 26

26 About the smallness constraint Let s state that smallness constraint again: S is bigger than a multiple of the data, and we assume cε 2 T F 2S T 1 + 2S s+2 S. Given a Hamiltonian and some data, and a time horizon T, this can be enforced by taking ε small. Given a Hamiltonian and some data and a value of ε, then this can be enforced by taking T small. For some Hamiltonians, it could be the case that F 2S 2 as S, faster than S. Then, given values of ε and T, the smallness constraint can be enforced by taking the data sufficiently small. So, the smallness constraint simultaneously considers the length of the time horizon, the strength of the coupling in the system, and the size of the data. David Ambrose Drexel University Some Existence Theory June 29, / 26

27 Thanks for your attention. David Ambrose Drexel University Some Existence Theory June 29, / 26