Mean Field Limits for Interacting Bose Gases and the Cauchy Problem for Gross-Pitaevskii Hierarchies. Thomas Chen University of Texas at Austin

Transcription

1 Mean Field Limits for Interacting Bose Gases and the Cauchy Problem for Gross-Pitaevskii Hierarchies Thomas Chen University of Texas at Austin Nataša Pavlović University of Texas at Austin Analysis and Applied Mathematics Seminar University of Toronto,

2 Outline of the talk 1. Introduction (a brief review of a derivation of the cubic NLS) 2. The Cauchy problem for Gross-Pitaevskii hierarchies 3. The conserved energy and an application 4. Higher order energies and applications 2

3 1 Introduction The mathematical analysis of interacting Bose gases has been an extremely active topic in recent years. 1. Proof of Bose-Einstein condensation Lieb-Seiringer-Yngvason; Aizenman-L-S-Solovej-Y 2. Derivation of nonlinear Schrödinger or Hartree eq. Hepp, Ginibre-Velo, Spohn Elgart-Erdös-Schlein-Yau, Erdös-Schlein-Yau, Adami-Bardos-Golse-Teta Fröhlich-Graffi-Schwarz, Fröhlich-Knowles-Pizzo, Anapolitanos-Sigal, Abou-Salem, Pickl Klainerman-Machedon, Kirkpatrick-Schlein-Staffilani Rodnianski-Schlein, Grillakis-Machedon-Margetis, Grillakis-Margetis 3

4 A brief review of the work of Erdös-Schlein-Yau [ ] on the derivation of the cubic NLS Step 1: From N-body Schrödinger to BBGKY hierarchy The starting point is a system of N bosons whose dynamics is generated by the Hamiltonian H N := N ( xj ) + 1 N V N (x i x j ), (1.1) j=1 1 i<j N on the Hilbert space H N = L 2 sym(r dn ), whose elements Ψ(x 1,..., x N ) are fully symmetric with respect to permutations of the arguments x j. Here V N (x) = N dβ V (N β x). 4

5 The solutions of the Schrödinger equation i t Ψ N = H N Ψ N with initial condition Ψ N,0 H N determine: the N-particle density matrix γ N (t, x N ; x N ) = Ψ N (t, x N )Ψ N (t, x N ), and its k-particle marginals γ (k) N (t, x k ; x k ) = dx N k γ N (t, x k, x N k ; x k, x N k ), Here for k = 1,..., N, which are positive definite and admissible a. x k = (x 1,..., x k ), x N k = (x k+1,..., x N ). a We call Γ N = (γ (k) N ) k admissible if γ (k) N 5 = Tr k+1γ (k+1) N for all k.

6 The BBGKY hierarchy is given by i t γ (k) N + 1 N = ( x k x k )γ (k) N [V N (x i x j ) V N (x i x j)] γ (k) N (1.2) + N k N 1 i<j k k i=1 Tr k+1 [V N (x i x k+1 ) V N (x i x k+1 )] γ (k+1) N (1.3) In the limit N, the sums weighted by combinatorial factors have the following size: In (1.2), k2 N 0 for every fixed k and sufficiently small β. In (1.3), N k N 1, for every fixed k and V N (x i x j ) b 0 δ(x i x j ), with b 0 = dx V (x). 6

7 Step 2: BBGKY hierarchy GP hierarchy Accordingly, in the limit N, one obtains the infinite GP hierarchy i t γ (k) = k ( xj x j ) γ (k) + b 0 j=1 k j=1 B j;k+1 γ (k+1) (1.4) where the contraction operator is given via ( ) B j;k+1 γ (k+1) (t, x 1,..., x k ; x 1,..., x k) (1.5) := dx k+1 dx k+1 [ δ(xj x k+1 )δ(x j x k+1) δ(x j x k+1 )δ(x j x k+1) ] γ (k+1) (t, x 1,..., x j,..., x k, x k+1 ; x 1,..., x k, x k+1). 7

8 Step 3: Factorized solutions of GP and NLS It is easy to see that γ (k) = φ φ k := k j=1 φ(t, x j ) φ(t, x j) is a solution of the GP if φ satisfies the cubic NLS with φ 0 L 2 (R d ). i t φ + x φ b 0 φ 2 φ = 0 8

9 Step 4: Uniqueness of solutions of GP hierarchies While the existence of factorized solutions can be easily verified, the proof of uniqueness of solutions of the GP hierarchy is the most difficult part in this analysis. 9

10 Roughly speaking, the method of Erdös, Schlein, and Yau for deriving the cubic NLS justifies the heuristic explained above and it consists of: (i) Deriving the Gross-Pitaevskii (GP) hierarchy as the limit as N of the BBGKY hierarchy, for asymptotically factorized initial data and for a scaling where the particle interaction potential tends to a delta distribution. (ii) Proving uniqueness of solutions for the GP hierarchy, which implies that for factorized initial data, the solutions of the GP hierarchy are determined by a cubic NLS. The proof of uniqueness is accomplished by using highly sophisticated estimates based on Feynman graph expansions. 10

11 A remark about ESY solutions of the GP hierarchy: Solutions of the GP hierarchy are studied in spaces of k-particle marginals {γ (k) γ (k) h 1 < } with norms γ (k) h α := Tr( S (k,α) γ (k) ), (1.6) where S (k,α) := k xj α x j α. j=1 11

12 An alternative method for proving uniqueness of the GP: In 2008 Klainerman and Machedon introduced an alternative method for proving uniqueness in a space of density matrices equipped with the Hilbert-Schmidt type Sobolev norm γ (k) H α k := ( Tr( S (k,α) γ (k) 2 ) ) 1 2. (1.7) The method is based on: a reformulation of the relevant combinatorics via the board game argument and the use of certain space-time norms. 12

13 The method of Klainerman and Machedon requires the assumption that the a priori space-time bound B j;k+1 γ (k+1) L 1 t Ḣk 1 < C k, (1.8) holds, with C independent of k. Subsequently: Kirkpatrick, Schlein and Staffilani proved the bound for the cubic GP in d = 2. [C., Pavlović, JFA 2011]: Proof of the bound for the quintic GP in d = 1, 2, for derivation of quintic defocusing NLS. But the case of cubic GP in d = 3 remained open. 13

14 2 The Cauchy problem for Gross-Pitaevskii hierarchies (joint work with N. Pavlović) The work of Klainerman and Machedon inspired us to study well-posedness for the Cauchy problem for focusing and defocusing GP hierarchy. Main problem: Hierarchical equations for γ (k) don t close; from Duhamel expansion, one gets higher and higher order terms γ (k+m). Accordingly, no fixed point argument available for lwp. 14

15 Why is it difficult to prove uniqueness? Let us fix a positive integer r. One can express the solution γ (r) to the GP hierarchy as: γ (r) (t r, ) = = tr 0 tr 0 =... e i(t r t r+1 ) (r) ± B r+1 (γ (r+1) (t r+1 )) dt r+1 tr+1 0 e i(t r t r+1 ) (r) ± B r+1 e i(t r+1 t r+2 ) (r+1) ± B r+2 (γ (r+2) (t r+2 )) dt r+1 dt r+2 where = B r+1 = tr 0... tr+n 1 0 r B j;r+1, j=1 t r+n = (t r, t r+1,..., t r+n ), J r (t r+n ) dt r+1...dt r+n, (2.1) J r (t r+n ) = e i(t r t r+1 ) (r) ± B r+1...e i(t r+(n 1) t r+n) (r+(n 1)) ± B r+n (γ (r+n) (t r+n )). The iterated Duhamel s formula has r(r + 1)...(r + n 1) terms. Problem: Factorially large number of terms, and the series involves γ (r+n) with arbitrarily large n.

16 2.1 Revisiting the GP hierarchy Recall, (k) ± = x k x k, with xk = k xj. j=1 First, we introduce a convenient notation: Γ = ( γ (k) ( t, x 1,..., x k ; x 1,..., x k) ) k N, ± Γ := ( (k) ± γ(k) ) k N, BΓ := ( B k+1 γ (k+1) ) k N. Then, the cubic GP hierarchy can be written as a i t Γ + ± Γ = µ BΓ. (2.2) a Moreover, for µ = 1 we refer to the GP hierarchy as defocusing, and for µ = 1 as focusing. 16

17 More generally, let p {2, 4}. We introduce the p-gp hierarchy as follows a where Here i t Γ + ± Γ = µ BΓ, (2.3) BΓ := ( B k+ p 2 γ(k+ p 2 ) ) k N. (2.4) B p k+ p γ(k+ 2 ) = B + 2 k+ p γ (k+ p 2 ) B k+ p γ (k+ p 2 ), (2.5) 2 2 where B + k+ p 2 γ (k+ p 2 ) = B k+ p 2 γ (k+ p 2 ) = k B + j;k+1,...,k+ p γ (k+ p 2 ), 2 j=1 k B j;k+1,...,k+ p γ (k+ p 2 ), 2 j=1 a We refer to (2.3) as the cubic GP hierarchy if p = 2, and as the quintic GP hierarchy if p = 4. 17

18 with ( B + j;k+1,...,k+ p 2 γ (k+ p 2 )) (t, x 1,..., x k ; x 1,..., x k) = γ (k+ p 2 ) (t, x 1,..., x j,..., x k, x j, x j ; x 1,..., x k, x j,, x j ), }{{}}{{} p p 2 2 ( B j;k+1,...,k+ p 2 γ (k+ p 2 )) (t, x 1,..., x k ; x 1,..., x k) = γ (k+ p 2 ) (t, x 1,..., x k, x j, x j; x 1,..., x j,... x k, x j,, x j). }{{}}{{} p p

19 2.2 Spaces Main insight: The equations for γ (k) don t close & no fixed point argument. Solution: Consider the space of sequences Γ and endow it with a suitable topology. Let G := L 2 (R dk R dk ) k=1 be the space of sequences of density matrices Γ := ( γ (k) ) k N. As a crucial ingredient of our arguments, we introduce Banach spaces H α ξ = { Γ G Γ H α ξ < } where Γ H α ξ := k N ξ k γ (k) H α (R dk R dk ). Properties: Finiteness: Γ H α ξ < C implies that γ (k) H α (R dk R dk ) < Cξ k. Interpretation: ξ 1 upper bound on typical H α -energy per particle. 19

20 2.3 Recent results for the GP hierarchy 1. Local in time existence and uniqueness (C., Pavlović) of solutions to focusing and defocusing GP hierarchies. 2. Blow-up (C., Pavlović, Tzirakis) of solutions to the focusing GP hierarchies in certain cases. 3. Higher order conserved energy functionals for solutions to the GP hierarchy (C., Pavlović). 4. Global existence of solutions to the GP hierarchy in certain cases (C., Pavlović). 20

21 2.4 Local in time existence and uniqueness [C., Pavlović, DCDS 2010] We prove local in time existence and uniqueness of solutions to the cubic and quintic GP hierarchy with focusing or defocusing interactions, in a subspace of Hξ α, for α A(d, p), which satisfy a spacetime bound BΓ L 1 t I H α ξ <, (2.6) for some ξ > 0. 21

22 The parameter α determines the regularity of the solution, and our results hold for α A(d, p) where ( 1 2, ) if d = 1 A(d, p) := ( d 2 1 2(p 1), ) if d 2 and (d, p) (3, 2) [ 1, ) if (d, p) = (3, 2), in dimensions d 1, and where p = 2 for the cubic, and p = 4 for the quintic GP hierarchy. 22

23 More precisely, we prove the following: Theorem 2.1. Let 0 < ξ < 1. Assume that α A(d, p) where d 1 and p {2, 4}. Then, for every Γ 0 H α ξ, there exist constants T > 0 and 0 < ξ < ξ such that there exists a unique solution Γ(t) Hξ α for t [0, T ] with BΓ L 1 t [0,T ] H α ξ <. Moreover, there exists a finite constant C(T, d, p, ξ, ξ ) such that the bound BΓ L 1 t I H α ξ C(T, d, p, ξ, ξ ) Γ 0 H α ξ (2.7) holds. 23

24 Flavor of the proof: Note that the GP hierarchy can be formally written as a system of integral equations Γ(t) = e it ± Γ 0 iµ t BΓ(t) = B e it ± Γ 0 iµ 0 t ds e i(t s) ± BΓ(s) (2.8) 0 ds B e i(t s) ± BΓ(s), (2.9) where (2.9) is obtained by applying the operator B on the linear non-homogeneous equation (2.8). We prove the local well-posedness result by applying the fixed point argument in the following space: W α ξ (I) := { Γ L t IHξ α BΓ L 2 t IHξ α }, (2.10) where I = [0, T ]. 24

25 Remark: While the approach via spacetime norms allows for new and shorter proofs of uniqueness in a subspace of Hξ α, these results do not, in general, imply uniqueness for the ESY solutions. Spaces of solutions used by ESY correspond to H α ξ = { Γ G Γ H α ξ < } (2.11) where Γ H α ξ := ξ k γ (k) h α, (2.12) k N with γ (k) h α := Tr( S (k,α) γ (k) ). (2.13) 25

26 New proof of existence of solutions Theorem 2.2. [C., Pavlović] Existence of solutions Γ(t) Hξ α of focusing and defocusing GP hierarchies for initial data Γ(0) Hξ α, for ξ < ξ sufficiently small. No requirement on spacetime norms. A posteriori, solution satisfies BΓ L 2 t H α ξ < C Γ(0) α H ξ. Proof idea: Consider cutoff initial data Γ N (0) = (γ (1),..., γ (N), 0, 0,... ) solve hierarchy equations (which then close!), obtaining Γ N (t) where γ (n) (t) = 0 for n > N. Then, prove that (Γ N (t)) N is Cauchy sequence in L t [0,T ] Hα ξ, and take limit N. Verify that the limit satisfies GP equations. 26

27 3 The conserved energy and applications (joint with N. Pavlović and N. Tzirakis; AIHP 2010) In this work we (A) Identify an observable corresponding to the average energy per particle and prove that it is conserved. (B) Prove, on the L 2 critical and supercritical level, that solutions of focusing GP hierarchies with a negative average energy per particle and finite variance blow up in finite time. 27

28 3.1 Conservation of average energy per particle Theorem 3.1. [C., Pavlović, Tzirakis] Assume Γ(t) H α ξ, α 1, solves p-gp for µ = ±1. Define k-particle energy and ξ-energy of GP, 0 < ξ < 1, [ k E k ( Γ(t) ) := Tr ( 1 ] 2 x j )γ (k) + µ [ p + 2 Tr B + k+ p 2 j=1 γ (k+ p 2 ) ] E ξ ( Γ(t) ) := k 1 ξ k E k ( Γ(t) ). Then, the ξ-energy is conserved, E ξ ( Γ(t) ) = E ξ ( Γ(0) ). In particular, admissibility a reduction to the one-particle density E ξ ( Γ(t) ) = ( kξ k ) E 1 ( Γ(t) ). k 1 a We call Γ = (γ (k) ) k N admissible if γ (k) = Tr k+1 γ (k+1) for all k N. 28

29 Explicit expression for one-particle energy: Let k p := 1 + p 2. Then, E 1 (Γ) = Tr[ 1 2 xγ (1) ] + µ p + 2 dx γ (kp) (x,..., x; x,..., x) }{{}}{{} k p k p For factorized states Γ(t) = ( φ(t) φ(t) k ) k N, E 1 ( Γ(t) ) = 1 2 φ(t) 2 L 2 + µ p + 2 φ(t) p+2 L p+2, conserved energy for NLS i t φ + φ + µ φ p φ = 0. 29

30 3.2 Blow-up of solutions to the focusing GP hierarchies Theorem 3.2. [C., Pavlović, Tzirakis] Let p p L 2 = 4 d. Assume that Γ(t) = ( γ (k) (t) ) k N is a positive definite solution of the focusing p-gp with Γ(0) H 1 ξ for some 0 < ξ < 1, and Tr( x 2 γ (1) (0) ) <. If E 1 ( Γ(0) ) < 0, then the solution Γ(t) blows up in finite time. 30

31 Glassey s argument in the case of an L 2 -critical or supercritical focusing NLS Consider a solution of i t φ = φ φ p φ with φ(0) = φ 0 H 1 (R d ) and p p L 2 = 4, such that d E[φ(t)] := 1 2 φ(t) 2 L 2 1 p + 2 φ(t) p+2 L p+2 = E[φ 0 ] < 0. Moreover, assume that x φ 0 L 2 <. Then the quantity V (t) := φ(t), x 2 φ(t) satisfies the virial identity 2 t V (t) = 16E[φ 0 ] 4d p p L 2 p + 2 φ(t) p+2 L p+2. (3.1) Hence, if E[φ 0 ] < 0, and p p L 2, this identity shows that V is a strictly concave function of t. But since V is also non-negative, we conclude that the solution can exit only for a finite amount of time. 31

32 Glassey s argument in the case of an L 2 -critical or supercritical focusing p-gp The quantity that will be relevant in reproducing Glassey s argument is given by V k ( Γ(t) ) := Tr( k j=1 x 2 jγ (k) (t) ). (3.2) Similarly as in our discussion of the conserved energy, we observe that a V k ( Γ(t) ) = k V 1 ( Γ(t) ). (3.3) a Again, this follows from the fact that γ (k) is symmetric in its variables, and from the admissibility of γ (k) (t) for all k N. 32

33 Goal: Calculate 2 t V 1 (t) and relate it to the conserved energy Denote by E K 1 (t) the kinetic part of the energy E 1 (t) and by E P 1 (t) the potential part of the energy E 1 (t) i.e. E K 1 ( Γ(t) ) = 1 2 Tr( ( x)γ (1) (t) ), E P 1 ( Γ(t) ) = µ p + 2 Tr( B+ 1;2,...,1+ p 2 γ (1+ p 2 ) (t) ). We calculate t 2 V 1 (t) and relate it to the conserved energy per particle as follows t 2 V 1 (t) = 8 du u 2 γ(u; u) + µ 4dp dx γ(x,..., X; X,..., X) p + 2 }{{}}{{} 1+ p 2 1+ p 2 = 16E1 K ( Γ(t) ) + 4dp E1 P ( Γ(t) ) = 16E 1 ( Γ(0) ) + 4dµ p p L 2 p + 2 dx γ(x,..., X; X,..., X). }{{}}{{} 1+ p 2 1+ p 2 Hence for the focusing (µ = 1) GP hierarchy with p p L 2, t 2 V 1 (t) 16E 1 ( Γ(0) ). However, the function V 1 (t) is nonnegative, so we conclude that if E 1 ( Γ(0) ) < 0, the solution can exist only for a finite amount of time.

34 4 Higher order energies and applications (joint with N. Pavlović) 1. We introduce a new family of higher order energy functionals and prove that they are conserved for solutions of the GP hierarchy 2. We prove a priori energy bounds on positive definite solutions Γ(t) H 1 ξ for: defocusing, energy-subcritical GP hierarchies for focusing, L 2 -subcritical GP hierarchies 3. As a corollary, we enhance local to global well-posedness of positive definite solutions Γ W 1 ξ([0, T ]) with initial data in H 1 ξ of p-gp hierarchies that are: defocusing and energy subcritical, or (de)focusing and L 2 -subcritical. 34

35 4.1 Higher order energy conservation Define the operators, for l N, K l := 1 2 (1 x l ) Tr l+1,...,l+ p 2 + µ p + 2 B+ l;l+1,...,l+ p 2 Affects only block of k p := 1 + p 2 variables, starting at index l γ (j) (x 1,..., x l,..., x l+ p,..., x j ; x 2 }{{} k p 1,..., x l,..., x l+ p 2 }{{} k p Relation to one-particle energy E 1 (Γ) (using admissibility):,..., x j) Tr 1,...,l,l+kp,,j [ Kl γ (j) ] = E 1(Γ) We assemble m successive blocks into the operator K (m) := K 1 K kp +1 K (m 1)kp

36 Theorem 4.1. (C., Pavlović) Assume Γ = (γ (j) ) H 1 ξ, for some 0 < ξ < 1, is an admissible solution of p-gp, µ = ±1. Then, K (m) ] := Tr Γ(t) 1,k p +1,2k p +1,...,(m 1)k p +1[ K (m) γ (mkp) (t) is conserved. 36

37 Proof. Determine t K (m), and show that all terms either Γ(t) vanish or mutually cancel, using admissibility of Γ(t), and the fact that the m blocks do not mix. Particles in GP hierarchy only interact via mean field with one another, but not directly. 37

38 4.2 A priori energy bounds We use higher order energy functionals to obtain three types of bounds: 1. A priori energy bounds for both focusing and defocusing energy-subcritical p-gp hierarchies. 2. A priori H 1 bounds for defocusing energy subcritical p-gp hierarchies. 3. A priori H 1 bounds for L 2 -subcritical focusing p-gp hierarchies. 38

39 A priori energy bounds for focusing and defocusing energy-subcritical GP Theorem 4.2. Let p < 4. If Γ(t) d 2 H1 ξ is a positive definite solution to the p-gp hierarchy (focusing or defocusing), then the a priori bound (2ξ) m K (m) Γ(t) Γ(t) H 1 (4.1) ξ m N holds for all ξ satisfying and all t R. ξ (1 + 2 p + 2 C Sob(d, p)) 1 k p ξ (4.2) 39

40 Proof. Let p < 4. We use the Sobolev inequality for the GP hierarchy, to bound the d 2 interaction energy by the kinetic energy: Tr 1 ( B 1;2,...,kp γ (k p) ) C Sob (d, p) Tr 1,...,kp ( S (k p,1) γ (k p) ). (4.3) To see this, we write γ (k p) as γ (k p) (x kp, x k p ) = j λ j φj (x kp ) φ j (x k p ), with respect to an orthonormal basis (φ j ) j, where λ j 0 and j λ j = 1. Then we have Tr 1 ( B + 1;2,...,k p γ (kp) ) = λ j dx φ j (x,..., x) 2. (4.4) }{{} j k p Now Sobolev (with q = k p and α = 1) implies that for p < 4 d 2 Tr 1 ( B + 1;2,...,k p γ (kp) ) C Sob λ j φ j 2 Hx 1 k p j C Sob λ j φ j 2 h 1 x k p j we have = C Sob Tr 1,...,kp ( S (k p,1) γ (k p) ).

41 Accordingly, (4.3) implies Tr 1 ( K 1 γ (k p) ) ( p + 2 C Sob(d, p) ) Tr 1,...,kp ( S (k p,1) γ (k p) ). (4.5) By iteration, we obtain that Therefore, for all ξ satisfying K (m) Γ ( p + 2 C Sob(d, p) ) m Tr 1,...,mkp ( S (mk p,1) γ (mk p) ). (2ξ) l K (l) Γ Hence, the claim follows. l l Γ H 1 ξ ( ( p + 2 C Sob(d, p) ) 1 kp ξ )l γ (l) h 1 l ξ ( p + 2 C Sob(d, p) ) 1 k p ξ. (4.6)

42 A priori H 1 bounds for defocusing energy subcritical GP Theorem 4.3. Assume that µ = +1 (defocusing p-gp hierarchy), p < 4, and that Γ(t) d 2 H1 ξ for t [0, T ], is a positive definite solution of the p-gp hierarchy with initial data Γ 0 H 1 ξ for ξ, ξ such that Then, the a priori bound ξ ( p + 2 C Sob(d, p) ) 1 k p ξ, (4.7) Γ(t) H 1 ξ m N (2ξ) m K (m) Γ(t) = m N(2ξ) m K (m) Γ 0 Γ 0 H 1 ξ (4.8) is satisfied for all t [0, T ]. 42

43 A priori H 1 bounds for focusing L 2 subcritical GP Theorem 4.4. Let p < p L 2 = 4 d and µ < 0 (L2 subcritical, focusing). Moreover, let α > α 0 := (k p 1)d 2k p and αk p < 1, where k p = 1 + p. Assume 2 µ < 1 4 (1 αk p) C 0 (α), (4.9) with C 0 (α) from generalized Sobolev inequality for γ (n), so that ( ) C 0 (α) D := D(α, p, d, µ ) = 1 µ > 0. (4.10) 1 4 (1 αk p) Let Γ(t) H 1 ξ be a positive definite solution of the focusing (µ < 0) p-gp hierarchy for t I, with initial data Γ(0) = Γ 0 H 1 ξ, where ξ 1 D ( p + 2 C Sob(d, p) ) 1 k p ξ. (4.11) 43

44 Then the a priori bound holds for all t I. Γ(t) H 1 ξ = m=1 m=1 (2D ξ) m K (m) Γ(t) (4.12) (2D ξ) m K (m) Γ 0 (4.13) Γ 0 H 1 ξ (4.14) 44

45 Enhancing local to global well-posedness Theorem 4.5. Assume one of the following two cases: Energy subcritical, defocusing p-gp hierarchy with p < 4 d 2 µ = +1. Moreover, ξ, ξ satisfy (4.7). and L 2 subcritical, focusing p-gp hierarchy with p < 4 d and µ < 0 satisfying (4.9). In addition, ξ, ξ satisfy (4.11). Then, there exists T > 0 such that for I j := [jt, (j + 1)T ], with j Z, there exists a unique global solution Γ j Z W 1 (I j, ξ) of the p-gp hierarchy with initial condition Γ(0) = Γ 0, satisfying Γ(t) H 1 ξ Γ(t) H 1 ξ C Γ 0 H 1 ξ (4.15) if Γ(t) is positive definite for all t R. 45

46 Physical meaning of GP hierarchy: Dynamics of infinitely many interacting bosons. Particles interact with each other via mean field, but not directly. Admits solutions corresponding to NLS, but richer than only NLS. Analysis of well-posedness of Cauchy problem for GP hierarchy of interest by itself: Quantum field theory with features of dispersive nonlinear PDEs 46