Quantum hydrodynamics with large data

Transcription

1 Quantum hydrodynamics with large data Pierangelo Marcati (joint work with P. Antonelli) Seminar on Compressible Fluids - Université Pierre et Marie Curie Laboratoire Jacques-Louis Lions March 29, 2010 P 2 Antonelli, Marcati () QHD with large data March 29, / 34

2 QHD system with collision terms where the initial data t ρ + div J = 0 t J + div J J ρ + P(ρ) +ρ V + f ( ρ, J, ρ, D 2 ( ρ) = 2 2 ρ ρ V = ρ ρ ) (1) ρ(0) = ρ 0, J(0) = J 0 (2) satisfy ρ0 H 1 (R 3 ), Λ 0 := J 0 ρ0 L 2 (R 3 ). and the pressure is given by P(ρ) = p 1 p+1 ρ(p+1)/2, 1 p < 5. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

3 Connection with Schrödinger equation (Noncollisional case f=0) Given a solution ψ to the Schrödinger equation i t ψ ψ = ψ p 1 ψ + V ψ V = ψ 2 ψ(0) = ψ 0. (3) ρ = ψ 2, J = Im(ψ ψ) (4) are (weak) solutions to QHD t ρ + div J ( = 0 ) ( t J + div J J ρ + P(ρ) + ρ V = 2 2 ρ ρ V = ρ ρ(0) = ρ 0 := ψ 0 2, J(0) = J 0 := Im(ψ 0 ψ 0 ). ρ ) (5) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

4 The collision term References to Nonlinear Schrödinger equation: Cazenave book, 2003,Tao book 2006, papers from Ginibre Velo in the 70s, 80 s, Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T. Ann. Math. 167(3), (2007)) The collision term can have for instance the form f = αj + ρ g(t, x, ρ, Λ, ρ, D 2 ρ), (6) where g is a nonlinear operator of ρ, Λ, ρ, D 2 ρ, under certain Carathéodory-type conditions. For simplicity from now on we study the case f ( ρ, J, ρ, D 2 ρ) = J. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

5 Weak Solutions of (1), (1) and Irrotationality. ρ L ([0, ); H 1 (R 3 )), Λ := J/ ρ L ([0, ); L 2 (R 3 )), 0 (ρ t η + J η)dxdt = R 3 ρ 0 η(0)dx; R 3 (7) (J t ζ + Λ Λ : ζ + P(ρ) div ζ ρ V ζ J ζ 0 R ρ ρ : ζ 2 ρ div ζ)dxdt = J 0 ζ(0)dx (8) 4 R 3 η C 0 ([0, ) R3 ), ζ C 0 ([0, ) R3 ; R 3 ), Generalized Irrotationality Condition in distributional sense J = 2 ρ Λ. (9) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

6 Main Theorem Theorem Let ψ 0 H 1 (R 3 ) and (Madelung Tranformations) ρ 0 := ψ 0 2, J 0 := Im(ψ 0 ψ 0 ). There exists a global weak solution (ρ, J) of QHD, with initial data (ρ(0), J(0)) = (ρ 0, J 0 ), such that ρ L ([0, ); Hloc 1 (R3 )) J L ([0, ); L 2 ρ loc (R3 )) and such that (7), (8), (9) hold. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

7 Some Motivations Fluidodynamical description of a quantum system : Madelung, E.: Quantentheorie in hydrodynamischer form. Z. Physik 40, 322 (1927) Semiconductor devices: Gardner, SIAM J. Appl. Math Superfluidity and Superconductivity: Landau, Phys. Rev. 1941, Khalatnikov, 1962, Dalfovo, Giorgini, Pitaevskii, Stringari, Rev. Mod. Phys. 1990, Feynman, R.P.: Superfluidity and Superconductivity. Rev. Mod. Phys. 29(2), 205 (1957) Two-fluid hydrodynamics for a trapped weakly interacting Bose gas : Zaremba, Nikkuni, Griffin, Stringari, Phys. Review A, P 2 Antonelli, Marcati () QHD with large data March 29, / 34

8 Source: Dalfovo, Giorgini, Pitaevskii, Stringari, Rev. Mod. Phys., 71, 3 (1999) (credit to Eric Cornell (JILA U.Colorado)) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

9 Generalized Irrotationality Condition Suppose everything is smooth and J = ρu, then (9) implies ρ u = 0, (10) which means the current velocity u is irrotational in ρdx. Possibility of occurrence of vortices in the nodal region {ρ = 0}. (Source: Barker, Microelectronic Engineering, 63, , 2002) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

10 Figure: Source: Josserand, Pomeau, Nonlinearity, 14, (2001) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

11 Known results Jungel, Mariani, Rial, 2001, M3AS: local existence of smooth solutions via the non standard NLS: i t ψ ψ = ψ p 1 ψ + V ψ + Ṽ ψ, (11) with Ṽ (ψ) = 2i log ψ. The higher order Sobolev regularity requires ψ the mass density bounded away from 0 and the phase to be small around a constant value. Li, M., 2004, CMP: Global existence. Higher order Sobolev regularity fot small perturbation around steady states, proof via nonstandard energy estimates under a Quantum Subsonic Condition. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

12 From QHD to nonlinear Schrödinger? Problem: (ρ, J), weak solutions of (5), s.t. ρ H 1 (R 3 ) and Λ := J ρ L 2 (R 3 ) find ψ H 1 (R 3 ) solution of (3). Answer: in general NO Obstructions : The WKB formulation : ψ = ρe is/ doesn t work, since in the vacuum {ρ = 0} the phase S cannot be defined. The velocity u = S is not defined. GMT, Federer, Ziemer, 1972: ρ H 1 (R 3 ) implies the nodal region {ρ = 0} may contain a nontrivial singular set (Quantum Dots). Pauli Problem: one cannot determine a wave function just by knowing its mass and momentum densities. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

13 Difficulty of using WKB - Searching the phase 1 Degenerate elliptic equation: div(ρ S) = div J. (12) Problem: degenerate elliptic equation (ρ 0); ρ doesn t satisfy Muckenhoupt s conditions. Quantum Hamilton-Jacobi eq: (u = S)! t S ρ u S = h(ρ) V +. (13) 2 2 ρ Transport equation (Ambrosio s theory): ρ t S + J S = ρh(ρ) ρv + 2 ρ ρ (14) 2 Problem: div (t,x) (ρ, J) = 0, but J is not in BV loc. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

14 The vacuum problem - Searching the phase - 2 Nelson s stochastic mechanics (description of quantum phenomena in term of diffusions: Nelson, Carlen, Guerra, Morato). Carlen, 1984: equivalence between Nelson s mechanics and quantum mechanics (Schrödinger equation). Definition of a stochastic process, with probability density ρdx, through a drift determined by the velocity v(t, x) := { Im ψ(t,x) ψ(t,x) ψ(t, x) 0 0 ψ(t, x) = 0. Problem: too much smoothness needed for ψ. (15) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

15 Searching the phase - 3 Aim: given ψ, find a function θ such that ψ = ρe iθ (similar to the problem of lifting in Sobolev spaces, see Bourgain, Brezis, Mironescu) From above discussion, this is difficult. Idea: find the unitary factor e iθ and work with it (similar to the polar factorization of L p spaces, see Brenier). P 2 Antonelli, Marcati () QHD with large data March 29, / 34

16 Lemma (trivial) Let B R be the ball centered at the origin with radius R. Let us define the set S R := {φ L 2 (B R ) s.t. φ L (B R ) 1} and consider the functional Φ[φ] := Re ψ(x)φ(x)dx B R Then the maximization problem sup φ SR Φ[φ] has a solution ˆφ S R such that ψ = ρ ˆφ a.e. in B R (and thus ˆφ = 1 for ρdx-a.e. x B R. Trivial because ˆφ is actually the Radon-Nikodym derivative of ψ w.r.t. ρ. The variational approach could be relevant if we ask for some smoothness of the maximizer (e.g. H 1 ) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

17 Lemma (Stability lemma)let ψ H 1 (R 3 ), φ L (R 3 ) s.t. ψ = ρφ a.e. in R 3, then ρ = Re(φ ψ) H 1 (R 3 ); Λ := Im(φ ψ) L 2 (R 3 ) (16) moreover ( trivial null form structure) 2 Re( j ψ k ψ) = 2 j ρ k ρ + Λ (j) Λ (k). (17) If {ψ n } H 1 (R 3 ), ψ n ψ in H 1 (R 3 ), then. Corollary ρ n ρ, Λ n Λ in L 2 (R 3 ) (18) (Stability of Irrotationality) Let ψ and {ψ n } in H 1 (R 3 ) as before, then 2i ρ Λ is H 1 (R 3 ) stable. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

18 Basic Idea - Restricted to f = -J Discretize time: take τ > 0 and split [0, ) in many subintervals [kτ, (k + 1)τ), k 0. Step A: solve the nonlinear Schrödinger-Poisson system (3) Step B: update the wave funtion in order to take into account collisions (phase shift) From the QHD point of view : Step A: solve the non-collisional QHD system Step B: solve the collision via an ODE: { d dt ρ = 0 d dt J + J = 0. Remark: The update mechanisms act only on the polar factor not on ρ P 2 Antonelli, Marcati () QHD with large data March 29, / 34

19 Formal Iteration Procedure Fractional Step Method Step k = 0. Take ψ 0 H 1 (R 3 ) and solve (3) in [0, τ) R 3. Factorize ψ τ (τ ) := ρ τ φ τ. Define ψ τ (τ+) := ρ τ φ (1 τ) τ. From step k 1 to step k, k 1: Take ψ τ (kτ+) (defined at previous step) and solve (3) in [kτ, (k + 1)τ) R 3 with initial datum ψ τ (kτ+) Factorize ψ τ ((k + 1)τ ) = ρ (k+1)τ φ (k+1)τ Define ψ τ ((k + 1)τ+) := ρ (k+1)τ φ (1 τ) (k+1)τ Formal since you actually need Approximate Polar Factorization Lemma. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

20 Lemma (Approximate Polar Factorization) Let ψ H 1 (R 3 ), and let τ, ε > 0. Then there exists ψ H 1 (R 3 ) such that ρ = ρ, Λ = (1 τ)λ + rε, where ρ := ψ, ρ := ψ, Λ := Im(φ ψ), Λ := Im( φ ψ), φ, φ are polar factors for ψ, ψ respectively, and Furthermore we have r ε L 2 (R 3 ) ε. ψ = ψ i τ φ Λ + r ε,τ, (19) where φ L (R 3 ) 1 and r ε,τ L 2 (R 3 ) C(τ ψ L 2 (R 3 ) + ε). P 2 Antonelli, Marcati () QHD with large data March 29, / 34

21 Define (ρ τ, J τ ) := ( ψ τ 2, Im(ψ τ ψ τ )). Lemma (Consistency) (ρ τ, J τ ) are approximate solutions of (1), in the sense that ρ τ t η + J τ ηdxdt + 0 R 3 ρ 0 η(0)dx = o(1), R 3 (20) J τ t ζ + Λ τ Λ τ : ζ + P(ρ τ ) div ζ ρ τ V τ ζ J τ ζ 0 R ρ τ ρ τ : ζ 2 4 ρτ div ζdxdt + J 0 ζ(0)dx = o(1) R 3 as τ 0, for any η C 0 ([0, ) R3 ), ζ C 0 ([0, ) R3 ; R 3 ). (21) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

22 Energy estimates Define the energy E τ (t) := R3 2 2 ρ τ Λτ 2 + f (ρ τ ) V τ 2 dx (22) Lemma (Dissipation of energy for approximate solutions) Let t [Nτ, (N + 1)τ), and let ρ τ, Λ τ be defined as above. Then E τ (t) τ N Λ τ (kτ ) 2 L 2 (R 3 ) + (1 + τ)e 0. (23) k=1 P 2 Antonelli, Marcati () QHD with large data March 29, / 34

23 Strichartz estimates for Schrödinger in R 3 Admissible pair (q, r) 1 q = 3 ( ) r (24) 2 q, 2 r 6 (25) Let U( ) be the free Schrödinger group then for any admissible pair (q, r), ( q, r) we have (Keel, Tao 1998) s<t R U( )u 0 L q t Lr x (R R3 ) u 0 L 2 (R 3 ) (26) U(t s)f (s)ds L q t Lr x (I R 3 ) F L q t L r x (I R 3 ) U(t s)f (s)ds L q t Lr x (I R 3 ) F L q t L r x (I R 3 ) (27) (28) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

24 Following the iteration procedure we need to estimate in Strichartz norms this unpleasent expression : ψ τ (t) =σ τ t U(t Nτ)σ τ Nτ U(τ) στ τ U(τ) ψ 0 i t Nτ φ t Λ(t) i τ στ t U(t Nτ)φ Nτ Λ τ (Nτ ) +... i τ στ t U(t Nτ) U(τ)φ τ Λ τ (τ ) iσ τ t t Nτ U(t s)f (s)ds iσ τ t U(t Nτ)σ τ Nτ Nτ (N 1)τ +... iσ τ t U(t Nτ)σ τ Nτ U(τ) στ τ U(Nτ s)f (s)ds τ where σ τ kτ := φ kτ φ (1 τ) kτ and F = ( ψ τ p 1 ψ τ + V τ ψ τ ). It is impossible to estimate the commutator [U(τ), σ τ kτ ] (low regularity!!). 0 U(τ s)f (s)ds, P 2 Antonelli, Marcati () QHD with large data March 29, / 34

25 Thanks to Approximate Polar Factorization Lemma, we have the following expression ψ τ (t) =U(t) ψ 0 i τ N U(t kτ) (φ k Λ τ (kτ )) k=1 N + τ U(t kτ)( φ k ψ k ) i + k=1 N U(t kτ)r k, k=1 t 0 U(t s)f (s)ds with φ k L 1, R k L 2 2 k τe 0. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

26 Strichartz estimates for ψ τ Lemma (Strichartz estimates for ψ τ ) Let 0 < T <, ψ τ be as before, then one has ψ τ L q t Lr x ([0,T ] R3 ) C(E 1 2 0, ρ 0 L 1 (R 3 ), T ) (29) for each admissible pair of exponents (q, r). P 2 Antonelli, Marcati () QHD with large data March 29, / 34

27 ψ τ L q t Lr x ([0,T 1] R 3 ) U(t) ψ 0 L q t Lr x ([0,T 1] R 3 ) + τ N U(t kτ) (φ k Λ τ (kτ )) L q t Lr x ([0,T 1 ] R 3 ) =: A + B + C + D. + + k=1 N U(t kτ)rk τ L q t Lr x ([0,T 1] R 3 ) k=1 t 0 U(t s)f (s)ds L q t Lr x ([0,T 1] R 3 ) P 2 Antonelli, Marcati () QHD with large data March 29, / 34

28 The estimate of A is straightforward, since The estimate of B follows from τ N k=1 U(t) ψ 0 L q t Lr x ([0,T 1] R 3 ) ψ 0 L 2 (R 3 ). U(t kτ) (φ k Λ τ (kτ )) L q t Lr x ([0,T 1] R 3 ) τ N k=1 The term C can be estimated in a similar way, namely Λ τ (kτ ) L 2 (R 3 ) T 1 E (30) N U(t kτ)rk τ L q t Lr x k=1 N rk τ L 2 (R 3 ) T 1 ψ 0 H 1 (R 3 ). (31) k=1 P 2 Antonelli, Marcati () QHD with large data March 29, / 34

29 To estimate D decompose F = F 1 + F 2 + F 3, where F 1 = ( ψ τ p 1 ψ τ ), F 2 = V τ ψ τ and F 3 = V τ ψ τ. There exists α > 0, depending on p, such that ψ τ p 1 ψ τ L q t L r x ([0,T 1 ] R 3 ) T α 1 ψ τ Ṡ1 ([0,T 1 ] R 3 ). (32) Choose (q 2, r 2 ) = (1, 2), hence by Hölder and Hardy-Littlewood-Sobolev V τ ψ τ L 1 t L 2 x ([0,T 1] R 3 ) T V τ L t L 2 x ψτ L 2 t L 6 x Choose (q 3, r 3 ) = ( 2 and Hölder V τ ψ τ T ψ τ 2 2 3ε, 2 1+2ε ε 1+2ε Lt Lx ([0,T 1 ] R 3 ) ε 2+3ε Lt Lx T ψ τ 2 L t L 2 x ψτ L 2 t L 6 x. ) and use again Hardy-Littlewood-Sobolev T V τ 2 1 3ε L t L 1 ε x ψ τ L t L 2 x ψ τ L t L 2 x T ψ τ 2 L t L 2 x ψτ ε 1+6ε Lt Lx P 2 Antonelli, Marcati () QHD with large data March 29, / 34

30 we summarize the previous estimates ψ τ Ṡ0 ([0,T 1 ] R 3 ) (1+T )E T α 1 ψ τ ṗ S 0 ([0,T 1 ] R 3 ) +T ψ 0 2 L 2 (R 3 ) ψτ Ṡ0 ([0,T 1 ] R 3 ). Lemma There exist T 1 (E 0, ψ 0 L 2 (R 3 ), T ) > 0, C 1 (E 0, ψ 0 L 2 (R 3 ), T ) > 0, s.t. ψ τ Ṡ0 ([0, T ] R 3 ) C 1(E 0, ψ 0 L 2 (R 3 ), T ) (33) for all 0 < T T 1 (E 0, ψ 0 L 2 (R 3 )), hence for all T > 0,. ψ τ Ṡ0 ([0,T ] R 3 ) ([ T C 1 (E 0, ψ 0 L 2, T ) T 1 ] ) + 1 = C( ψ 0 L 2, E 0, T ). P 2 Antonelli, Marcati () QHD with large data March 29, / 34

31 Smoothing estimates for Schrödinger From Constantin, Saut (1988) we have the following smoothing estimates U( )u 0 L 2 ([0,T ];H 1/2 loc (R3 )) u 0 L 2 (R 3 ) (34) t U(t s)f (s)ds L 0 2 ([0,T ]:H 1/2 loc (R3 )) F L 1 t L2 x ([0,T ] R3 ). (35) Use expression for ψ τ and Strichartz estimates: we find ψ τ L 2 ([0,T ];H 1/2 loc (R3 )) C( ψ 0 L 2 (R 3 ), T, E 0 ) (36) thus { ψ τ } relatively compact in L 2 ([0, T ]; L 2 (R 3 )). P 2 Antonelli, Marcati () QHD with large data March 29, / 34

32 Compactness We use Rakotoson-Temam (Appl. Math. Letters, 2001) ( Aubin-Lions type Lemma ). {ψ τ } uniform bounded in L 2 ([0, T ]; H 3/2 loc (R3 )) and H 1 loc (R3 ) compactly embedded in H 3/2 loc (R3 ). Equi-integrability property from the Energy estimates P 2 Antonelli, Marcati () QHD with large data March 29, / 34

33 Proposition {ψ τ } L 2 ([0, T ]; H 1 loc (R3 )) is relatively compact, there exists ψ L 2 ([0, T ]; H 1 loc (R3 )), s.t. (up to subsequences) ψ τ ψ L 2 ([0, T ]; H 1 loc (R3 )) (37) ρ τ ρ L 2 ([0, T ]; H 1 (R 3 )) (38) Λ τ Λ L 2 ([0, T ]; L 2 (R 3 )), (39) where ρ, Λ are the hydrodynamic quantities defined via ψ through the Madelung transform. Then (ρ, J) is a weak solution to the QHD but ψ is not a solution of any Schrödinger equation. P 2 Antonelli, Marcati () QHD with large data March 29, / 34

34 2-D QHD Theorem Let ρ 0, J 0 be such that there exists ψ 0 H 1 (R 2 ) satisfying Furthermore, let us assume that and that ρ 0 = ψ 0 2, J 0 = Im(ψ 0 ψ 0 ). R 2 ρ 0 log ρ 0 dx <, (40) V (0, x) := 1 log x y ρ 0 (y)dy (41) 2π R 2 satisfies V (0, ) L r (R 2 ), for some 2 < r <. Then, for 0 < T < there exists a finite energy weak solution of QHD in [0, T ) R 2. P 2 Antonelli, Marcati () QHD with large data March 29, / 34