Quantum Hydrodynamic Systems and applications to superfluidity at finite temperatures

Transcription

1 Quantum Hydrodynamic Systems and applications to superfluidity at finite temperatures Paolo Antonelli 1 Pierangelo Marcati 2 1 Gran Sasso Science Institute, L Aquila 2 Gran Sasso Science Institute and University of L Aquila Mathematical Challenges in Quantum Mechanics, Bressanone, 8 13 February, 2016 Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

2 Superfluidity Figure: Superfluid Helium 4 He source: Alfred Leitner - superfluid liquid Helium Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

3 Superfluidity Figure: Superfluid Helium 4 He source: Alfred Leitner - superfluid liquid Helium frictionless flow through narrow capillarities; irrotational outside the nodal region, v = 0 in {ρ > 0}; quantized vortices: m C v dl = 2π n, n Z. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

4 Quantum Hydrodynamics (QHD) t ρ + div J = 0 ( J J t J + div ρ ) ( ) + P(ρ) = 2 ρ 2 ρ, ρ with (t, x) R + R 3, and initial data ρ(0) = ρ 0, J(0) = J 0. Mass (charge) density ρ, momentum (current) density J. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

5 Quantum Hydrodynamics (QHD) { t ρ + div(ρu) = 0 t (ρu) + div(ρu u) + P(ρ) = 0 with (t, x) R + R 3, and initial data ρ(0) = ρ 0, J(0) = J 0. Mass (charge) density ρ, momentum (current) density J = ρu, velocity field u. Compressible Euler system Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

6 Quantum Hydrodynamics (QHD) t ρ + div(ρu) = 0 t (ρu) + div(ρu u) + P(ρ) = 2 2 ρ ( ) ρ ρ with (t, x) R + R 3, and initial data ρ(0) = ρ 0, J(0) = J 0. Mass (charge) density ρ, momentum (current) density J = ρu, velocity field u. Compressible Euler system+quantum Bohm s potential Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

7 Quantum Hydrodynamics (QHD) t ρ + div(ρu) = 0 t (ρu) + div(ρu u) + P(ρ) = 2 2 ρ ( ) ρ ρ with (t, x) R + R 3, and initial data ρ(0) = ρ 0, J(0) = J 0. Mass (charge) density ρ, momentum (current) density J = ρu, velocity field u. Compressible Euler system+quantum Bohm s potential ( ) 1 ρ 2 ρ = 1 ρ 4 div(ρ 2 log ρ) = 1 4 ρ div( ρ ρ). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

8 Finite energy weak solutions t ρ + div J = 0 ( J J t J + div ρ ) ( ) + P(ρ) = 2 ρ 2 ρ, ρ Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

9 Finite energy weak solutions t ρ + div J = 0 ( J J t J + div ρ ) + P(ρ) = 2 4 ρ 2 div( ρ ρ), Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

10 Finite energy weak solutions, t ρ + div J = 0 t J + div ( Λ Λ + 2 ρ ρ ) + P(ρ) = 2 4 ρ, with Λ = J ρ. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

11 Finite energy weak solutions, t ρ + div J = 0 t J + div ( Λ Λ + 2 ρ ρ ) + P(ρ) = 2 4 ρ, with Λ = J ρ. 2 Energy: E[ρ, J] = 2 ρ J 2 2 ρ + f (ρ) dx, P(ρ) = ρf (ρ) f (ρ) = γ 1 γ ργ, 1 γ < 3. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

12 Finite energy weak solutions, t ρ + div J = 0 t J + div ( Λ Λ + 2 ρ ρ ) + P(ρ) = 2 4 ρ, with Λ = J ρ. 2 Energy: E[ρ, Λ] = 2 ρ Λ 2 + f (ρ) dx, P(ρ) = ρf (ρ) f (ρ) = γ 1 γ ργ, 1 γ < 3. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

13 Finite energy weak solutions, t ρ + div J = 0 t J + div ( Λ Λ + 2 ρ ρ ) + P(ρ) = 2 4 ρ, with Λ = J ρ. 2 Energy: E[ρ, Λ] = 2 ρ Λ 2 + f (ρ) dx, P(ρ) = ρf (ρ) f (ρ) = γ 1 γ ργ, 1 γ < 3. Aim: find ( ρ, Λ) such that ρ := ( ρ) 2, J := ρλ is a finite energy weak solution. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

14 Global existence of F.E.W.S. Theorem For any ψ 0 H 1 (R 3 ), let ρ 0 := ψ 0 2, J 0 := Im( ψ 0 ψ 0 ), then there exist ( ρ, Λ) such that (ρ, J) is a finite energy weak solution for the QHD system. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

15 Global existence of F.E.W.S. Theorem For any ψ 0 H 1 (R 3 ), let ρ 0 := ψ 0 2, J 0 := Im( ψ 0 ψ 0 ), then there exist ( ρ, Λ) such that (ρ, J) is a finite energy weak solution for the QHD system. Remark No further regularity and/or smallness assumptions; Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

16 Global existence of F.E.W.S. Theorem For any ψ 0 H 1 (R 3 ), let ρ 0 := ψ 0 2, J 0 := Im( ψ 0 ψ 0 ), then there exist ( ρ, Λ) such that (ρ, J) is a finite energy weak solution for the QHD system. Remark No further regularity and/or smallness assumptions; Vacuum (quantized vortices); Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

17 Global existence of F.E.W.S. Theorem For any ψ 0 H 1 (R 3 ), let ρ 0 := ψ 0 2, J 0 := Im( ψ 0 ψ 0 ), then there exist ( ρ, Λ) such that (ρ, J) is a finite energy weak solution for the QHD system. Remark No further regularity and/or smallness assumptions; Vacuum (quantized vortices); No need to define the velocity field in the vacuum region; Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

18 Global existence of F.E.W.S. Theorem For any ψ 0 H 1 (R 3 ), let ρ 0 := ψ 0 2, J 0 := Im( ψ 0 ψ 0 ), then there exist ( ρ, Λ) such that (ρ, J) is a finite energy weak solution for the QHD system. Remark No further regularity and/or smallness assumptions; Vacuum (quantized vortices); No need to define the velocity field in the vacuum region; No uniqueness! Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

19 Analogy with NLS - WKB i t ψ = 2 2m ψ + f ( ψ 2 )ψ ψ(0) = ψ 0. Energy 2 E[ψ] = 2 ψ 2 + f ( ψ 2 ) dx Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

20 Analogy with NLS - WKB i t ψ = 2 2m ψ + f ( ψ 2 )ψ ψ(0) = ψ 0. Energy 2 E[ψ] = 2 ψ 2 + f ( ψ 2 ) dx WKB ansatz: ψ = ρe is/, then (ρ, S) satisfy t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 ρ 2 ρ Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

21 WKB ansatz t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 2 ρ ρ Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

22 WKB ansatz t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 2 ρ ρ u = S Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

23 WKB ansatz t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 2 ρ ρ ( ) u = S t u + (u )u + f (ρ) = 2 ρ 2 ρ Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

24 WKB ansatz t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 2 ρ ρ ( ) u = S t u + (u )u + f (ρ) = 2 ρ 2 ρ J = ρu = ρ S Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

25 WKB ansatz t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 2 ρ ρ ( ) u = S t u + (u )u + f (ρ) = 2 ρ 2 ρ J = ρu = ρ S (ρ, J) solves (QHD) & E[ψ] = E[ρ, J] Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

26 WKB ansatz t ρ + div(ρ S) = 0 t S S 2 + f (ρ) = 2 2 ρ ρ ( ) u = S t u + (u )u + f (ρ) = 2 ρ 2 ρ J = ρu = ρ S (ρ, J) solves (QHD) & E[ψ] = E[ρ, J] 2 ψ 2 = 2 ρ 2 + ρ S 2 = 2 ρ 2 + J 2 ρ Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

27 Mathematical problems of WKB Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

28 Mathematical problems of WKB vacuum: WKB ansatz only valid when ψ(t, x) 0 S not defined in {ψ = 0} Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

29 Mathematical problems of WKB vacuum: WKB ansatz only valid when ψ(t, x) 0 S not defined in {ψ = 0} regularity issue: the set {ψ = 0} when ψ H 1 (R 3 ) may be irregular (Federer, Ziemer) Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

30 Mathematical problems of WKB vacuum: WKB ansatz only valid when ψ(t, x) 0 S not defined in {ψ = 0} regularity issue: the set {ψ = 0} when ψ H 1 (R 3 ) may be irregular (Federer, Ziemer) irrotationality: u = 0, no vortices are taken into account in the WKB description Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

31 Polar Factorization ψ H 1 (R 3 ), define P(ψ) = { φ L s.t. φ L 1, ψ = ψ φ a.e. R 3}. φ P(ψ), then φ = 1 ρ dx-a.e. in R 3 and it is uniquely defined ρ dx a.e. in R 3. Not uniquely defined in {ρ = 0}! We call (any) φ P(ψ) polar factor associated to ψ. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

32 Polar Factorization ψ H 1 (R 3 ), define P(ψ) = { φ L s.t. φ L 1, ψ = ψ φ a.e. R 3}. φ P(ψ), then φ = 1 ρ dx-a.e. in R 3 and it is uniquely defined ρ dx a.e. in R 3. Not uniquely defined in {ρ = 0}! We call (any) φ P(ψ) polar factor associated to ψ. WKB ansatz: ψ = ρe is/, equations for (ρ, S) Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

33 Polar Factorization ψ H 1 (R 3 ), define P(ψ) = { φ L s.t. φ L 1, ψ = ψ φ a.e. R 3}. φ P(ψ), then φ = 1 ρ dx-a.e. in R 3 and it is uniquely defined ρ dx a.e. in R 3. Not uniquely defined in {ρ = 0}! We call (any) φ P(ψ) polar factor associated to ψ. WKB ansatz: ψ = ρe is/, equations for (ρ, S) Polar factorisation: ψ = ρφ, (QHD) Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

34 Lemma (Polar factorisation) Let φ L (R 3 ) be such that ψ = ψ φ a.e. and φ L (R 3 ) 1. Then ρ = Re( φ ψ) a.e., Λ := Im( φ ψ) a.e. and 2 Re( ψ ψ) = 2 ρ ρ + Λ Λ a.e. Furthermore the decomposition is H 1 stable, i.e. if {ψ n } H 1 (R 3 ) s.t. ψ n ψ in H 1, then ρ n ρ, Λ n Λ in L 2 (R 3 ). Remark In general we only have φ n φ weak in L. ψ W 1,1 loc (R3 ), ψ = 0 a.e. in ψ 1 ({0}). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

35 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

36 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

37 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, where J := Im( ψ ψ) is the current density. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

38 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, where J := Im( ψ ψ) is the current density. t J + 2 div ( Re( ψ ψ) ) + P(ρ) = 2 4 ρ. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

39 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, where J := Im( ψ ψ) is the current density. t J + 2 div ( Re( ψ ψ) ) + P(ρ) = 2 4 ρ. We have 2 Re( ψ ψ) = 2 ρ ρ + Λ Λ. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

40 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, where J := Im( ψ ψ) is the current density. t J + div(λ Λ + 2 ρ ρ) + P(ρ) = 2 4 ρ. We have 2 Re( ψ ψ) = 2 ρ ρ + Λ Λ. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

41 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, where J := Im( ψ ψ) is the current density. t J + div(λ Λ) + P(ρ) = 2 4 ρ div( ρ ρ). We have 2 Re( ψ ψ) = 2 ρ ρ + Λ Λ. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

42 QHD through Polar Fact. (Madelung transformation) Moments associated to the wave function. ψ soln. to NLS ρ := ψ 2 mass density t ρ + div J = 0, where J := Im( ψ ψ) is the current density. t J + div(λ Λ) + P(ρ) = 2 4 ρ div( ρ ρ). We have 2 Re( ψ ψ) = 2 ρ ρ + Λ Λ. Rigorously: density argument, continuity of NLS w.r.t. initial data, H 1 stability of polar factorisation. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

43 Energy and generalized irrotationality condition E[ 2 ρ, Λ] = 2 ρ Λ 2 + f (ρ) dx Finite energy weak solutions to (QHD) satisfy J = 2 ρ Λ, for a.e. t. If the solutions are smooth (e.g. the velocity field can be defined), then the generalized irrotationality condition is equivalent to ρ u = 0, a.e. i.e. the velocity field is irrotational ρ dx a.e. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

44 Energy and generalized irrotationality condition E[ 2 ρ, Λ] = 2 ψ 2 + f ( ψ 2 ) dx Finite energy weak solutions to (QHD) satisfy J = 2 ρ Λ, for a.e. t. If the solutions are smooth (e.g. the velocity field can be defined), then the generalized irrotationality condition is equivalent to ρ u = 0, a.e. i.e. the velocity field is irrotational ρ dx a.e. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

45 Energy and generalized irrotationality condition E[ 2 ρ, Λ] = 2 ψ 2 + f ( ψ 2 ) dx = E[ψ] Finite energy weak solutions to (QHD) satisfy J = 2 ρ Λ, for a.e. t. If the solutions are smooth (e.g. the velocity field can be defined), then the generalized irrotationality condition is equivalent to ρ u = 0, a.e. i.e. the velocity field is irrotational ρ dx a.e. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

46 Energy and generalized irrotationality condition E[ 2 ρ, Λ] = 2 ψ 2 + f ( ψ 2 ) dx = E[ψ 0 ] Finite energy weak solutions to (QHD) satisfy J = 2 ρ Λ, for a.e. t. If the solutions are smooth (e.g. the velocity field can be defined), then the generalized irrotationality condition is equivalent to ρ u = 0, a.e. i.e. the velocity field is irrotational ρ dx a.e. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

47 Energy and generalized irrotationality condition E[ 2 ρ, Λ] = 2 ψ 2 + f ( ψ 2 ) dx = E[ ρ 0, Λ 0 ] Finite energy weak solutions to (QHD) satisfy J = 2 ρ Λ, for a.e. t. If the solutions are smooth (e.g. the velocity field can be defined), then the generalized irrotationality condition is equivalent to ρ u = 0, a.e. i.e. the velocity field is irrotational ρ dx a.e. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

48 QHD with collisions t ρ + div J = 0 ( J J t J + div ρ V = ρ ) ( ) + P(ρ) + ρ V + J = 2 ρ 2 ρ ρ Momentum relaxation term introduced to phenomenologically model collisions between electrons in the semiconductor device (Bløtekjær, Baccarani, Wordeman). Energy: E[ρ, J] = R3 2 2 ρ J 2 2 ρ + f (ρ) V 2 dx, dissipates along the flow of solutions E(t) + t 0 R3 J 2 ρ dxdt = E(0). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

49 The collision term J destroys the analogy with NLS, i t ψ = 1 2 ψ + V ψ + f ( ψ 2 )ψ + arg(ψ)ψ V = ψ 2 No good Cauchy theory for this equation. Previous results for QHD with collisions: [Jüngel, Mariani Rial, M3AS, 2002]: local existence for regular initial data, bounded away from zero. [Li, Marcati, CMP, 2004]: local regular solutions, global subsonic solutions. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

50 Theorem Let ψ 0 H 1 (R 3 ) and let ρ 0 := ψ 0 2, J 0 := Im( ψ 0 ψ 0 ). Then there exists a global in time finite energy weak solution (ρ, J) to the QHD system with collisions with initial data (ρ 0, J 0 ). The solution satisfies ρ L (R + : H 1 (R 3 )), Λ L (R + ; L 2 (R 3 )) L 2 (R + ; L 2 (R 3 )) and ρ L q ([0, T ]; W 1,r (R 3 )), Λ L q ([0, T ]; L r (R 3 )), for any 0 < T <, where (q, r) is any arbitrary (Strichartz) admissible pair for Schrödinger in R 3. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

51 Strategy of proof 1 find a sequence of approximate solutions ( ρ τ, Λ τ ); Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

52 Strategy of proof 1 find a sequence of approximate solutions ( ρ τ, Λ τ ); 2 show the sequence converges (compactness); Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

53 Strategy of proof 1 find a sequence of approximate solutions ( ρ τ, Λ τ ); 2 show the sequence converges (compactness); 3 show the limit is a weak solution to the QHD with collisions (consistency). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

54 Strategy of proof 1 find a sequence of approximate solutions ( ρ τ, Λ τ ); 2 show the sequence converges (compactness); 3 show the limit is a weak solution to the QHD with collisions (consistency). Difficulties Good definition of approximate solutions; Prove sufficient a priori estimates to get the compactness. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

55 Fractional step solve the QHD without collisions (NLS) update with collisions Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

56 Fractional step solve the QHD without collisions (NLS) update with collisions t J + J = 0 Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

57 Fractional step solve the QHD without collisions (NLS) update with collisions t J + J = 0 J new = e τ J old (1 τ)j old Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

58 Fractional step solve the QHD without collisions (NLS) update with collisions t J + J = 0 J new = e τ J old (1 τ)j old S new = (1 τ)s old, Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

59 Fractional step solve the QHD without collisions (NLS) update with collisions t J + J = 0 J new = e τ J old (1 τ)j old S new = (1 τ)s old, ψ new = (φ old ) 1 τ ρ old Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

60 More exactly... Lemma (Updating) Let ψ o H 1 (R 3 ) and let ε, τ > 0 be two arbitrary, small real numbers. Then there exists ψ n H 1 (R 3 ) s.t. ρ n = ρ o, Λ n = (1 τ)λ o + r ε, where and r ε L 2 ε, ψ n = ψ o i τ φ Λ o + r ε,τ, with φ L 1, r ε,τ L 2 C(τ ψ o L 2 + ε). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

61 Consistency of the approximate solutions Proposition Let us assume there exist ρ L 2 loc ([0, T ); H 1 loc (R3 )), Λ L 2 loc ([0, T ); L2 loc (R3 )), 0 < T < such that ρ τ ρ in L 2 loc ([0, T ) : H1 loc (R3 )) Λ τ Λ in L 2 loc ([0, T ); L2 loc (R3 )), where ( ρ τ, Λ τ ) is the sequence of approximate solutions constructed above. Then ( ρ, Λ) defines a finite energy weak solution to the QHD system with collisions. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

62 Compactness of approximate solutions With the fractional step we get a sequence {ψ τ } (and consequently {( ρ τ, Λ τ )}). Energy dissipation for approximate solutions: E τ (t) + τ 2 [t/τ] 1 k=0 Λ τ (kτ ) 2 dx (1 + τ)e 0. Up to passing to subsequences, ψ τ ψ in L (R + : H 1 (R 3 )). Not sufficient for the quadratic terms ρ τ ρ τ + Λ τ Λ τ = Re( ψ τ ψ τ ) Need further compactness: Use dispersion Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

63 Dispersive estimates for Schrödinger semigroup Strichartz estimates (Strichartz, Ginibre-Velo, Keel-Tao) (q, r) are admissible if 2 q, 2 r 6 and 1 q = 3 ( ) r e i 2 t f L q t Lr x f L 2 t e i 2 (t s) F (s) ds L q t 0 Lr x F L q t L r x Local smoothing estimates (Constantin-Saut, Sjölin, Vega) e i 2 t f L 2 ([0,T ];H 1/2 loc (R3 )) f L 2 t e i 2 (t s) F (s) ds L 0 2 ([0,T ];H 1/2 loc (R3 )) F L 1 t L2 x Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

64 Compactness estimates From the updating Lemma ψ τ (t) =e i t 2 ψ 0 i i τ [t/τ] 1 k=0 t 0 e i (t s) 2 (V τ ψ τ + f ( ψ τ 2 )ψ τ )(s) ds e i (t kτ) 2 ( φ k,τ Λτ (kτ ) + r k,τ ). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

65 Compactness estimates From the updating Lemma ψ τ (t) =e i t 2 ψ 0 i i τ [t/τ] 1 k=0 t 0 e i (t s) 2 (V τ ψ τ + f ( ψ τ 2 )ψ τ )(s) ds e i (t kτ) 2 ( φ k,τ Λτ (kτ ) + r k,τ ). By Strichartz estimates (with a standard bootstrap argument) ψ τ L q t Lr x ([0,T ] R3 ) C(E 0, M 0, T ) Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

66 Compactness estimates From the updating Lemma ψ τ (t) =e i t 2 ψ 0 i i τ [t/τ] 1 k=0 t 0 e i (t s) 2 (V τ ψ τ + f ( ψ τ 2 )ψ τ )(s) ds e i (t kτ) 2 ( φ k,τ Λτ (kτ ) + r k,τ ). By Strichartz estimates (with a standard bootstrap argument) ψ τ L q t Lr x ([0,T ] R3 ) C(E 0, M 0, T ) and by using this + local smoothing ψ τ L 2 ([0,T ]:H 1/2 loc (R3 )) C(E 0, M 0, T ), for any 0 < T <. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

67 Aubin-Lions type lemma Proposition For any 0 < T <, ψ τ ψ in L 2 ([0, T ]; L 2 loc (R3 )), up to passing to subsequences. In particular, ρ τ ρ, Λ τ Λ in L 2 ([0, T ]; L 2 loc (R3 )). By the consistency of the sequence of approximate solutions, (ρ, J) := (( ρ) 2, ρλ) is a finite energy weak solution to the QHD system with collisions in [0, T ] R 3, for any 0 < T <. Thus the global existence theorem follows. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

68 Superfluidity at finite temperatures (work in progress with P. Marcati, M. D Amico) Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

69 Landau two fluid model Khalatnikov, An introduction to the theory of Superfluidity Griffin, Nikuni, Zaremba, Bose-condensed gases at finite temperatures + P s (ρ s ) + ρ s V ext = 1 2 ρ s t ρ n + div(ρ n v n ) = Γ 21 t (ρ n v n ) + div(ρ n v n v n ) t ρ s + div(ρ s v s ) = Γ 12 t (ρ s v s ) + div(ρ s v s v s ) + P n (ρ n ) + ρ n V ext = div entropy eqn. ( ρs ρs ) Q 12 ( ( 2η D v n 1 )) 3 1 Tr D v n Q 21 Superfluidity near the λ point/bec at finite temperatures. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

70 Landau two fluid model - (very) simplified V ext = 0, Γ 12 = Γ 21 = 0, Q 21 = 0, Q 12 = 1 τ ρ 1(v 1 v 2 ). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

71 Landau two fluid model - (very) simplified V ext = 0, Γ 12 = Γ 21 = 0, Q 21 = 0, Q 12 = 1 τ ρ 1(v 1 Qv 2 ). t ρ 1 + div J 1 = 0 ( ) J1 J 1 t J 1 + div + P 1 (ρ 1 ) = 1 ( ) 2 ρ ρ1 1 1 ρ1 τ (J 1 ρ 1 Qv 2 ) ρ 1 t ρ 2 + div(ρ 2 v 2 ) = 0 t (ρ 2 v 2 ) + div(ρ 2 v 2 v 2 ) + P 2 (ρ 2 ) = η v η div v 2, Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

72 Landau two fluid model - (very) simplified V ext = 0, Γ 12 = Γ 21 = 0, Q 21 = 0, Q 12 = 1 τ ρ 1(v 1 Qv 2 ). t ρ 1 + div J 1 = 0 ( ) J1 J 1 t J 1 + div + P 1 (ρ 1 ) = 1 ( ) 2 ρ ρ1 1 1 ρ1 τ (J 1 ρ 1 Qv 2 ) ρ 1 t ρ 2 + div(ρ 2 v 2 ) = 0 t (ρ 2 v 2 ) + div(ρ 2 v 2 v 2 ) + P 2 (ρ 2 ) = η v η div v 2, 1 t 2 ρ 2 v f 2 (ρ 2 ) dx + η 0 v div v 2 3 dx 1 2 ρ 2,0 v 2,0 2 + f 2 (ρ 2,0 ) dx. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

73 Landau two fluid model - (very) simplified V ext = 0, Γ 12 = Γ 21 = 0, Q 21 = 0, Q 12 = 1 τ ρ 1(v 1 Qv 2 ). t ρ 1 + div J 1 = 0 ( ) J1 J 1 t J 1 + div + P 1 (ρ 1 ) = 1 ( ) 2 ρ ρ1 1 1 ρ1 τ (J 1 ρ 1 Qv 2 ) ρ 1 t ρ 2 + div(ρ 2 v 2 ) = 0 t (ρ 2 v 2 ) + div(ρ 2 v 2 v 2 ) + P 2 (ρ 2 ) = η v η div v 2, 1 t 2 ρ 2 v f 2 (ρ 2 ) dx + η 0 v div v 2 3 dx 1 2 ρ 2,0 v 2,0 2 + f 2 (ρ 2,0 ) dx. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

74 Superfluid part at NLS level i t ψ = 1 2 ψ + Ṽ ψ + f 1( ψ 2 )ψ, (1) where Ṽ s.t. Ṽ = Qv 2. Theorem (Ortner, Süli, 2012) Ṽ = V + V p, where for a.e. t R, V (t) C (R 3 ); V p L 2 t Wx 1,6 and V p L 2 t Wx 1,6 Ṽ L 2 t L 6 v 2 x L 2 t L 6; x α V L 2 t L x C Ṽ L 2, for all α 1. t L6 x Ingredients: GWP for (1), Strichartz and local smoothing estimates for (1), fractional step. Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

75 Construction of the fundamental solution of the Schrödinger equation Theorem (Fujiwara, J. Anal. Math. 1979) Assume t R, V (t, ) C (R d ); α N d, α 2, sup (t,x) R R d α V (t, x) C. Then there exists unitary operator U(t, s) such that U(t, s)f is the solution to i t u = 1 u + Vu 2 u(s) = f. Problem: V is such that α V L 2 t L x (R R d )! (work in progress) Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

76 Towards a more realistic two-fluids model... Equation for the order parameter (Khalatnikov) i t ψ = 2 2m ψ+f ( ψ 2 )ψ iλm [ ( 1 i ) 2 2 m v n ψ + f 1( ψ 2 )ψ Total mass and momentum density evolutions t (ρ s + ρ n ) + div(ρ s v s + ρ n v n ) = 0 t (ρ s v s + ρ n v n ) + div(ρ s v s v s + ρ n v n v n + pi) = ( div η D v n 2 ) 3 η div v ni +entropy ] Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

77 Conclusions existence of finite energy weak solutions for quantum fluids models: no further regularity or smallness assumptions; no need to define the velocity field: polar factorisation ( ρ, Λ); vacuum quantized vortices; Future perspectives uniqueness/stability; (more physical) two-fluid models; quantum plasma physics (Quantum MHD). Antonelli, Marcati (GSSI) QHD and Superfluidity / 31

78 References A., Marcati, On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics, CMP 2009 A. Marcati, The Quantum Hydrodynamic System in two Space Dimensions, ARMA 2012 A., Marcati, Finite Energy Global Solutions to a Two-Fluid Model Arising in Superfluidity, Bull. Acad. Sinica Antonelli, Marcati (GSSI) QHD and Superfluidity / 31