Hydrodynamic Limit of the Gross-Pitaevskii equation Kung-Chien Wu Department of Pure Mathematics and Mathematical Statistics University of Cambridge, UK and Institute of Mathematics, Academia Sinica, Taiwan June 26, 2012 Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Outline • Introduction • Wave Group • Main Theorem and Proof Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Outline • Introduction • Wave Group • Main Theorem and Proof Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Gross-Pitaevskii equation Time scaled Gross-Pitaevskii equation i ε α ∂ t ψ ε + ε 2 α 2 ∆ ψ ε − 1 ε 2 ( | ψ ε | 2 − ρ 0 ) ψ ε = 0 . Madelung transform (1927) ψ ε = R exp( iS /ε α ) GP becomes ∂ t R + R 2 ∆ S + ∇ R · ∇ S = 0 , 2 |∇ S | 2 + R 2 − ρ 0 = ε 2 α ∂ t S + 1 ∆ R R . ε 2 2 Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Gross-Pitaevskii equation Time scaled Gross-Pitaevskii equation i ε α ∂ t ψ ε + ε 2 α 2 ∆ ψ ε − 1 ε 2 ( | ψ ε | 2 − ρ 0 ) ψ ε = 0 . Madelung transform (1927) ψ ε = R exp( iS /ε α ) GP becomes ∂ t R + R 2 ∆ S + ∇ R · ∇ S = 0 , 2 |∇ S | 2 + R 2 − ρ 0 = ε 2 α ∂ t S + 1 ∆ R R . ε 2 2 Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Gross-Pitaevskii equation Time scaled Gross-Pitaevskii equation i ε α ∂ t ψ ε + ε 2 α 2 ∆ ψ ε − 1 ε 2 ( | ψ ε | 2 − ρ 0 ) ψ ε = 0 . Madelung transform (1927) ψ ε = R exp( iS /ε α ) GP becomes ∂ t R + R 2 ∆ S + ∇ R · ∇ S = 0 , 2 |∇ S | 2 + R 2 − ρ 0 = ε 2 α ∂ t S + 1 ∆ R R . ε 2 2 Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure Hydrodynamic Variables ρ ε = R 2 = | ψ ε | 2 i ε α u ε = ∇ S = 2 | ψ ε | 2 ( ψ ε ∇ ψ ε − ψ ε ∇ ψ ε ) ϕ ε = ρ ε − ρ 0 J ε = ρ ε u ε , , ε Hydrodynamic structure of GP ∂ t ρ ε + ∇ · ( ρ ε u ε ) = 0 , � ∆ √ ρ ε � ε 2 ∇ ( ρ ε − ρ 0 ) = ε 2 α ∂ t u ε + ( u ε · ∇ ) u ε + 1 2 ∇ √ ρ ε . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure Hydrodynamic Variables ρ ε = R 2 = | ψ ε | 2 i ε α u ε = ∇ S = 2 | ψ ε | 2 ( ψ ε ∇ ψ ε − ψ ε ∇ ψ ε ) ϕ ε = ρ ε − ρ 0 J ε = ρ ε u ε , , ε Hydrodynamic structure of GP ∂ t ρ ε + ∇ · ( ρ ε u ε ) = 0 , � ∆ √ ρ ε � ε 2 ∇ ( ρ ε − ρ 0 ) = ε 2 α ∂ t u ε + ( u ε · ∇ ) u ε + 1 2 ∇ √ ρ ε . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure The hydrodynamic Euler equation ( ρ ε , J ε ) ∂ t ρ ε + ∇ · J ε = 0 , � J ε ⊗ J ε � ρ ε ∇ 2 log ρ ε � 2 ∇ ( ϕ ε ) 2 = ε 2 α � + 1 ερ 0 ∇ ϕ ε + 1 ∂ t J ε + ∇ · 4 ∇ · . ρ ε • J ε 0 → J 0 = ρ 0 v 0 , ϕ ε 0 → 0, and ∇ · ( ρ 0 v 0 ) = 0. Hydrodynamic Limit ( ε → 0 ) Lake equations (anelastic system) with nonconstant density ρ 0 � � ∇ · ρ 0 u = 0 , ∂ t ( ρ 0 u ) + ∇ · ( ρ 0 u ⊗ u ) + ρ 0 ∇ π = 0 , ρ 0 u ( x , 0) = ρ 0 v 0 . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure The hydrodynamic Euler equation ( ρ ε , J ε ) ∂ t ρ ε + ∇ · J ε = 0 , � J ε ⊗ J ε � ρ ε ∇ 2 log ρ ε � 2 ∇ ( ϕ ε ) 2 = ε 2 α � + 1 ερ 0 ∇ ϕ ε + 1 ∂ t J ε + ∇ · 4 ∇ · . ρ ε • J ε 0 → J 0 = ρ 0 v 0 , ϕ ε 0 → 0, and ∇ · ( ρ 0 v 0 ) = 0. Hydrodynamic Limit ( ε → 0 ) Lake equations (anelastic system) with nonconstant density ρ 0 � � ∇ · ρ 0 u = 0 , ∂ t ( ρ 0 u ) + ∇ · ( ρ 0 u ⊗ u ) + ρ 0 ∇ π = 0 , ρ 0 u ( x , 0) = ρ 0 v 0 . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure The hydrodynamic Euler equation ( ρ ε , J ε ) ∂ t ρ ε + ∇ · J ε = 0 , � J ε ⊗ J ε � ρ ε ∇ 2 log ρ ε � 2 ∇ ( ϕ ε ) 2 = ε 2 α � + 1 ερ 0 ∇ ϕ ε + 1 ∂ t J ε + ∇ · 4 ∇ · . ρ ε • J ε 0 → J 0 = ρ 0 v 0 , ϕ ε 0 → 0, and ∇ · ( ρ 0 v 0 ) = 0. Hydrodynamic Limit ( ε → 0 ) Lake equations (anelastic system) with nonconstant density ρ 0 � � ∇ · ρ 0 u = 0 , ∂ t ( ρ 0 u ) + ∇ · ( ρ 0 u ⊗ u ) + ρ 0 ∇ π = 0 , ρ 0 u ( x , 0) = ρ 0 v 0 . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure Dispersive limit of the Schr¨ odinger type equations M. Puel (CPDE, 02); A. J¨ ungel, S. Wang (CPDE, 03); F. H. Lin, P. Zhang (CMP, 05); T. C. Lin, P. Zhang (CMP, 06); C.K. Lin, K.C. Wu (JMPA, to appear). Question : How about ∇ · J 0 � = 0 and ϕ ε 0 → ϕ 0 . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Hydrodynamic Structure Dispersive limit of the Schr¨ odinger type equations M. Puel (CPDE, 02); A. J¨ ungel, S. Wang (CPDE, 03); F. H. Lin, P. Zhang (CMP, 05); T. C. Lin, P. Zhang (CMP, 06); C.K. Lin, K.C. Wu (JMPA, to appear). Question : How about ∇ · J 0 � = 0 and ϕ ε 0 → ϕ 0 . Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Review Previous Work Incompressible limit of the Navier-Stokes or Euler system • classical solution: S. Klainerman, A. Majda (CPAM, 81). • weak solutions: P.L. Lion, N. Masmoudi (JMPA, 98). Incompressible limit with nonconstant density • D. Bresch, M. Gisclon, C. K. Lin (M2AN, 05). • D. Bresch, B. Desjardins, G. M´ etivier (06). • N. Masmoudi (JMPA, 07). • E. Feireisl, J. M´ alek, A. Novotn´ y, I. Straskraba (CPDE, 08). Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Review Previous Work Incompressible limit of the Navier-Stokes or Euler system • classical solution: S. Klainerman, A. Majda (CPAM, 81). • weak solutions: P.L. Lion, N. Masmoudi (JMPA, 98). Incompressible limit with nonconstant density • D. Bresch, M. Gisclon, C. K. Lin (M2AN, 05). • D. Bresch, B. Desjardins, G. M´ etivier (06). • N. Masmoudi (JMPA, 07). • E. Feireisl, J. M´ alek, A. Novotn´ y, I. Straskraba (CPDE, 08). Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Helmholtz Decomposition Let f ∈ L 2 1 /ρ 0 ( T n ), the weighted Helmholtz decomposition f = H ρ 0 [ f ] ⊕ H ⊥ ρ 0 [ f ] with H ⊥ div H ρ 0 [ f ] = 0 , ρ 0 [ f ] = ρ 0 ∇ Ψ . where Ψ ∈ D 1 , 2 ( T n ) is the unique solution of the problem � � ∀ ϕ ∈ D 1 , 2 ( T n ) . T n ρ 0 ∇ Ψ · ∇ ϕ dx = T n f · ∇ ϕ dx , D 1 , 2 ( T n ) : completion of C ∞ 0 ( T n ) w.r.t. �∇ · � L 2 1 /ρ 0 ( T n ) . L 2 1 /ρ 0 ( T n ) : weighted Hilbert space with the scalar product � T n v · w dx < v , w > 1 /ρ 0 = . ρ 0 Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Helmholtz Decomposition Let f ∈ L 2 1 /ρ 0 ( T n ), the weighted Helmholtz decomposition f = H ρ 0 [ f ] ⊕ H ⊥ ρ 0 [ f ] with H ⊥ div H ρ 0 [ f ] = 0 , ρ 0 [ f ] = ρ 0 ∇ Ψ . where Ψ ∈ D 1 , 2 ( T n ) is the unique solution of the problem � � ∀ ϕ ∈ D 1 , 2 ( T n ) . T n ρ 0 ∇ Ψ · ∇ ϕ dx = T n f · ∇ ϕ dx , D 1 , 2 ( T n ) : completion of C ∞ 0 ( T n ) w.r.t. �∇ · � L 2 1 /ρ 0 ( T n ) . L 2 1 /ρ 0 ( T n ) : weighted Hilbert space with the scalar product � T n v · w dx < v , w > 1 /ρ 0 = . ρ 0 Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Helmholtz Decomposition (A) Conservation of charge ∂ ∂ t ρ ε + ∇ · J ε = 0 . (B) Conservation of momentum (current) � � ∂ t J ε + 1 ∂ ( ∇ ψ ε ⊗ ∇ ψ ε + ∇ ψ ε ⊗ ∇ ψ ε ) − ∇ 2 ( | ψ ε | 2 ) 2 ε 2 α ∇ · +1 2 ∇ ( ϕ ε ) 2 + 1 ερ 0 ∇ ϕ ε = 0 . Define J ε = H ρ 0 [ J ε ] + H ⊥ ρ 0 [ J ε ] = H ρ 0 [ J ε ] + ρ 0 ∇ w ε , Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
Helmholtz Decomposition (A) Conservation of charge ∂ ∂ t ρ ε + ∇ · J ε = 0 . (B) Conservation of momentum (current) � � ∂ t J ε + 1 ∂ ( ∇ ψ ε ⊗ ∇ ψ ε + ∇ ψ ε ⊗ ∇ ψ ε ) − ∇ 2 ( | ψ ε | 2 ) 2 ε 2 α ∇ · +1 2 ∇ ( ϕ ε ) 2 + 1 ερ 0 ∇ ϕ ε = 0 . Define J ε = H ρ 0 [ J ε ] + H ⊥ ρ 0 [ J ε ] = H ρ 0 [ J ε ] + ρ 0 ∇ w ε , Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
the equation can be rewritten as ε∂ t ϕ ε + div ( ρ 0 ∇ w ε ) = 0 , ε∂ t ( √ ρ 0 ∇ w ε ) + √ ρ 0 ∇ ϕ ε = ε √ ρ 0 F ε , 1 where F ε = − ε 2 α � ∇ ψ ε ⊗ ∇ ψ ε + ∇ ψ ε ⊗ ∇ ψ ε � 2 H ⊥ ρ 0 ∇ · ρ 0 ∇ ( ϕ ε ) 2 + ε 2 α − 1 ρ 0 ∇ ∆ ρ ε . 2 H ⊥ 4 H ⊥ It is obvious that ∂ t ϕ ε and ∂ t ( √ ρ 0 ∇ w ε ) are of order O (1 /ε ) and are highly oscillatory as ε → 0. So we have to introduce the wave group in order to filter out the fast oscillating wave. Kung-Chien Wu Hydrodynamic Limit of the Gross-Pitaevskii equation
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