Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Université P. et M. Curie Paris 6, Laboratoire Jacques-Louis Lions & Institut Universitaire de France Jean-Michel Coron’s 60th birthday, June 20, 2016
The Ginzburg-Landau equations u : Ω ⊂ R 2 → C − ∆ u = u ε 2 ( 1 − | u | 2 ) Ginzburg-Landau equation (GL) ∂ t u = ∆ u + u ε 2 ( 1 − | u | 2 ) parabolic GL equation (PGL) i ∂ t u = ∆ u + u ε 2 ( 1 − | u | 2 ) Gross-Pitaevskii equation (GP) Associated energy � |∇ u | 2 + ( 1 − | u | 2 ) 2 E ε ( u ) = 1 2 2 ε 2 Ω Models: superconductivity, superfluidity, Bose-Einstein condensates, nonlinear optics
Vortices ◮ in general | u | ≤ 1, | u | ≃ 1 = superconducting/superfluid phase, | u | ≃ 0 = normal phase ◮ u has zeroes with nonzero degrees = vortices ◮ u = ρ e i ϕ , characteristic length scale of { ρ < 1 } is ε = vortex core size ◮ degree of the vortex at x 0 : � 1 ∂ϕ ∂τ = d ∈ Z 2 π ∂ B ( x 0 , r ) ◮ In the limit ε → 0 vortices become points , (or curves in dimension 3).
Solutions of (GL), bounded number N of vortices ◮ minimal energy min E ε = π N | log ε | + min W + o ( 1 ) as ε → 0 ◮ u ε minimizing E ε has vortices all of degree + 1 (or all − 1) which converge to a minimizer of � W (( x 1 , d 1 ) , . . . , ( x N , d N )) = − π d i d j log | x i − x j | + boundary terms... i � = j “renormalized energy" , Kirchhoff-Onsager energy (in the whole plane) [Bethuel-Brezis-Hélein ’94] ◮ Some boundary condition needed to obtain nontrivial minimizers ◮ nonminimizing solutions: u ε has vortices which converge to a critical point of W : ∇ i W ( { x i } ) = 0 ∀ i = 1 , · · · N [Bethuel-Brezis-Hélein ’94] ◮ stable solutions converge to stable critical points of W [S. ’05]
Solutions of (GL), bounded number N of vortices ◮ minimal energy min E ε = π N | log ε | + min W + o ( 1 ) as ε → 0 ◮ u ε minimizing E ε has vortices all of degree + 1 (or all − 1) which converge to a minimizer of � W (( x 1 , d 1 ) , . . . , ( x N , d N )) = − π d i d j log | x i − x j | + boundary terms... i � = j “renormalized energy" , Kirchhoff-Onsager energy (in the whole plane) [Bethuel-Brezis-Hélein ’94] ◮ Some boundary condition needed to obtain nontrivial minimizers ◮ nonminimizing solutions: u ε has vortices which converge to a critical point of W : ∇ i W ( { x i } ) = 0 ∀ i = 1 , · · · N [Bethuel-Brezis-Hélein ’94] ◮ stable solutions converge to stable critical points of W [S. ’05]
Dynamics, bounded number N of vortices ◮ For well-prepared initial data, d i = ± 1, solutions to (PGL) have vortices which converge (after some time-rescaling) to solutions to dx i dt = −∇ i W ( x 1 , . . . , x N ) [Lin ’96, Jerrard-Soner ’98, Lin-Xin ’99, Spirn ’02, Sandier-S ’04] ◮ For well-prepared initial data, d i = ± 1, solutions to (GP) dx i ∇ ⊥ = ( − ∂ 2 , ∂ 1 ) dt = −∇ ⊥ i W ( x 1 , . . . , x N ) [Colliander-Jerrard ’98, Spirn ’03, Bethuel-Jerrard-Smets ’08] ◮ All these hold up to collision time ◮ For (PGL), extensions beyond collision time and for ill-prepared data [Bethuel-Orlandi-Smets ’05-07, S. ’07]
A word about dimension 3 (or higher) ◮ Leading order of the energy becomes π | d | L | log ε | where L = length (or area) of vortex line (integer multiplicity rectifiable current) ◮ Minimizers/solutions to (GL) converge to length minimizing / stationary currents (= straight lines) [Rivière ’95, Lin-Rivière ’01, Sandier ’01, Bethuel-Brezis-Orlandi ’01, Jerrard-Soner ’02, Alberti-Baldo-Orlandi ’03, Bourgain-Brezis-Mironescu ’04] ◮ (PGL) → mean curvature motion (Brakke) [Bethuel-Orlandi-Smets ’06] ◮ (GP) → binormal flow (partial results) [Jerrard ’02]
Vorticity ◮ In the case N ε → ∞ , describe the vortices via the vorticity : supercurrent � a , b � := 1 2 ( a ¯ j ε := � iu ε , ∇ u ε � b + ¯ ab ) vorticity µ ε := curl j ε ◮ ≃ vorticity in fluids, but quantized: µ ε ≃ 2 π � i d i δ a ε i µ ε 2 π N ε → µ signed measure, or probability measure, ◮
Mean-field limit for stationary solutions If u ε is a solution to (GL) and N ε ≫ 1 then µ ε / N ε → µ solution to h = − ∆ − 1 µ µ ∇ h = 0 in a suitable weak sense ( ≃ Delort): T µ := −∇ h ⊗ ∇ h + 1 2 |∇ h | 2 δ j i Weak relation is div T µ = 0 in “finite parts" [Sandier-S ’04] � h is constant on the support of µ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ����� ����� ������������ ������������ ����� ����� ������������ ������������ ������������ ������������ ����� ����� ����� ����� ������������ ������������ ����� ����� ������������ ������������ ����� ����� c 1 ������������ ������������ c 2 ����� ����� ������������ ������������ ����� ����� ������������ ������������ ����� ����� ������������ ������������ ������������ ������������ ����� ����� ������������ ������������ ����� ����� ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ Ω
Dynamics in the case N ε ≫ 1 Back to | log ε | ∂ t u = ∆ u + u N ε ε 2 ( 1 − | u | 2 ) in R 2 (PGL) iN ε ∂ t u = ∆ u + u ε 2 ( 1 − | u | 2 ) in R 2 (GP) ◮ For (GP), by Madelung transform, the limit dynamics is expected to be the 2D incompressible Euler equation. Vorticity form ∂ t µ − div ( µ ∇ ⊥ h ) = 0 h = − ∆ − 1 µ (EV) ◮ For (PGL), formal model proposed by [Chapman-Rubinstein-Schatzman ’96], [E ’95]: if µ ≥ 0 h = − ∆ − 1 µ ∂ t µ − div ( µ ∇ h ) = 0 (CRSE)
Dynamics in the case N ε ≫ 1 Back to | log ε | ∂ t u = ∆ u + u N ε ε 2 ( 1 − | u | 2 ) in R 2 (PGL) iN ε ∂ t u = ∆ u + u ε 2 ( 1 − | u | 2 ) in R 2 (GP) ◮ For (GP), by Madelung transform, the limit dynamics is expected to be the 2D incompressible Euler equation. Vorticity form ∂ t µ − div ( µ ∇ ⊥ h ) = 0 h = − ∆ − 1 µ (EV) ◮ For (PGL), formal model proposed by [Chapman-Rubinstein-Schatzman ’96], [E ’95]: if µ ≥ 0 h = − ∆ − 1 µ ∂ t µ − div ( µ ∇ h ) = 0 (CRSE)
Study of the Chapman-Rubinstein-Schatzman-E equation ◮ [Lin-Zhang ’00, Du-Zhang ’03, Masmoudi-Zhang ’05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L ∞ solutions, which decay in 1 / t (uses pseudo-differential operators) ◮ [Ambrosio-S ’08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of � Ω |∇ ∆ − 1 µ | 2 for the 2-Wasserstein metric (à la [Otto, F ( µ ) = 1 2 Ambrosio-Gigli-Savaré]). ◮ [S-Vazquez ’13] PDE approach in all dimension. Existence via limits in fractional diffusion ∂ t µ + div ( µ ∇ ∆ − s µ ) when s → 1, uniqueness in the class L ∞ , propagation of regularity, asymptotic self-similar profile µ ( t ) = 1 π t 1 B √ t
Study of the Chapman-Rubinstein-Schatzman-E equation ◮ [Lin-Zhang ’00, Du-Zhang ’03, Masmoudi-Zhang ’05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L ∞ solutions, which decay in 1 / t (uses pseudo-differential operators) ◮ [Ambrosio-S ’08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of � Ω |∇ ∆ − 1 µ | 2 for the 2-Wasserstein metric (à la [Otto, F ( µ ) = 1 2 Ambrosio-Gigli-Savaré]). ◮ [S-Vazquez ’13] PDE approach in all dimension. Existence via limits in fractional diffusion ∂ t µ + div ( µ ∇ ∆ − s µ ) when s → 1, uniqueness in the class L ∞ , propagation of regularity, asymptotic self-similar profile µ ( t ) = 1 π t 1 B √ t
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