Blow-up of neutron stars in the HartreeFock theory Dinh-Thi Nguyen - PowerPoint PPT Presentation

Introduction Time-Independent Time-Dependent Reference Blow-up of neutron stars in the Hartree–Fock theory Dinh-Thi Nguyen Mathematisches Institut Ludwig–Maximilians–Universit¨ at M¨ unchen The Analysis of Complex Quantum Systems Large Coulomb Systems and Related Matters CIRM, Marseille 2019 D.-T Nguyen Blow-up of neutron stars 1 / 12

Introduction Time-Independent Time-Dependent Reference N -particles Schr¨ odinger operator N –particles fermionic Hamiltonian ( � = c = 1) N � − ∆ x i + m 2 − κ | x i − x j | − 1 � � H N = i =1 1 ≤ i < j ≤ N 0 ≤ m is the neutron mass, κ = Gm 2 with G the gravitational constant. Quantum Energy (internal spin degree of freedom q ≥ 1) N � � E Q L 2 ( R 3 , C q ) , � ψ N � L 2 = 1 � N = inf � ψ N , H N ψ N � : ψ N ∈ . D.-T Nguyen Blow-up of neutron stars 2 / 12

Introduction Time-Independent Time-Dependent Reference N -particles Schr¨ odinger operator N –particles fermionic Hamiltonian ( � = c = 1) N � − ∆ x i + m 2 − κ | x i − x j | − 1 � � H N = i =1 1 ≤ i < j ≤ N 0 ≤ m is the neutron mass, κ = Gm 2 with G the gravitational constant. Quantum Energy (internal spin degree of freedom q ≥ 1) N � � E Q L 2 ( R 3 , C q ) , � ψ N � L 2 = 1 � N = inf � ψ N , H N ψ N � : ψ N ∈ . Questions 2 1 Behaviors of E Q 3 ր τ c . N and ground states when N → ∞ and κ N 2 Blow-up profile with minimal Chandrasekhar limit mass of finite time blow-up solutions of the evolution equation. D.-T Nguyen Blow-up of neutron stars 2 / 12

Introduction Time-Independent Time-Dependent Reference Chandraskhar limit mass Hardy–Littlewood–Sobolev inequality ( σ f ≈ 1 . 092) �� ρ ( x ) ρ ( y ) L 1 ≥ 1 4 2 ∀ 0 ≤ ρ ∈ L 1 ∩ L 4 3 ( R 3 ) . σ f � ρ � 3 3 � ρ � 3 | x − y | d x d y , 4 2 L ◮ There exists a unique optimizer Q , which can be chosen non-negative radially symmetric decreasing and satisfies �� Q ( x ) Q ( y ) Q ( x ) d x = 1 � � 4 3 d x = σ f Q ( x ) | x − y | d x d y = 1 . 2 ◮ Q has compact support and solves the Lane–Emden equation (Lieb–Oxford ’81) 4 �� Q ⋆ 1 � − 2 � 1 3 = 3 σ f Q ( x ) . |·| 3 + D.-T Nguyen Blow-up of neutron stars 3 / 12

Introduction Time-Independent Time-Dependent Reference Chandraskhar limit mass Hardy–Littlewood–Sobolev inequality ( σ f ≈ 1 . 092) �� ρ ( x ) ρ ( y ) L 1 ≥ 1 4 2 ∀ 0 ≤ ρ ∈ L 1 ∩ L 4 3 ( R 3 ) . σ f � ρ � 3 3 � ρ � 3 | x − y | d x d y , 4 2 L ◮ There exists a unique optimizer Q , which can be chosen non-negative radially symmetric decreasing and satisfies �� Q ( x ) Q ( y ) Q ( x ) d x = 1 � � 4 3 d x = σ f Q ( x ) | x − y | d x d y = 1 . 2 ◮ Q has compact support and solves the Lane–Emden equation (Lieb–Oxford ’81) 4 �� Q ⋆ 1 � − 2 � 1 3 = 3 σ f Q ( x ) . |·| 3 + � τ c � 6 π 2 � 1 � 3 2 with τ c = K cl 3 . σ f and K cl = 3 ◮ Chandrasekhar limit mass : κ 4 q D.-T Nguyen Blow-up of neutron stars 3 / 12

Introduction Time-Independent Time-Dependent Reference Subcritical regime N � − ∆ x i + m 2 − κ � � | x i − x j | − 1 H N = i =1 1 ≤ i < j ≤ N Theorem (Lieb–Yau ’87) 2 3 < τ c . Then Fix τ = κ N � 4 N →∞ N − 1 E Q � � N = E Ch E Ch 3 ( R 3 ) , lim (1) := inf τ ( ρ ) : 0 ≤ ρ ∈ L ρ = 1 τ where �� ρ ( x ) ρ ( y ) | p | 2 + m 2 d p d x − τ q � � � E Ch τ ( ρ ) = | x − y | d x d y . � 1 � 6 π 2 ρ ( x ) (2 π ) 3 2 3 | p | < q Futhermore, if τ > τ c then E Ch (1) = −∞ . τ D.-T Nguyen Blow-up of neutron stars 4 / 12

Introduction Time-Independent Time-Dependent Reference The Hartree–Fock Variational Problem ◮ The Hartree–Fock functional �� ρ γ ( x ) ρ γ ( y ) − | γ ( x , y ) | 2 − ∆ + m 2 γ − κ � E HF ( γ ) = Tr d x d y . 2 | x − y | ◮ The set of Hartree–Fock states √ K HF = { γ = γ ∗ ∈ S 1 ( L 2 ( R 3 ; C q )) : Tr 1 − ∆ γ < + ∞ , 0 ≤ γ ≤ 1 } . Theorem (Lenzmann–Lewin ’10) Fix q ≥ 1. Suppose m > 0 and κ < 4 /π . Then for all 0 < N < N HF ( κ ) � τ c � 3 2 ), there exists a minimizer for ( ∼ κ → 0 κ E HF ( N ) = inf {E HF ( γ ) : γ ∈ K HF , Tr γ = N } . Furthermore, for every 0 < τ = κ N 2 / 3 < τ c , N →∞ N − 1 E HF ( N ) = E Ch lim (1) . τ D.-T Nguyen Blow-up of neutron stars 5 / 12

Introduction Time-Independent Time-Dependent Reference The First Main Result Theorem 1 (arXiv:1903.10062) 2 3 = τ c − N − β with 0 < β < 1 / 9. Then Fix q ≥ 1, m > 0. Let τ N := κ N � Λ = 3 1 � 1 2 N − 1 E HF ( N ) = ( τ c − τ N ) 3 . 2 � 2Λ + o (1) N →∞ � , 4 m R 3 Q ( x ) K cl Let γ N be a minimizer for E HF ( N ) and let ρ γ N ( x ) = γ N ( x , x ). Then there exists a sequence { y N } ⊂ R 3 such that, up to subsequence, 3 1 1 3 x + y N ) = Λ 3 Q (Λ x ) 2 ρ γ N (( τ c − τ N ) 2 N N →∞ ( τ c − τ N ) lim 4 strongly in L r ( R 3 ) for 1 ≤ r < 4 3 ( R 3 ). 3 and weakly in L D.-T Nguyen Blow-up of neutron stars 6 / 12

Introduction Time-Independent Time-Dependent Reference The First Main Result Theorem 1 (arXiv:1903.10062) 2 3 = τ c − N − β with 0 < β < 1 / 9. Then Fix q ≥ 1, m > 0. Let τ N := κ N � Λ = 3 1 � 1 2 N − 1 E HF ( N ) = ( τ c − τ N ) 3 . 2 � 2Λ + o (1) N →∞ � , 4 m R 3 Q ( x ) K cl Let γ N be a minimizer for E HF ( N ) and let ρ γ N ( x ) = γ N ( x , x ). Then there exists a sequence { y N } ⊂ R 3 such that, up to subsequence, 3 1 1 3 x + y N ) = Λ 3 Q (Λ x ) 2 ρ γ N (( τ c − τ N ) 2 N N →∞ ( τ c − τ N ) lim 4 strongly in L r ( R 3 ) for 1 ≤ r < 4 3 ( R 3 ). 3 and weakly in L 4 3 ( R 3 ) convergence requires to prove the Lieb–Thirring ◮ Attaining the L inequality with the sharp constant. D.-T Nguyen Blow-up of neutron stars 6 / 12

Introduction Time-Independent Time-Dependent Reference The Hartree–Fock Evolution Equation �� � ρ γ t ⋆ 1 � + κγ t ( x , y ) � − ∆ + m 2 − κ i ∂ t γ t = | x − y | , γ t . |·| ◮ The set of initial data K HF := { γ = γ ∗ ∈ X HF : 0 ≤ γ ≤ 1 } , where √ � � γ ∈ S 1 ( L 2 ( R 3 , C q )) : Tr X HF := 1 − ∆ γ < ∞ ◮ γ 0 ∈ K HF ⇒ unique solution γ t ∈ C 0 ([0 , T ); K HF ) ∩ C 1 ([0 , T ); X ′ HF ). ◮ Blow-up alternative: either T = + ∞ (global) or 0 < T < + ∞ √ (local/blow-up) and lim t ր T Tr − ∆ γ t = + ∞ . ◮ Conservation of mass Tr γ t = Tr γ 0 and energy E HF ( γ t ) = E HF ( γ 0 ), �� ρ γ ( x ) ρ γ ( y ) − γ ( x , y ) − ∆ + m 2 γ − κ � E HF ( γ ) = Tr d x d y . 2 | x − y | D.-T Nguyen Blow-up of neutron stars 7 / 12

Introduction Time-Independent Time-Dependent Reference Known Results Theorems (Fr¨ ohlich–Lenzmann ’07) 1 The solution exists globally provided that the initial data γ 0 satisfies � 3 � τ cr 2 Tr γ 0 < with 0 < τ cr < τ c . κ 2 If solutions γ t are radial and blow up at time 0 < T < + ∞ then � 3 2 , � τ cr � lim inf ρ γ t ( x ) ≥ ∀ R > 0 . κ t ր T | x |≤ R D.-T Nguyen Blow-up of neutron stars 8 / 12

Introduction Time-Independent Time-Dependent Reference The Second Main Result Theorem 2 Fix q ≥ 1 and m ≥ 0. Let γ t be general non-radial blow-up solutions of √ the HF equation, i.e. lim t ր T Tr − ∆ γ t = + ∞ . 1 There exists a function z t : [0 , T ) → R 3 such that � 3 2 , � τ c � lim inf ρ γ t ( x ) ≥ ∀ R > 0 . κ t ր T | x − z t |≤ R D.-T Nguyen Blow-up of neutron stars 9 / 12

Introduction Time-Independent Time-Dependent Reference The Second Main Result Theorem 2 Fix q ≥ 1 and m ≥ 0. Let γ t be general non-radial blow-up solutions of √ the HF equation, i.e. lim t ր T Tr − ∆ γ t = + ∞ . 1 There exists a function z t : [0 , T ) → R 3 such that � 3 2 , � τ c � lim inf ρ γ t ( x ) ≥ ∀ R > 0 . κ t ր T | x − z t |≤ R � τ c � 3 2 . ◮ Solutions are global provided Tr γ 0 < κ D.-T Nguyen Blow-up of neutron stars 9 / 12

Introduction Time-Independent Time-Dependent Reference The Second Main Result Theorem 2 Fix q ≥ 1 and m ≥ 0. Let γ t be general non-radial blow-up solutions of √ the HF equation, i.e. lim t ր T Tr − ∆ γ t = + ∞ . 1 There exists a function z t : [0 , T ) → R 3 such that � 3 2 , � τ c � lim inf ρ γ t ( x ) ≥ ∀ R > 0 . κ t ր T | x − z t |≤ R � τ c � 3 2 . ◮ Solutions are global provided Tr γ 0 < κ � τ c � 3 2 | ≪ 1 s.t γ t blows up in finite time. ◮ There exists γ 0 with | Tr γ 0 − κ D.-T Nguyen Blow-up of neutron stars 9 / 12