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¨

• dinger operator

N–particles fermionic Hamiltonian ( = c = 1)

HN =

N

• i=1
• −∆xi + m2 − κ
• 1≤i<j≤N

|xi − xj|−1 0 ≤ m is the neutron mass, κ = Gm2 with G the gravitational constant.

Quantum Energy (internal spin degree of freedom q ≥ 1)

E Q

N = inf

• ψN, HNψN : ψN ∈

N

• L2(R3, Cq), ψNL2 = 1
• .

D.-T Nguyen Blow-up of neutron stars 2 / 12

Introduction Time-Independent Time-Dependent Reference

N-particles Schr¨

• dinger operator

N–particles fermionic Hamiltonian ( = c = 1)

HN =

N

• i=1
• −∆xi + m2 − κ
• 1≤i<j≤N

|xi − xj|−1 0 ≤ m is the neutron mass, κ = Gm2 with G the gravitational constant.

Quantum Energy (internal spin degree of freedom q ≥ 1)

E Q

N = inf

• ψN, HNψN : ψN ∈

N

• L2(R3, Cq), ψNL2 = 1
• .

Questions

1 Behaviors of E Q

N and ground states when N → ∞ and κN

2 3 ր τc. 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 (σf ≈ 1.092)

σf ρ

4 3

L

4 3 ρ 2 3

L1 ≥ 1

2

ρ(x)ρ(y)

|x − y| dxdy, ∀0 ≤ ρ ∈ L1 ∩ L

4 3 (R3).

◮ There exists a unique optimizer Q, which can be chosen non-negative radially symmetric decreasing and satisfies σf

• Q(x)

4 3 dx =

• Q(x)dx = 1

2

Q(x)Q(y)

|x − y| dxdy = 1. ◮ Q has compact support and solves the Lane–Emden equation (Lieb–Oxford ’81) 4 3σf Q(x)

1 3 =

• Q ⋆ 1

|·|

• − 2

3

• +

.

D.-T Nguyen Blow-up of neutron stars 3 / 12

Introduction Time-Independent Time-Dependent Reference

Chandraskhar limit mass

Hardy–Littlewood–Sobolev inequality (σf ≈ 1.092)

σf ρ

4 3

L

4 3 ρ 2 3

L1 ≥ 1

2

ρ(x)ρ(y)

|x − y| dxdy, ∀0 ≤ ρ ∈ L1 ∩ L

4 3 (R3).

◮ There exists a unique optimizer Q, which can be chosen non-negative radially symmetric decreasing and satisfies σf

• Q(x)

4 3 dx =

• Q(x)dx = 1

2

Q(x)Q(y)

|x − y| dxdy = 1. ◮ Q has compact support and solves the Lane–Emden equation (Lieb–Oxford ’81) 4 3σf Q(x)

1 3 =

• Q ⋆ 1

|·|

• − 2

3

• +

. ◮ Chandrasekhar limit mass:

τc

κ

3

2 with τc = Kcl

σf and Kcl = 3 4

6π2

q

1

3 . D.-T Nguyen Blow-up of neutron stars 3 / 12

Introduction Time-Independent Time-Dependent Reference

Subcritical regime

HN =

N

• i=1
• −∆xi + m2 − κ
• 1≤i<j≤N

|xi − xj|−1

Theorem (Lieb–Yau ’87)

Fix τ = κN

2 3 < τc. Then

lim

N→∞ N−1E Q N = E Ch τ

(1) := inf

• ECh

τ (ρ) : 0 ≤ ρ ∈ L

4 3 (R3),

• ρ = 1
• where

ECh

τ (ρ) =

• q

(2π)3

• |p|<

6π2ρ(x)

q

1

3

• |p|2 + m2dpdx − τ

2

ρ(x)ρ(y)

|x − y| dxdy. Futhermore, if τ > τc then E Ch

τ

(1) = −∞.

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 EHF(γ) = Tr

• −∆ + m2γ − κ

2

ργ(x)ργ(y) − |γ(x, y)|2

|x − y| dxdy. ◮ The set of Hartree–Fock states KHF = {γ = γ∗ ∈ S1(L2(R3; Cq)) : 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 < NHF(κ) (∼κ→0

τc

κ

3

2 ), there exists a minimizer for

EHF(N) = inf {EHF(γ) : γ ∈ KHF, Tr γ = N} . Furthermore, for every 0 < τ = κN2/3 < τc, lim

N→∞ N−1EHF(N) = E Ch τ

(1).

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)

Fix q ≥ 1, m > 0. Let τN := κN

2 3 = τc − N−β with 0 < β < 1/9. Then

N−1EHF(N) = (τc − τN)

1 2 2Λ + o(1)N→∞

,

Λ = 3 4m

• 1

Kcl

• R3 Q(x)

2 3 .

Let γN be a minimizer for EHF(N) and let ργN(x) = γN(x, x). Then there exists a sequence {yN} ⊂ R3 such that, up to subsequence, lim

N→∞(τc − τN)

3 2 ργN((τc − τN) 1 2 N 1 3 x + yN) = Λ3Q(Λx)

strongly in Lr(R3) for 1 ≤ r < 4

3 and weakly in L

4 3 (R3). 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)

Fix q ≥ 1, m > 0. Let τN := κN

2 3 = τc − N−β with 0 < β < 1/9. Then

N−1EHF(N) = (τc − τN)

1 2 2Λ + o(1)N→∞

,

Λ = 3 4m

• 1

Kcl

• R3 Q(x)

2 3 .

Let γN be a minimizer for EHF(N) and let ργN(x) = γN(x, x). Then there exists a sequence {yN} ⊂ R3 such that, up to subsequence, lim

N→∞(τc − τN)

3 2 ργN((τc − τN) 1 2 N 1 3 x + yN) = Λ3Q(Λx)

strongly in Lr(R3) for 1 ≤ r < 4

3 and weakly in L

4 3 (R3).

◮ Attaining the L

4 3 (R3) convergence requires to prove the Lieb–Thirring

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

i∂tγt =

• −∆ + m2 − κ
• ργt ⋆ 1

|·|

• + κγt(x, y)

|x − y| , γt

• .

◮ The set of initial data KHF := {γ = γ∗ ∈ XHF : 0 ≤ γ ≤ 1}, where XHF :=

• γ ∈ S1(L2(R3, Cq)) : Tr

√ 1 − ∆γ < ∞

• ◮ γ0 ∈ KHF ⇒ unique solution γt ∈ C0([0, T); KHF) ∩ C1([0, T); X ′

HF).

◮ Blow-up alternative: either T = +∞ (global) or 0 < T < +∞ (local/blow-up) and limtրT Tr √ −∆γt = +∞. ◮ Conservation of mass Tr γt = Tr γ0 and energy EHF(γt) = EHF(γ0), EHF(γ) = Tr

• −∆ + m2γ − κ

2

ργ(x)ργ(y) − γ(x, y)

|x − y| dxdy.

D.-T Nguyen Blow-up of neutron stars 7 / 12

Introduction Time-Independent Time-Dependent Reference

Known Results

Theorems (Fr¨

• hlich–Lenzmann ’07)

1 The solution exists globally provided that the initial data γ0 satisfies

Tr γ0 <

τcr

κ

3

2

with 0 < τcr < τc.

2 If solutions γt are radial and blow up at time 0 < T < +∞ then

lim inf

tրT

• |x|≤R

ργt(x) ≥

τcr

κ

3

2 ,

∀R > 0.

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 γt be general non-radial blow-up solutions of the HF equation, i.e. limtրT Tr √ −∆γt = +∞.

1 There exists a function zt : [0, T) → R3 such that

lim inf

tրT

• |x−zt|≤R

ργt(x) ≥

τc

κ

3

2 ,

∀R > 0.

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 γt be general non-radial blow-up solutions of the HF equation, i.e. limtրT Tr √ −∆γt = +∞.

1 There exists a function zt : [0, T) → R3 such that

lim inf

tրT

• |x−zt|≤R

ργt(x) ≥

τc

κ

3

2 ,

∀R > 0. ◮ Solutions are global provided Tr γ0 <

τc

κ

3

2 . 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 γt be general non-radial blow-up solutions of the HF equation, i.e. limtրT Tr √ −∆γt = +∞.

1 There exists a function zt : [0, T) → R3 such that

lim inf

tրT

• |x−zt|≤R

ργt(x) ≥

τc

κ

3

2 ,

∀R > 0. ◮ Solutions are global provided Tr γ0 <

τc

κ

3

2 .

◮ There exists γ0 with | Tr γ0 −

τc

κ

3

2 | ≪ 1 s.t γt blows up in finite time. 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 γt be general non-radial blow-up solutions of the HF equation, i.e. limtրT Tr √ −∆γt = +∞.

1 There exists a function zt : [0, T) → R3 such that

lim inf

tրT

• |x−zt|≤R

ργt(x) ≥

τc

κ

3

2 ,

∀R > 0.

2 Assuming Tr γ0 =

τc

κ

3

2 . Let ℓt = (Tr

√ −∆γt)−1κ−2τ 3

c ,

˜ γt(x, y) = ℓ3

t γt(ℓtx, ℓty) and ρ˜ γt(x) = ˜

γt(x, x). Then lim

tրT ρ˜ γt(· − zt) =

τc

κ

3

2 Q

strongly in Lr(R3) for 1 ≤ r < 3

• 2. Here Q is the Lane–Emden solution.

◮ Solutions are global provided Tr γ0 <

τc

κ

3

2 .

◮ There exists γ0 with | Tr γ0 −

τc

κ

3

2 | ≪ 1 s.t γt blows up in finite time. D.-T Nguyen Blow-up of neutron stars 9 / 12

Introduction Time-Independent Time-Dependent Reference

Proof of Mass Concentration.

Let {tn} ⊂ R+ with tn ր T and define ˜ γtn(x, y) = ℓ3

tnγtn(ℓtnx, ℓtny) where

ℓtn := (Tr √ −∆γtn)−1κ−2τ 3

c → 0. Then Tr

√ 1 − ∆˜ γtn < +∞. Our goal is to prove that Mcr := lim

R→∞ lim sup n→∞ sup z∈R3

• |x−z|≤R

ρ˜

γtn ≥

τc

κ

3

2 .

◮ Tr γ0 ≥ Mcr > 0 since EHF(˜ γtn)|m=0 = ℓtnEHF(γtn)|m=0 → 0. ◮ Assuming Mcr <

τc

κ

3

2

Mcr <

τc

κ

3

2

Mcr <

τc

κ

3

2 . If ˜

γtn → γ strongly in trace-class then lim inf

n→∞ EHF(˜

γtn)|m=0 ≥ Kclργ

4 3

L

4 3 − κ

2

ργ(x)ργ(y)

|x − y| dxdy > 0.

D.-T Nguyen Blow-up of neutron stars 10 / 12

Introduction Time-Independent Time-Dependent Reference

Proof of Mass Concentration.

Let {tn} ⊂ R+ with tn ր T and define ˜ γtn(x, y) = ℓ3

tnγtn(ℓtnx, ℓtny) where

ℓtn := (Tr √ −∆γtn)−1κ−2τ 3

c → 0. Then Tr

√ 1 − ∆˜ γtn < +∞. Our goal is to prove that Mcr := lim

R→∞ lim sup n→∞ sup z∈R3

• |x−z|≤R

ρ˜

γtn ≥

τc

κ

3

2 .

◮ Tr γ0 ≥ Mcr > 0 since EHF(˜ γtn)|m=0 = ℓtnEHF(γtn)|m=0 → 0. ◮ Assuming Mcr <

τc

κ

3

2

Mcr <

τc

κ

3

2

Mcr <

τc

κ

3

2 . If ˜

γtn → γ strongly in trace-class then lim inf

n→∞ EHF(˜

γtn)|m=0 ≥ Kclργ

4 3

L

4 3 − κ

2

ργ(x)ργ(y)

|x − y| dxdy > 0.

Lemma (Kinetic energy of weak limit of solutions)

If {˜ γtn} ⊂ KHF and ρ˜

γtn ⇀ ργ weakly in L

4 3 (R3) then

lim inf

n→∞ Tr

√ −∆˜ γtn ≥ Kclργ

4 3

L

4 3 . D.-T Nguyen Blow-up of neutron stars 10 / 12

Introduction Time-Independent Time-Dependent Reference

Proof of Mass Concentration (continue).

Fix χ1, χ2 : R → [0, 1] such that χ2

1 + χ2 2 = 1, χ1 = 1 if |x| ≤ 1, χ1 = 0 if

|x| ≥ 2, and define ˜ γ(i)

tn = χi(·/R)˜

γtnχi(·/R), i ∈ {1, 2}. Then lim

n→∞ EHF(˜

γtn)|m=0 ≥ lim

R→∞ lim n→∞ EHF(˜

γ(1)

tn )|m=0 + lim R→∞ lim n→∞ EHF(˜

γ(2)

tn )|m=0.

D.-T Nguyen Blow-up of neutron stars 11 / 12

Introduction Time-Independent Time-Dependent Reference

Proof of Mass Concentration (continue).

Fix χ1, χ2 : R → [0, 1] such that χ2

1 + χ2 2 = 1, χ1 = 1 if |x| ≤ 1, χ1 = 0 if

|x| ≥ 2, and define ˜ γ(i)

tn = χi(·/R)˜

γtnχi(·/R), i ∈ {1, 2}. Then lim

n→∞ EHF(˜

γtn)|m=0 ≥ lim

R→∞ lim n→∞ EHF(˜

γ(1)

tn )|m=0 + lim R→∞ lim n→∞ EHF(˜

γ(2)

tn )|m=0.

(A1) Either ˜ γ(2)

tn → γ(2) strongly, then lim R→∞ lim n→∞ EHF(˜

γ(2)

tn )|m=0 > 0,

(A2) or {˜ γ(2)

tn } is not relatively compact. Then there exists γ(2) ∈ KHF with

0 < Tr γ(2) < Tr γ0 − Tr γ(1) such that χ1(·/R)˜ γ(2)

tn χ1(·/R) → γ(2)

strongly in trace-class and, with ˜ γ(3)

tn = χ2(·/R)˜

γ(2)

tn χ2(·/R),

lim

R→∞ lim n→∞ Tr ˜

γ(3)

tn = Tr γ0 − Tr γ(1) − Tr γ(2).

D.-T Nguyen Blow-up of neutron stars 11 / 12

Introduction Time-Independent Time-Dependent Reference

Proof of Mass Concentration (continue).

Fix χ1, χ2 : R → [0, 1] such that χ2

1 + χ2 2 = 1, χ1 = 1 if |x| ≤ 1, χ1 = 0 if

|x| ≥ 2, and define ˜ γ(i)

tn = χi(·/R)˜

γtnχi(·/R), i ∈ {1, 2}. Then lim

n→∞ EHF(˜

γtn)|m=0 ≥ lim

R→∞ lim n→∞ EHF(˜

γ(1)

tn )|m=0 + lim R→∞ lim n→∞ EHF(˜

γ(2)

tn )|m=0.

(A1) Either ˜ γ(2)

tn → γ(2) strongly, then lim R→∞ lim n→∞ EHF(˜

γ(2)

tn )|m=0 > 0,

(A2) or {˜ γ(2)

tn } is not relatively compact. Then there exists γ(2) ∈ KHF with

0 < Tr γ(2) < Tr γ0 − Tr γ(1) such that χ1(·/R)˜ γ(2)

tn χ1(·/R) → γ(2)

strongly in trace-class and, with ˜ γ(3)

tn = χ2(·/R)˜

γ(2)

tn χ2(·/R),

lim

R→∞ lim n→∞ Tr ˜

γ(3)

tn = Tr γ0 − Tr γ(1) − Tr γ(2).

Lieb–Thirring-type inequality (0 < Kcr < Kcl)

Tr √ −∆γ ≥ Kcrργ

4 3

L

4 3 ,

∀γ ∈ KHF.

D.-T Nguyen Blow-up of neutron stars 11 / 12

Introduction Time-Independent Time-Dependent Reference

Reference

• J. Fr¨
• hlich, E. Lenzmann, Blow-Up for Nonlinear Wave Equations

describing Boson Stars, CPAM (2007).

• J. Fr¨
• hlich, E. Lenzmann, Dynamical Collapse of White Dwarfs in

Hartree- and Hartree–Fock Theory, CMP (2007).

• C. Hainzl, E. Lenzmann, M. Lewin, B. Schlein, On Blowup for

Time-Dependent Generalized Hartree–Fock Equations, AHP (2010).

• C. Hainzl, B. Schlein, Stellar Collapse in the Time Dependent

Hartree–Fock Approximation, CMP (2009).

• E. Lenzmann, M. Lewin, Minimizers for the Hartree–Fock–Bogoliubov

theory of the neutron stars and white dwarfs, DMJ (2010). E.H. Lieb, H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, CMP (1987).

THANK YOU!

D.-T Nguyen Blow-up of neutron stars 12 / 12