[PPT] - Global well-posedness of the NLS system for infinitely many fermions PowerPoint Presentation

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Global well-posedness of the NLS system for infinitely many fermions

Younghun Hong University of Texas at Austin Joint work with Thomas Chen and Nataˇ sa Pavlovi´ c (UT Austin) Western States Meeting, 2016

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Hartree-Fock equation The Hartree-Fock equation is the system of N coupled equations,        i∂tu1 = (−∆ + w ∗ ρ − Xt)u1, · · · i∂tuN = (−∆ + w ∗ ρ − Xt)uN, (1) for orthonormal functions u1, · · · , uN in L2, where w = interaction potential, ρ(t, x) =

N

j=1

|uj(t, x)|2=total particle density. The exchange term Xt is the integral operator with kernel Xt(x; x′) = w(x − x′)

N

j=1

uj(t, x)uj(t, x′).

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The one-particle density matrix γ(t) =

N

j=1

|uj(t)uj(t)|. The system (1) can be written as a single operator-valued PDE, i∂tγ = [−∆ + w ∗ ργ − Xt, γ], where [A, B] = AB − BA, ργ(t, x) = γ(t, x, x)=density function and the exchange term is the integral operator with kernel Xt(x; x′) = w(x − x′)γ(t, x, x′). Orthonormality {uj}N

j=1 implies 0 ≤ γ ≤ 1.

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Derivation

[Narnhofer-Sewell ’84], [Bardos-Erdos-Golse-Mauser-Yau ’02], [B-G-Gottlieb-M ’03], [Elgart-Erdos-Schlein-Yau ’04], [Benedikter-Porta-Schlein ’14].

Conservation laws

N = Trγ =

N

j=1
|φj|2dx = the number of particles,

E(γ) = Tr(−∆)γ + 1 2

w(x − x′)
ργ(x)ργ(x′) − |γ(x, x′)|2

dxdx′ =

N

j=1
|∇φj|2dx + 1

2

1≤j,k≤N
w(x − x′)
|φj(x)|2|φk(x′)|2 − · · ·
dxdx′

= energy.

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GWP in the energy space

[Bove-da Prato-Fano ’74, ’76], [Brezzi-Markowich ’91], [Zagatti ’92].

Modified scattering (w =

1 |x|)

[Wada ’02], [Ikeda ’12].

Blow-up (pseudo-relativistic,

√ −∆ + m) [Fr¨

hlich-Lenzmann ’07], [Hainzl-Schlein ’09],

[Hainzl-Lewin-Lenzmann-Schlein ’10]

Sometimes, for simplicity, ignoring the exchange term, the Hartree

equation for fermions is considered, i∂tγ = [−∆ + w ∗ ργ, γ].

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Hartree equation for infinitely many fermions: “formulation by Mathieu Lewin and Julien Sabin” We consider the Hartree equation for fermions, i∂tγ = [−∆ + w ∗ ργ, γ]. It admits stationary solutions γf whose particle number N = Tr(γ) is infinity. For a reasonable f : [0, ∞) → R, a Fourier multiplier γf = f(−∆) is solves the equation.

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(1) Fermi gas at zero temperature: f(x) = 1(x≤µ) = ⇒ Π−

µ = γf = 1(−∆≤µ).

(2) Fermi gas at positive temperature: f(x) = 1 e

x−µ T

+ 1 = ⇒ γf = 1 e

−∆−µ T

+ 1 . (3) Bose gas at positive temperature: f(x) = 1 e

x−µ T

− 1 = ⇒ γf = 1 e

−∆−µ T

− 1 . (4) Boltzmann gas at positive temperature: f(x) = e− x−µ

T

= ⇒ γf = e

∆+µ T

.

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We are interested in the dynamics of the perturbation Q = γ − γf. Indeed, it is easy to check that Q solves the perturbed Hartree equation i∂tQ = [−∆ + w ∗ ρQ, γf + Q]. (2)

M. Lewin and J. Sabin [CMP’15]: Zero temperature: Cauchy problem for

Q in d ≥ 2 is globally well-posed in suitable space of solutions, for symmetric w ∈ L1(Rd) ∩ L∞(Rd) with w ≥ 0 ( w ≥ −ǫ for 2D).

M. Lewin and J. Sabin [CMP’15]: Positive temperature: Cauchy problem

for Q in d = 1, 2, 3 is globally well-posed in suitable space of solutions, for ∇w ∈ L1(R3) ∩ L∞(R3) for d = 3 and w ≥ 0. For d = 1, 2, w ≥ −Cd.

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Some key ingredients in [L-S]: Zero temperature case γf = Π−

µ = 1(−∆≤µ)

Conservation of the relative energy:

E(Q) := Tr0(−∆ − µ)Q + 1 2

Rd(w ∗ ρQ)ρQdx.

For w ≥ 0, E(Q) is positive and conserved.

Lieb-Thirring inequality of [Frank-Lewin-Lieb-Seiringer ’13]:

Tr0(−∆ − µ)Q ≥ KLT

Rd
ρΠ−

µ + ρQ

1+ 2

d −

ρΠ−

µ

1+ 2

d − 2+d

d

ρΠ−

µ

2

d ρQ

dx.

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Positive temperature case

Relative entropy H(γ, γf) [Lewin-Sabin, Lett. Math. Phys. ’14].
Klein’s inequality

H(γ, γf) ≥ CTr(1 − ∆)(γ − γf)2.

Conservation of the relative free energy

F(γ, γf) = H(γ, γf) + 1 2

Rd(w ∗ ργ)ργdx.
Strichartz estimates for density functions [Frank-Lewin-Lieb-Seiringer

’14] (Sp Schatten class) ρeit∆γ0e−it∆Lp

t∈RLq x γ0

S

2q q+1 ,

where 2

p + d q = d and 1 ≤ q ≤ d+2 d . Later, by [Frank-Sabin ’14], it is

extended to the optimal range of q < d+1

d−1.

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A remark on the Strichartz estimates for density functions When γ0 = N

j=1 |φjφj| for some orthonormal set {φj}N j=1 in L2,

N
j=1

|eit∆φj|2

Lp

t∈RLq x

N

q+1 2q ,

with q+1

2q ≤ 1.

It has better summability than what follows from the triangle inequality and Strichartz estimates,

N
j=1

|eit∆φj|2

Lp

t∈RLq x

≤

N

j=1
|eit∆φj|2
Lp

t∈RLq x

=

N

j=1

eit∆φj2

L2p

t∈RL2q x

N
j=1

φj2

L2 = N.

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Question: Can we include a singular potential such as w = δ? Perhaps, we need better smoothing (Strichartz) estimates...

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The perturbed Hartree/NLS(w = δ) equation of operator kernels Recall that Q

Hilbert-Schmidt operator

Q(x, x′)
perator kernel (function)

; QHS = Q(x, x′)L2

x,x′ .

The perturbed Hartree/NLS is equivalent to i∂tQ(t, x, x′) = − (∆x − ∆x′)Q(t, x, x′) + (w ∗ ρQ(t, x) − w ∗ ρQ(t, x′))(γf(x − x′) + Q(t, x, x′)). In the integral form, Q(t) = eit(∆x−∆x′ )Q0 − i t ei(t−s)(∆x−∆x′ ) (w ∗ ρQ(s, x) − w ∗ ρQ(s, x′))(γf(x − x′) + Q(s))

ds.

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New Strichartz estimates (I) For simplicity, let d = 3. We call (q, r) admissible if 2

q + 3 r = 3 2 and

2 ≤ q, r ≤ ∞. Theorem (Strichartz estimates for operator kernels; Chen-H.-Pavlovi´ c). For admissible (q, r) and (˜ q, ˜ r), we have eit(∆x−∆x′ )γ0Lq

t∈RLr xL2 x′ γ0L2 x,x′ ,

t

ei(t−s)(∆x−∆x′ )R(s)ds

Lq

t∈RLr xL2 x′

R(t)L˜

q′ t∈RL˜ r′ x L2 x′ .

The same bounds hold with interchanged x and x′.

Proof. It follows from the dispersive estimate with a frozen variable, that

is, eit(∆x−∆x′ )γ0Lp

xL2 x′ = eit∆xγ0Lp xL2 x′ ≤ eit∆xγ0L2 x′ Lp x

|t|−d( 1

2 − 1 p )γ0L2 x′ Lp′ x ≤ |t|−d( 1 2 − 1 p )γ0Lp′ x L2 x′

for p ≥ 2, and the standard argument of Keel and Tao.

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Application By the Strichartz estimates,

∇x

1+ǫ 2 ∇x′ 1+ǫ 2

t ei(t−s)(∆x−∆x′ ) ρQ(s, x)Q(s, x, x′)

ds
L2

t L6 xL2 x′ ∩L2 t L6 x′ L2 x

∇x

1+ǫ 2 ∇x′ 1+ǫ 2

ρQ(t, x)Q(t, x, x′)
L1

t L2 x,x′

ρQL2

t L3 x∇x 1+ǫ 2 ∇x′ 1+ǫ 2 Q(t, x, x′)

L2

t L6 xL2 x′

+ |∇|

1+ǫ 2 ρQL2 t L2 x∇x′ 1+ǫ 2 Q(t, x, x′)

L2

t L∞ x L2 x′

|∇|

1 2 ρQL2 t L2 x∇x 1+ǫ 2 ∇x′ 1+ǫ 2 Q(t, x, x′)

L2

t L6 xL2 x′ ∩L2 t L6 x′ L2 x

. The same inequality holds with interchanged x and x′.

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New Strichartz estimates (II) For simplicity, we assume that d = 3 and ǫ > 0 is small. Theorem (Strichartz estimates for density functions; Chen-H.-Pavlovi´ c). |∇|

1 2 ρeit∆γ0e−it∆L2 t∈RHǫ x ∇x 1+ǫ 2 ∇x′ 1+ǫ 2 γ0L2 x,x′ ,

(3)

|∇|

1 2 ρ

t ei(t−s)∆R(s)e−i(t−s)∆ds

L2

t∈RHǫ x

∇x

1+ǫ 2 ∇x′ 1+ǫ 2 R(t)L1 t∈RL2 x,x′ .

Remark. (3) is equivalent to

|∇|

1 2 ρeit∆γ0e−it∆L2 t∈RHǫ x ∇ 1+ǫ 2 γ0∇ 1+ǫ 2 HS.

When γ0 = N

j=1 |φjφj| for some orthonormal set {φj}N j=1 in H

1+ǫ 2 ,

|∇|

1 2

N
j=1

|eit∆φ|2

L2

t∈RHǫ x

N 1/2. More regularity implies better summability!

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Proof. The proof is based on the space-time Fourier transform as in

Klainerman-Machedon’s null form estimates and Bougain’s bilinear estimates.

ργ(ξ) = Fx
γ(x, x)
∼ Fx

R6 ˆ

γ(ξ1, ξ′

1)eix·(ξ1+ξ′

1)dξ1dξ′

1

∼
R6 ˆ

γ(ξ1, ξ′

1)δ(ξ − ξ1 − ξ′ 1)dξ1dξ′ 1 =

R3 ˆ

γ(ξ1, ξ − ξ1)dξ1. ⇒ (ρeit∆γ0e−it∆)∼(ξ) ∼ Ft

R3 eit(|ξ1|2−|ξ−ξ1|2) ˆ

γ0(ξ1, ξ − ξ1)dξ1

∼
R3 δ(τ − |ξ1|2 + |ξ − ξ1|2) ˆ

γ0(ξ1, ξ − ξ1)dξ1. By Plancherel, ⇒|∇|1/2ρeit∆γ0e−it∆2

L2

t∈RHǫ x

∼

R
R3 |ξ|ξ2ǫ
R3 δ(τ − |ξ1|2 + |ξ − ξ1|2) ˆ

γ0(ξ1, ξ − ξ1)dξ1

2

dξdτ.

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By Cauchy-Schwartz, ⇒|∇|1/2ρeit∆γ0e−it∆2

L2

t∈RHǫ x

≤

R
R3

R3

|ξ|ξ2ǫδ(τ − |ξ1|2 + |ξ − ξ1|2) ξ11+ǫξ − ξ11+ǫ dξ1

·

R3 δ(τ − |ξ1|2 + |ξ − ξ1|2)ξ11+ǫξ − ξ11+ǫ| ˆ

γ0(ξ1, ξ − ξ1)|2dξ1

dξdτ

≤ sup

τ,ξ R3

|ξ|ξ2ǫδ(τ − |ξ1|2 + |ξ − ξ1|2) ξ11+ǫξ − ξ11+ǫ dξ1

∇x

1+ǫ 2 ∇x′ 1+ǫ 2 γ0L2 x,x′ .

Everything is reduced to the following integral estimate

R3

|ξ|ξ2ǫδ(τ + |ξ|1 − 2ξ · ξ1) ξ11+ǫξ − ξ11+ǫ dξ1 ≤ C. Integrating ξ1 ∈ R3 in the ξ-direction, it becomes

P

ξ2ǫ ξ11+ǫξ − ξ11+ǫ dσ(ξ1) ≤ C, where P is a two-dimensional hyperplane, which is not hard to show.

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Applications of two new Strichartz estimates

LWP of the perturbed NLS equation in Hα for any α > 1

2, where Hα is

the Banach space equipped with the norm QHα = ∇xα∇x′αQ(x, x′)L2

x,x′ .

GWP of the perturbed NLS equation in the relative energy space “when

γf = Π−

µ = 1(−∆≤µ) (zero temperature case).”

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GWP of the NLS system in the zero temperature case The relative energy is defined by E(Q) := Tr0(−∆ − µ)Q + 1 2

Rd(ρQ)2dx.

The relative energy space is defined by K :=

Q = γ − Π−

µ ∈ X : 0 ≤ γ ≤ 1

in the Banach space

QX = QOp +

±

|∆ + µ|

1 2 Q±±|∆ + µ| 1 2 S1,

where Q±± = Π±

µ QΠ± µ , Π+ µ = 1(−∆≥µ) and Π− µ = 1(−∆≤µ).

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In K, the kinetic energy of the relative energy controls the X norm, E(Q) ≥ Tr0(−∆ − µ)Q =

±

Tr0(−∆ − µ)Q±± =

±

|∆ + µ|

1 2 Q±±|∆ + µ| 1 2 S1.

Theorem (Chen-H.-Pavlovi´ c). For initial data Q0 ∈ K, there exists a unique global solution Q(t) in K to the equation i∂tQ = [−∆ + ρQ, Π−

µ + Q].

Moreover, the relative energy is conserved, E(Q(t)) = E(Q0).

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Thank you for your attention !

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