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

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 1

Hartree-Fock equation The Hartree-Fock equation is the system of N coupled equations,  i∂ t u 1 = ( − ∆ + w ∗ ρ − X t ) u 1 ,    · · · (1)    i∂ t u N = ( − ∆ + w ∗ ρ − X t ) u N , for orthonormal functions u 1 , · · · , u N in L 2 , where w = interaction potential , N � | u j ( t, x ) | 2 =total particle density . ρ ( t, x ) = j =1 The exchange term X t is the integral operator with kernel N � X t ( x ; x ′ ) = w ( x − x ′ ) u j ( t, x ) u j ( t, x ′ ) . j =1 2

The one-particle density matrix N � γ ( t ) = | u j ( t ) �� u j ( t ) | . j =1 The system (1) can be written as a single operator-valued PDE, i∂ t γ = [ − ∆ + w ∗ ρ γ − X t , γ ] , where [ A, B ] = AB − BA, ρ γ ( t, x ) = γ ( t, x, x ) =density function and the exchange term is the integral operator with kernel X t ( x ; x ′ ) = w ( x − x ′ ) γ ( t, x, x ′ ) . Orthonormality { u j } N j =1 implies 0 ≤ γ ≤ 1 . 3

• 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 � | φ j | 2 dx = the number of particles , N = Tr γ = j =1 �� � ρ γ ( x ) ρ γ ( x ′ ) − | γ ( x, x ′ ) | 2 � E ( γ ) = Tr ( − ∆) γ + 1 w ( x − x ′ ) dxdx ′ 2 � � � � N � � |∇ φ j | 2 dx + 1 | φ j ( x ) | 2 | φ k ( x ′ ) | 2 − · · · w ( x − x ′ ) dxdx ′ = 2 j =1 1 ≤ j,k ≤ N = energy . 4

• GWP in the energy space [Bove-da Prato-Fano ’74, ’76], [Brezzi-Markowich ’91], [Zagatti ’92]. 1 • Modified scattering ( w = | x | ) [Wada ’02], [Ikeda ’12]. √ • Blow-up (pseudo-relativistic, − ∆ + m ) [Fr¨ ohlich-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∂ t γ = [ − ∆ + w ∗ ρ γ , γ ] . 5

Hartree equation for infinitely many fermions: “formulation by Mathieu Lewin and Julien Sabin” We consider the Hartree equation for fermions, i∂ t γ = [ − ∆ + w ∗ ρ γ , γ ] . It admits stationary solutions γ f whose particle number N = Tr ( γ ) is infinity. For a reasonable f : [0 , ∞ ) → R , a Fourier multiplier γ f = f ( − ∆) is solves the equation. 6

(1) Fermi gas at zero temperature: ⇒ Π − f ( x ) = 1 ( x ≤ µ ) = µ = γ f = 1 ( − ∆ ≤ µ ) . (2) Fermi gas at positive temperature: 1 1 f ( x ) = = ⇒ γ f = . − ∆ − µ x − µ + 1 + 1 e e T T (3) Bose gas at positive temperature: 1 1 f ( x ) = = ⇒ γ f = . x − µ − ∆ − µ − 1 e e − 1 T T (4) Boltzmann gas at positive temperature: f ( x ) = e − x − µ ∆+ µ = ⇒ γ f = e . T T 7

We are interested in the dynamics of the perturbation Q = γ − γ f . Indeed, it is easy to check that Q solves the perturbed Hartree equation i∂ t Q = [ − ∆ + w ∗ ρ Q , γ f + 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 ∈ L 1 ( R d ) ∩ L ∞ ( R d ) 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 ∈ L 1 ( R 3 ) ∩ L ∞ ( R 3 ) for d = 3 and � w ≥ 0 . For d = 1 , 2 , � w ≥ − C d . 8

Some key ingredients in [L-S]: Zero temperature case γ f = Π − µ = 1 ( − ∆ ≤ µ ) • Conservation of the relative energy: � E ( Q ) := Tr 0 ( − ∆ − µ ) Q + 1 R d ( w ∗ ρ Q ) ρ Q dx. 2 w ≥ 0 , E ( Q ) is positive and conserved. For � • Lieb-Thirring inequality of [Frank-Lewin-Lieb-Seiringer ’13]: Tr 0 ( − ∆ − µ ) Q � �� � � 1+ 2 � � 1+ 2 � � 2 d − d − 2+ d d ρ Q ≥ K LT µ + ρ Q ρ Π − ρ Π − ρ Π − dx. d µ µ R d 9

Positive temperature case • Relative entropy H ( γ, γ f ) [Lewin-Sabin, Lett. Math. Phys. ’14]. • Klein’s inequality H ( γ, γ f ) ≥ C Tr (1 − ∆)( γ − γ f ) 2 . • Conservation of the relative free energy � F ( γ, γ f ) = H ( γ, γ f ) + 1 R d ( w ∗ ρ γ ) ρ γ dx. 2 • Strichartz estimates for density functions [Frank-Lewin-Lieb-Seiringer ’14] ( S p Schatten class) � ρ e it ∆ γ 0 e − it ∆ � L p x � � γ 0 � q +1 , t ∈ R L q 2 q S 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 . 10

A remark on the Strichartz estimates for density functions When γ 0 = � N j =1 | φ j �� φ j | for some orthonormal set { φ j } N j =1 in L 2 , � | e it ∆ φ j | 2 � N � � � q +1 2 q , with q +1 � N 2 q ≤ 1 . � � L p t ∈ R L q x j =1 It has better summability than what follows from the triangle inequality and Strichartz estimates, � | e it ∆ φ j | 2 � � � | e it ∆ φ j | 2 � N N N � � � � � � � � e it ∆ φ j � 2 ≤ = � � � L 2 p t ∈ R L 2 q L p t ∈ R L q L p t ∈ R L q x x x j =1 j =1 j =1 N � � φ j � 2 L 2 = N. � j =1 11

Question: Can we include a singular potential such as w = δ ? Perhaps, we need better smoothing (Strichartz) estimates... 12

The perturbed Hartree/NLS( w = δ ) equation of operator kernels Recall that Q ( x, x ′ ) ; � Q � HS = � Q ( x, x ′ ) � L 2 Q � x,x ′ . Hilbert-Schmidt operator operator kernel (function) The perturbed Hartree/NLS is equivalent to i∂ t Q ( t, x, x ′ ) = − (∆ x − ∆ x ′ ) Q ( t, x, x ′ ) + ( w ∗ ρ Q ( t, x ) − w ∗ ρ Q ( t, x ′ ))( γ f ( x − x ′ ) + Q ( t, x, x ′ )) . In the integral form, Q ( t ) = e it (∆ x − ∆ x ′ ) Q 0 � t e i ( t − s )(∆ x − ∆ x ′ ) � � ( w ∗ ρ Q ( s, x ) − w ∗ ρ Q ( s, x ′ ))( γ f ( x − x ′ ) + Q ( s )) − i ds. 0 13

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 � e it (∆ x − ∆ x ′ ) γ 0 � L q x ′ � � γ 0 � L 2 x,x ′ , x L 2 t ∈ R L r � t � � � � e i ( t − s )(∆ x − ∆ x ′ ) R ( s ) ds � � R ( t ) � L ˜ x ′ . � � q ′ r ′ x L 2 t ∈ R L ˜ L q x L 2 t ∈ R L r 0 x ′ The same bounds hold with interchanged x and x ′ . Proof. It follows from the dispersive estimate with a frozen variable, that is, � e it (∆ x − ∆ x ′ ) γ 0 � L p x ′ = � e it ∆ x γ 0 � L p x ′ ≤ � e it ∆ x γ 0 � L 2 x ′ L p x L 2 x L 2 x � | t | − d ( 1 2 − 1 x ≤ | t | − d ( 1 2 − 1 p ) � γ 0 � L 2 p ) � γ 0 � L p ′ x ′ L p ′ x L 2 x ′ for p ≥ 2 , and the standard argument of Keel and Tao. 14

Application By the Strichartz estimates, � t � e i ( t − s )(∆ x − ∆ x ′ ) � � � � � 1+ ǫ 1+ ǫ ρ Q ( s, x ) Q ( s, x, x ′ ) 2 �∇ x ′ � � �∇ x � ds � 2 L 2 t L 6 x L 2 x ′ ∩ L 2 t L 6 x ′ L 2 0 x � �� � � � 1+ ǫ 1+ ǫ ρ Q ( t, x ) Q ( t, x, x ′ ) 2 �∇ x ′ � � � �∇ x � � 2 L 1 t L 2 x,x ′ � � 1+ ǫ 1+ ǫ 2 �∇ x ′ � 2 Q ( t, x, x ′ ) � � ρ Q � L 2 x ��∇ x � � t L 3 L 2 x L 2 t L 6 x ′ � � 1+ ǫ 1+ ǫ 2 Q ( t, x, x ′ ) 2 ρ Q � L 2 + �|∇| x ��∇ x ′ � � t L 2 L 2 x L 2 t L ∞ x ′ � � 1 1+ ǫ 1+ ǫ 2 Q ( t, x, x ′ ) 2 �∇ x ′ � 2 ρ Q � L 2 � �|∇| x ��∇ x � . � t L 2 L 2 x L 2 x ′ ∩ L 2 t L 6 t L 6 x ′ L 2 x The same inequality holds with interchanged x and x ′ . 15

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+ ǫ 1+ ǫ 1 2 ρ e it ∆ γ 0 e − it ∆ � L 2 2 �∇ x ′ � 2 γ 0 � L 2 �|∇| x � ��∇ x � x,x ′ , (3) t ∈ R H ǫ � � t � �� � � 1 1+ ǫ 1+ ǫ e i ( t − s )∆ R ( s ) e − i ( t − s )∆ ds 2 �∇ x ′ � 2 ρ 2 R ( t ) � L 1 � |∇| � ��∇ x � x,x ′ . � t ∈ R L 2 L 2 t ∈ R H ǫ 0 x Remark. (3) is equivalent to 1+ ǫ 1+ ǫ 1 2 � HS . 2 ρ e it ∆ γ 0 e − it ∆ � L 2 2 γ 0 �∇� �|∇| x � ��∇� t ∈ R H ǫ When γ 0 = � N 1+ ǫ j =1 | φ j �� φ j | for some orthonormal set { φ j } N 2 , j =1 in H � | e it ∆ φ | 2 �� � N � � � 1 � N 1 / 2 . � |∇| � 2 L 2 t ∈ R H ǫ x j =1 More regularity implies better summability! 16