For arbitrarily large times T > 0, we prove the uniform-in- propagation of semiclassical regularity for the solutions to the Hartree–Fock equation with singular interactions of the form $V(x)=±x−a$ with $a∈(0,12)$. As a by-product of this result, we extend to arbitrarily long times the derivation of the Hartree–Fock and the Vlasov equations from the many-body dynamics provided in the work of Chong et al. [arXiv:2103.10946 (2021)].

Consider the time-dependent Hartree–Fock equation

$iℏ∂tρ=Hρ,ρ$
(1)

describing the evolution of a positive self-adjoint trace class operator ρ = ρ(t) acting on $L2(R3)$. Here, $ℏ=h2π$ is the reduced Planck constant, $⋅,⋅$ denotes the commutator $A,B=AB−BA$, and Hρ is the Hamiltonian operator given by

$Hρ=−ℏ2Δ2+Vρ−h3Xρ,$
(2)

where Vρ is the mean-field potential and $Xρ$ is the exchange operator. The mean-field potential is defined as the multiplication operator by the function Vρ(x) = (K*ρ) (x), where $K:R3→R$ is the potential associated with some two-body interaction and ρ(x) is the spatial density, defined as the rescaled diagonal of the integral kernel ρ(·, ·) of the operator ρ, given by

$ρ(x)=diag(ρ)(x):=h3ρ(x,x).$

The exchange operator is defined to be the operator with kernel

$Xρ(x,y)=K(x−y)ρ(x,y).$

In this work, we normalize ρ so that

$h3Tr(ρ)=1andC∞:=ρ∞,$
(3)

where $⋅∞$ is the operator norm. These quantities are preserved by Eq. (1). Furthermore, we assume that the constant in (3) does not depend on . With this scaling, we see that ρ satisfies $∫R3ρ(x)ⅆx=h3Tr(ρ)=1$. In the absence of $Xρ$, we refer to Eq. (1) as the Hartree equation. All the results presented in this work hold for both the Hartree and the Hartree–Fock equation.

In the case when is fixed, say, = 1, the well-posedness theories for the Hartree and the Hartree–Fock equations are well known. For the case of the Hartree equation with the Coulomb potential, one can find the proof of the global-in-time well-posedness in L2 and the propagation of higher Hs regularity for the wave function in Ref. 1, which builds on the earlier works.2–4 The case of density operators in Schatten spaces but with an infinite trace was studied in Ref. 5. In the case of the Hartree–Fock equation, well-posedness in H2 was proved in Ref. 6 for bounded interactions and, then, in Refs. 7 and 8 for more singular potentials, including the case of the Coulomb potential. However, these works do not provide satisfactory estimates when is small and tending toward zero. Obtaining uniform-in- estimates is crucial not only for understanding, of course, the errors in the semiclassical limit → 0, as in Refs. 9–14, but also to create adapted numerical schemes15 and to understand the joint mean-field and semiclassical limit.16–19

In this paper, we are interested in proving the global-in-time propagation of regularity uniformly in the semiclassical parameter for solutions to the Hartree–Fock equation (1) when the interaction potential K is the inverse power law potential,

$K(x)=±1xa with a∈0,12.$

In particular, $K∈Lb,∞(R3)$, where $b=3a+1$ and $Lb,∞(R3)$ denotes the weak $Lb$ space on $R3$. Our main motivation is to extend the results of the local-in-time regularity obtained in our previous paper19 to global-in-time results, leading to the global-in-time mean-field and semiclassical limits for fermions from the N body Schrödinger equation to the Hartree–Fock and Vlasov equation.

Before stating our main result, we introduce the function spaces that we will be working with. First, we define the semiclassical phase space Lebesgue norms by

$ρLp:=h3pρp=h3pTr(ρp)1p$

for p ∈ [1, ], with the obvious modification for p = . Here, $ρp$ denotes the Schatten norm of order p and $A=A*A$ denotes the absolute value of the operator A with adjoint A*. Let p = −iℏ∇ be the momentum operator and

$mn:=1+pn,n∈N.$

The moment of order n and the weighted semiclassical Lebesgue norms with the operator weight $mn$ are given by

$Mn:=h3Tr(pnρ)andρLp(mn):=ρmnLp.$

In order to consider the quantum analog of Sobolev norms, we introduce the following operators:

$∇xρ:=∇,ρand∇ξρ:=xiℏ,ρ.$

Then, the semiclassical homogeneous Sobolev norms are defined by

$ρẆ1,p:=∑k=13(∇xkρLp+∇ξkρLp),$

with the corresponding inhomogeneous Sobolev norms given by $ρW1,p:=ρLp+ρẆ1,p$ and the weighted semiclassical Sobolev norms with the operator weight $mn$ given by

$ρW1,p(mn):=ρmnW1,p.$

Our main result states the global-in-time propagation of the regularity in terms of these norms.

Theorem 1.1.
Let$a∈(0,12)$,$n∈2N$be an even integer, andρbe a solution to the Hartree–Fock equation (1) with initial datum$ρin∈L∞(mn)$satisfying (3) such that
$ρin∈W1,2(mn)∩W1,q(mn)$
(4)
for$q∈2,∞$and with moments of order strictly larger than$31−a(n+a+1)$bounded uniformly in. Then,
$ρ∈Lloc∞(R+,W1,2(mn)∩W1,q(mn)∩L∞(mn))$
(5)
uniformly in ∈ (0, 1).

Note that Theorem 1.1 extends to arbitrarily long times the local-in-time theory studied in Ref. 19, Theorem 3.1 for $a∈(0,12)$. As a corollary, Theorem 1.1 entails the global-in-time derivation of the Hartree–Fock and the Vlasov equations from the many-body Schrödinger equation in the mean-field regime for mixed states, thus extending from local to global-in-time results (Theorems 3.2 and 3.3) in Ref. 19. The crucial regularity conditions needed to perform the joint mean-field and semiclassical limit in Ref. 19 when $a∈(0,12)$ are, indeed, of the form (5) with $q>61+2a$ (take p = 2 and q = q1 in the beginning of Ref. 19, Sec. 10.1).

As in Refs. 10 and 13, the key ingredient to getting long-time estimates in this work is the usage of quantum moments, used to bound the semiclassical weighted Lebesgue norms. In this paper, we prove the global-in-time propagation of quantum moments for the solution of the Hartree–Fock equation (1) when $a∈0,45$ and show that if $a∈(0,12)$, then the global-in-time bound on the moments, combined with a Grönwall-type argument, proves the uniform-in- propagation of regularity in weighted semiclassical Sobolev spaces.

This paper is structured as follows: in Sec. II, we recall the result obtained in Ref. 10 about the global-in-time propagation of quantum moments for the Hartree equation when $a∈(0,45]$ and extend it to solutions of the Hartree–Fock equation (1), while Sec. III is devoted to the Proof of Theorem 1.1.

Theorem 2.1.
Let$a∈(0,45]$,$n∈2N$, andρbe a solution of the Hartree–Fock equation with initial condition$ρin∈L1∩L∞$with moments of ordernbounded uniformly in. Then, there exists a continuous function$Φn∈C0(R+)$independent ofsuch that for any$t∈R+$,
$Mn(t)≤Φn(t).$

Remark 2.1.

If$a∈(45,2)$, then we can still get a short-time estimate when$a≤an=2nn+3$(see Remark 2.3). In particular,anis larger than 1 as soon asn ≥ 3. More generally, for anya ∈ (0, 2), the propagation of moment of ordernholds for any even$n≥na=3a2−a$.

Remark 2.2.

The proof of the theorem can be used to get an explicit function Φn(t). It has a polynomial growth in time when$a<45$and an exponential growth in time when$a=45$.

In this section, we recall the main ingredients of the proof of the propagation of moments for the Hartree equation obtained in Ref. 10, Theorem 3. This provides us a guide for the extension to the case of the Hartree–Fock equation addressed in Theorem 2.1.

Recall that for any density operator ρ (i.e., any positive trace class operator with trace one), there exists $J⊂N$, a sequence of functions $(ψj)j∈J$ orthonormal in $L2(R3)$, and a positive summable sequence $(λj)j∈J$ such that ρ can be written as

$ρ=∑j∈Jλj|ψj〉〈ψj|.$

For any even integer $n∈2N$, we define the moment density of order n by

$ρn(x):=h3∑j∈Jλjpn2ψj(x)2=diag(pn2ρ⋅pn2)(x)$
(6)

so that the moment of order n, previously defined, can be rewritten as $Mn=ρnL1$. Note that we also have $Mn=ρpn2L22$. With this notation, inequality (38) in Ref. 10 reads

$h3Tr(1iℏVρ,pnρ)≤CMn12sup(j,k,l)∈N3j+k+l=n/2−1ρ2jLα12ρ2kLβ12ρ2lLγ12,$
(7)

where

$1α′+1β′+1γ′=1b$

and α′, β′, γ′ are the Hölder conjugates of α, β, γ, respectively. The semiclassical kinetic inequality (Ref. 10, Theorem 6), which is a generalization of the Lieb–Thirring inequality for the nth order moment density, tells us that for any $(k,n)∈(2N)2$ verifying k ∈ [0, n], we have that

$ρkLp≤CρL∞1p′Mn1p with p=pn,k:=3+n3+k.$
(8)

Combining this inequality with inequality (7) implies the existence of some positive constants Θ, Θ0, and Θ2 such that the following estimate [see Ref. 10, inequality (44)] holds:

$h3Tr1iℏVρ,pnρ≤CρL∞Θ2Mn−2Θ0MnΘ,$

where Θ ≤ 1 when $a≤45$. This leads to the boundedness of moments by a Grönwall-type argument, together with the uniform boundedness of M2 due to the conservation of energy and an induction on n. Moreover, when $a<45$, then Θ < 1, thus proving a polynomial growth in time in this latter case.

We now consider the Hartree–Fock equation. By estimating the exchange (operator) term with a similar strategy, we get the analog of Ref. 10, Theorem 3. The analog of inequality (7) for the exchange term is given by the following lemma.

Lemma 2.1.
Denote the components ofpby$pxl$or simplyplforl ∈ {1, 2, 3}. Suppose that$n∈N$, and define$Q⊂[12,∞]4$, the set of (q1, q2, q3, q4) verifying$1q1+1q2∈(0,2)$,$1q3+1q4∈(0,2)$, and
$1q1′+1q2′+1q3′+1q4′=2b.$
(9)
Then, there exists a constantCindependent ofsuch that
$h3ℏTrh3Xρ,plnρ≤Csupk1,…k4infQρk1Lq112ρk2Lq212ρk3Lq312ρk4Lq412,$
(10)
where the supremum is taken over all the integers$(k1,…,k4)∈N4$such thatk1 + ⋯ + k4 = 2(n − 1).

Proof.
Let us begin by making the observation that $[Xρ,pln]ρ$ is trace class. By the Leibniz formula, the kernel of $plnXρρ$ is given by
$plnXρρ(x,y)=∑k=0nnk∫R3pxln−kK(x−z)pxlkρ(x,z)ρ(z,y)ⅆz=∑k=0n∑ℓ=0n−knkn−kℓ(−1)n−k∫R3K(x−z)(pxlkpzlℓρ(x,z))pzln−k−ℓρ(z,y)ⅆz=:∑k=0n∑ℓ=0n−knkn−kℓ(−1)n−k+1An,k,l(x,y).$
It is clear that each An,k,l is a product of two Hilbert–Schmidt operators, and so is trace class, provided that ρ is sufficiently regular. Since $Xρplnρ=An,0,0$ is also trace class, it follows that $[Xρ,pln]ρ$ is, indeed, trace class.
Let $In:=Tr(Xρ,plnρ)$. Then, by Ref. 20, VI.7, Theorem 17, we may express the trace of $Xρ,plnρ$ in terms of its kernel as follows:
$In=∬R6K(x−y)ρ(y,x)pxln(ρ)(x,y)−pxln(K(x−y)ρ(x,y))ρ(y,x)ⅆxⅆy.$
Therefore, using the Leibniz formula and, then, diagonalizing the self-adjoint compact operator ρ yield
$In=−∑k=0n−1nk∬R6(pxln−kK)(x−y)(pxlkρ(x,y))ρ(y,x)ⅆxⅆy=−∑(i,j)∈J2∑k=0n−1nkλiλj∫R3(pln−kK)*(ψīψj)plk(ψi)ψj̄,$
and so
$Iniℏ=∑(i,j)∈J2∑k=0n−1nkλiλj∫R3(∂lK)*pln−1−k(ψīψj)plk(ψi)ψj̄.$
Let In,k denote the integral in the above formula. We need to balance the powers of pl in In,k. For simplicity of notation, we assume that $n=2ñ∈2N$. The case when n is odd is similar. Then, if $k≤ñ$, we have
$In,k=∫R3(∂lK)*plñ−1(ψīψj)plñ−k(plk(ψi)ψj̄)=∑ℓ=0ñ−k∑ℓ̃=0ñ−1ñ−kℓñ−1ℓ̃In,k,ℓ,ℓ̃,$
where
$In,k,ℓ,ℓ̃=∫R3(∂lK)*(plℓ̃(ψī)plñ−1−ℓ̃(ψj))plñ−ℓ(ψi)plℓ(ψj̄).$
Now, note that by the Cauchy–Schwarz inequality for sums and Definition (6) for the moment density, we have that
$h2d∑(i,j)∈J2λiλjIn,k,ℓ,ℓ̃≤∫R3∂lK*ρ2ℓ̃12ρ2(ñ−1−ℓ̃)12ρ2(ñ−ℓ)12ρ2ℓ12≤CKρ2ℓ̃Lq112ρ2(ñ−1−ℓ̃)Lq212ρ2(ñ−ℓ)Lq312ρ2ℓLq412,$
where the last inequality follows from the Hardy–Littlewood–Sobolev inequality and the Hölder inequality. Similarly, when $k>ñ$, we have that
$In,k:=∫R3plk−ñ((∂lK)*pln−k−1(ψīψj)ψj̄)plñ(ψi)=∑ℓ=0k−ñ∑ℓ̃=0ñ−1−ℓk−ñℓñ−1−ℓℓ̃In,k,ℓ,ℓ̃,$
where
$In,k,ℓ,ℓ̃=∫R3(∂lK)*(plℓ̃(ψī)plñ−1−ℓ−ℓ̃(ψj))plℓ(ψj̄)plñ(ψi).$
Mimicking the estimates we obtained above for the case $k≤ñ$ completes the proof of the lemma.□

Now, combining the semiclassical kinetic interpolation inequality (8) for n = 2 and another $n∈N$, we get the following inequalities (see Ref. 13, Proposition 3.2).

Lemma 2.2.
Let 0 ≤ knand$pn,k′=(nk)′(1+3n)=3+nn−k$. Then, for anyp ∈ [1, pn,k], ifk ≥ 2 or if$p≥p2,0=53$, it holds that
$ρkLp≤Cn,k1p′ρL∞1p′M2θ2Mnθn$
with$θ2=n−kn−2−3+nn−21p′$and$θn=k−2n−2+5n−21p′$. Ifk = 0 andpp2,0, then
$ρLp≤C1p′ρL∞1p′M01−52p′M232p′.$

These two inequalities can be merged into a single inequality in terms of the non-homogeneous moments $1+Mn=h3Tr(ρmn)$.

Corollary 2.1.
For anyn ≥ 2,k ∈ [0, n], andp ∈ [1, pn,k], there exists a constantC > 0 such that for any compact operatorρ, we have the following estimate:
$ρkLp≤CρL∞1p′(1+M2)θ2(1+Mn)θn$
(11)
with$θn=k−2n−2+5n−21p′$.

Remark 2.3.
If$(qj)j=1,…,4$verify$qj∈[1,pn,kj]$and$∑j=14kj=2(n−1)$, then$qj′∈[pn,kj′,∞]$, so
$0≤∑j=141qj′≤∑j=14n−kjn+3=2(n+1)n+3.$
Hence, we can find such a family verifying (9) as soon as$2(n+1)n+3≥2b$or, equivalently, as soon as$a≤2nn+3=:an$.

We can now complete the Proof of Theorem 2.1 using a similar strategy as to the one explained in Sec. II A for the Hartree equation. Instead of proceeding by induction and bounding the time derivative of Mn by a product involving Mn and Mn−2, we directly estimate it by a product involving Mn and M2. This method allows us to slightly improve the result of Ref. 10, Theorem 3, even in the case of the Hartree equation, as it allows us to propagate moments of high order locally in time for any a ∈ (0, 2), while Ref. 10, Theorem 3 only covers the case $a∈(0,87)$ in dimension 3.

Proof of Theorem 2.1.
Taking the time derivative of moments and using the cyclicity of the trace yield
$iℏddtTr(ρpln)=−Tr(Vρ,plnρ)−Tr(Xρ,plnρ).$
Then, by inequality (7), which also holds for $pln$ in place of |p|n, and inequality (10), we deduce that
$h3ddtTr(ρpln)≤Cn,asupk1+⋯+k4=2(n−1)infQρk1Lq112ρk2Lq212ρk3Lq312ρk4Lq412$
with the notations of Lemma 2.1. Using the interpolation formula (11) for each of the terms in the right-hand side of the above inequality, we get
$h3iℏTr(h3Xρ,plnρ)≤Cn,aρL∞1b(1+M2)Θ2(1+Mn)Θn$
with $Θ2=n+1n−2−n+3n−21b$ and
$Θn=2n−102(n−2)+5n−21b=1+1n−25b−3.$
Moreover, by Ref. 19, Lemma 6.3, Mn can be further bounded by
$C−1∑l=13Tr(ρ(1+pln))≤Mn≤C∑l=13Tr(ρ(1+pln))$
for some C > 0. In particular,
$Θn≤1⟺b≥53⟺a≤45,$
which yields the result by Grönwall’s lemma.□

Note that bounds on the moments imply bounds on the semiclassical Schatten norms for any $p∈2,∞$. More precisely, we have the following proposition.

Proposition 3.1.
Letρbe a positive trace class operator. Then, for any$p∈2,∞$, there exists a constantC > 0 such that
$ρmnLp≤CρL∞1p′(1+Mnp)1p.$

Proof.
Using the fact that ρ is a bounded operator, we get that
$ρmnLp≤ρL∞1p′ρ1pmnLp≤ρL∞1p′ρmnp2L22p=ρL∞1p′h3Tr(ρmnp)1p,$
where the second inequality follows from the Araki–Lieb–Thirring inequality.□

For p = , we control $ρmnL∞$ by means of a Grönwall argument. We will need the following commutator estimates in the spirit of Refs. 10 and 19, but improved to lead to large time estimates.

Proposition 3.2
(weighted commutator estimate). Let$a∈(0,12)$,$n∈N$, andq ∈ [1, ]. Then, for anyɛ ∈ (0, 1) andn1 > n + a + 1, there exists a constantC > 0 such that for every compact self-adjoint operatorsρandμ,
$1ℏ∇Vρ,pjnμLq≤Cρmn1Lr±εμmnLq,$
(12)
$1ℏVρ,pjnμLq≤C(ρmn1Lr±ε+M0)μmnLq,$
(13)
where$r=31−a$and$⋅Lr±ε$stands for the norm
$⋅Lr±ε:=⋅Lr+ε∩Lr−ε=⋅Lr+ε+⋅Lr−ε.$

Proof of Proposition 3.2.
We first proceed as in Refs. 10 and 19 and write
$1ℏ∇Vρ,pjnμLq≤∑ℓ=0n−1Cℓgℓpjn−1−ℓμLq≤∑ℓ=0n−1CℓgℓL∞μmnLq,$
where g is the function defined by $gℓ(x)=(pjℓ(∂j∇Vρ))(x)$ and $Cℓ=∑k=ℓn−1kℓ$. Noting that
$gℓ=−∂j∇K*(pjℓρ)=−∂j∇K*diag(adpjℓ(ρ)),$
where adA(X) := [A, X], and using the fact that $∂j∇K∈L3a+2+ϵ+L3a+2−ϵ$ for any ϵ > 0 sufficiently small, then Young’s inequality yields
$gℓL∞≤CKdiag(adpjℓ(ρ))Lr±ε$
for any ɛ ∈ (0, 1). By Proposition 6.4 and Lemma 6.5 of Ref. 19, it implies that
$gℓL∞≤CKadpjℓ(ρ)mn0Lr±ε≤2ℓ+1CKρmn0+n−1Lr±ε$
with n0 > 3/r′. This proves inequality (12). When ∇Vρ is replaced by Vρ, then we just replace ∇K by K, and so we need to find a L bound for the function
$gℓ=−∂jK*(pjℓρ).$
When > 0, we just write $gℓ=iℏ∂j2K*(pjℓ−1ρ)$ and, then, use the same estimates as for ∇Vρ. If = 0, then g = −jK*ρ is bounded using Young’s inequality by
$gℓL∞≤ρLb′±ε≤ρLr+ρL1$
since $b′≤r$. By Ref. 19, Proposition 6.4, $ρLr≤ρmn0Lr$ with n0 > 3/r′ = a + 2.□

Proposition 3.3.
Let$a∈(0,12)$andρbe a solution to the Hartree–Fock equation (1). Assume that$ρin∈L∞(mn)$for some even$n∈2N$with moments of order$(n+a+1)31−a+ε$bounded uniformly infor some$ε∈0,1$. Then,
$ρ∈Lloc∞(R+,L∞(mn)).$

Proof.
By Ref. 19, Lemma 6.2 with q, we have
$ddtρmnL∞≤1ℏVρ,mnρL∞+1ℏh3Xρ,mnρL∞.$
(14)
Next, we use Proposition 3.2 to bound the first term on the right-hand side of inequality (14),
$1ℏVρ,mnρL∞≲(1+ρmn1Lr±ε)ρmnL∞,$
where $r=31−a$ and where we use the standard notation xy to denote xCy for some constant C ≤ 0, which is independent of x, y, and h. Since ɛ ≤ 1 and r ≥ 3, we see that rɛ ≥ 2, which allows us to apply Proposition 3.1 to get
$1ℏVρ,mnρL∞≲(1+ρL∞+Mn1(r+ε))ρmnL∞.$
(15)
As for the second term on the right-hand side of inequality (14), we first observe that by Ref. 19, Lemma 6.4,
$ℏ32−a∇xρmnLp=ℏ12−ap,ρmnLp≤2ℏ12−aρmn+1Lp,$
and then we apply Ref. 19, Proposition 6.8 with $mn=1+|p|n$ to get
$1ℏh3Xρ,mnρL∞≲ℏ12−aρmn+1L2ρmnL∞.$
Proposition 3.1 allows us to bound $ρmn+1L2$ in terms of M2(n+1) and obtain the following inequality for the left-hand side of Eq. (14):
$ddtρmnL∞≲(1+ρL∞+Mn1(r+ε)+M2(n+1))ρmnL∞,$
which gives a bound on $ρmnL∞$ by means of Grönwall’s lemma and the control on moments established in Theorem 2.1.□

Summarizing Theorem 2.1 and Propositions 3.1 and 3.3 implies the following result.

Corollary 3.1.
Let$a∈(0,12)$andρbe a solution to the Hartree–Fock equation (1). For$n∈2N$, let$ρin∈L∞(mn)$with the moment of order$(n+a+1)(31−a+ε)$bounded uniformly infor someɛ ∈ (0, 1). Then, for anyq ∈ [2, ], we have that
$ρ∈Lloc∞(R+,Lq(mn)).$

Proof of Theorem 1.1.
To propagate quantum Sobolev norms, we proceed as in Ref. 19, Sec. 6 and consider the inequalities
$ddt∇xρmnLq≤1ℏ(Vρ,mn∇xρLq+∇Vρ,mnρLq)+1ℏ∇Vρ,ρmnLq+1ℏh3Xρ,mn∇xρLq+1ℏh3X∇xρ,ρmnLq$
and
$ddt∇ξρmnLq≤1ℏVρ,mn∇ξρLq+∇xρmnLq+1ℏh3Xρ,mn∇ξρLq+1ℏh3X∇ξρ,ρmnLq,$
and we define $Nq=Nq(t):=ρW1,q(mn)$. Now, when $q≤61+2a$, Ref. 19, Proposition 6.5 yields an estimate of the form
$1ℏ∇Vρ,ρmnLq≤ρLr±ε∩L2∇x(ρmn)Lq$
(16)
with $r=31−a$. Then, using the above inequality, Proposition 3.2, and, similarly as in the proof of the previous proposition, Ref. 19, Propositions 6.8 and 6.9 to bound the terms involving the exchange term, we obtain for any $q≤61+2a$,
$ddtNq≲(1+ρmn1Lr±ε)Nq+ρLr±ε∩L2Nq+h12−aρmn+1L2Nq+h31q+12−1bρmnL∞N2$
with n1 > n + a + 1. Note that for $a∈(0,12)$, the quantity $1q+12−1b$ is positive for every $q∈2,∞$. Hence, for h ∈ (0, 1), we have that
$ddt(N2+Nq)≲(1+ρmn1Lr±ε+ρmnL∞+ρmn+1L2)(N2+Nq),$
which proves Theorem 1.1 for $q≤61+2a$ by Corollary 3.1 and Grönwall’s lemma. The limitation on q is due to the fact that we do not want to have any ∇ρ in the right-hand side of inequality (16). However, now that we obtained the boundedness of N2, we can propagate higher norms of the form $Nq2$ when $q2>61+2a$. Indeed, Ref. 19, Proposition 6.5 also yields, for instance,
$1ℏ∇Vρ,ρmnL∞≤ρL21−θρH1θ∇x(ρmn)L∞,$
with $θ=3b−12=a+12∈(0,1)$ and $ρL2$ being controlled by N2. All q2 ∈ (2, ) follow in the same way, finishing the proof of the theorem.□

J.J.C. was supported by the NSF through the RTG under Grant No. DMS- RTG 184031. C.S. acknowledges the NCCR SwissMAP and the support of the SNSF through the Eccellenza project under Grant No. PCEFP2_181153.

The authors have no conflicts to disclose.

Laurent Lafleche: Writing – original draft (equal); Writing – review & editing (equal). Jacky J. Chong: Writing – original draft (equal); Writing – review & editing (equal). Chiara Saffirio: Writing – original draft (equal); Writing – review & editing (equal).

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