# Relaxation limit from the Quantum Navier–Stokes equations to the Quantum
Drift–Diffusion equation

Paolo Antonelli  GSSI-Gran Sasso Science Institute (Italy)
<EMAIL_ADDRESS>,  Giada Cianfarani Carnevale  Dipartimento di
Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi
dell’Aquila (Italy<EMAIL_ADDRESS>,  Corrado
Lattanzio  Dipartimento di Ingegneria e Scienze dell’Informazione e
Matematica, Università degli Studi dell’Aquila (Italy<EMAIL_ADDRESS>and
Stefano Spirito  Dipartimento di Ingegneria e Scienze dell’Informazione e
Matematica, Università degli Studi dell’Aquila (Italy)
<EMAIL_ADDRESS>
###### Abstract.

The relaxation-time limit from the Quantum-Navier-Stokes-Poisson system to the
quantum drift-diffusion equation is performed in the framework of finite
energy weak solutions. No assumptions on the limiting solution are made. The
proof exploits the suitably scaled a priori bounds inferred by the energy and
BD entropy estimates. Moreover, it is shown how from those estimates the
Fisher entropy and free energy estimates associated to the diffusive evolution
are recovered in the limit. As a byproduct, our main result also provides an
alternative proof for the existence of finite energy weak solutions to the
quantum drift-diffusion equation.

##  1\. Introduction

This paper studies the relaxation-time limit for the Quantum Navier-Stokes-
Poisson (QNSP) system with linear damping, towards the quantum drift-diffusion
equation. More precisely, in the three dimensional torus $\mathbb{T}^{3}$, we
consider a compressible, viscous fluid, whose dynamics is prescribed by

$\displaystyle\partial_{t}\rho+\operatorname{div}(\rho u)=0$ (1.1)
$\displaystyle\partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes
u)-\operatorname{div}(\rho Du)+\nabla\rho^{\gamma}+\rho\nabla
V=2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\xi\rho u$
$\displaystyle-\Delta V=\rho-g.$

Here the unknowns $\rho$, $u$, and $V$ denote the particle density, the
velocity field, and the electrostatic potential respectively. The function $g$
is given and represents the doping profile.
The system arises in the macroscopic description of electron transport in
nanoscale semiconductor devices [23], where quantum-mechanical effects must be
taken into account. In this context the dissipative term $-\xi\rho u$
describes collisions between electrons and the semiconductor crystal lattice
(see, for instance, [7]), and $\tau=1/\xi$ is the relaxation time. The
advantage of using macroscopic models for quantum fluids, with respect to
kinetic models, is their reduced complexity, especially from a computational
point of view [36]. Moreover, hydrodynamic models correctly describe high
field phenomena or submicronic devices. However, in certain regimes, as in
particular for low carrier densities and small electric fields, these models
can be further reduced to some simpler ones. In the context of semiconductor
devices for instance, quantum transport of electrons can be effectively
described by the quantum drift-diffusion (QDD) equation [37], given by

$\displaystyle{\partial}_{t}\rho+\operatorname{div}\left(2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\nabla\rho^{\gamma}-\rho\nabla
V\right)=0$ (1.2) $\displaystyle-\Delta V=\rho-g.$

The (QDD) equation can be formally recovered from system (1.1) as a relaxation
limit. Precisely, by rescaling the time as follows

$t^{\prime}=\epsilon
t,\quad(\rho^{\epsilon},u^{\epsilon})(t^{\prime},x)=\left(\rho,u\right)\left(\frac{t^{\prime}}{\epsilon},x\right),$
(1.3)

where $\epsilon:=1/\xi$, the scaled system reads

$\displaystyle\partial_{t}\rho_{\epsilon}+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}u_{\epsilon})=0$
(1.4)
$\displaystyle\partial_{t}(\rho_{\epsilon}u_{\epsilon})+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}u_{\epsilon}\otimes
u_{\epsilon})-\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}Du_{\epsilon})+\frac{1}{\epsilon}\nabla\rho_{\epsilon}^{\gamma}+\frac{1}{\epsilon}\rho_{\epsilon}\nabla
V_{\epsilon}=\frac{1}{\epsilon}2\rho_{\epsilon}\nabla\left(\frac{\Delta\sqrt{\rho_{\epsilon}}}{\sqrt{\rho_{\epsilon}}}\right)-\frac{1}{\epsilon^{2}}\rho_{\epsilon}u_{\epsilon}$
$\displaystyle-\Delta V_{\epsilon}=\rho_{\epsilon}-g.$

Thus in the limit $\epsilon\to 0$, we formally obtain that

$\lim_{\epsilon\to
0}\rho_{\epsilon}\frac{u_{\epsilon}}{\epsilon}=2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\nabla\rho^{\gamma}-\rho\nabla
V$ (1.5)

and therefore the (QDD) equation.

The main purpose of our paper is to rigorously prove the above limit, that is
to prove that scaled finite energy weak solutions to (1.1) converge to finite
energy weak solutions to (1.2). To this aim, in the following we shall refer
to (1.4) with initial datum $(\rho^{0},u^{0})$ and doping profile $g$ possibly
depending in a suitable way on the relaxation parameter $\epsilon$ as well.

######  Theorem 1.1.

Let $(\rho_{\epsilon},u_{\epsilon},V_{\epsilon})$ be a weak solution of
$\eqref{eq:qns_intro}$ in the sense of Definition 2.1 with data
$(\rho^{0}_{\epsilon},u^{0}_{\epsilon},g_{\epsilon})$ satisfying

$\displaystyle\\{\rho_{\epsilon}^{0}\\}_{\epsilon}\mbox{ is bounded in
}L^{1}\cap L^{\gamma}(\mathbb{T}^{3})\ \hbox{such that}\
\rho_{\epsilon}^{0}\to\rho^{0}\ \hbox{in}\ L^{q}(\mathbb{T}^{3}),\,q<3$
$\displaystyle\\{\nabla\sqrt{\rho_{\epsilon}^{0}}\\}_{\epsilon}\mbox{ is
bounded in }L^{2}(\mathbb{T}^{3}),$
$\displaystyle\\{\sqrt{\rho_{\epsilon}^{0}}u_{\epsilon}^{0}\\}_{\epsilon}\mbox{
is bounded in }L^{2}(\mathbb{T}^{3}),$
$\displaystyle\\{g_{\epsilon}\\}_{\epsilon}\mbox{ is bounded in
}L^{2}(\mathbb{T}^{3})\ \hbox{such that}\ g_{\epsilon}\rightharpoonup g\
\hbox{in}\ L^{2}(\mathbb{T}^{3}).$

Then, up to subsequences, there exists $\rho\geq 0$ and $V$ such that

$\displaystyle\sqrt{\rho_{\epsilon}}\rightarrow\sqrt{\rho}\textrm{ strongly in
}L^{2}((0,T);H^{1}(\mathbb{T}^{3}))$ $\displaystyle\nabla
V_{\epsilon}\rightarrow\nabla V\textrm{ strongly in
}C([0,T);L^{2}(\mathbb{T}^{3})),$

and $(\rho,V)$ is a finite energy weak solution of $\rho$ of (1.2) with
initial datum $\rho(0)=\rho^{0}$, in the sense of Definition 2.8. Namely there
exist $\Lambda,\,\mathcal{S}\in L^{2}((0,T)\times\mathbb{T}^{3})$ such that

$\displaystyle\sqrt{\rho}\Lambda$
$\displaystyle=2\operatorname{div}(\sqrt{\rho}\nabla^{2}\sqrt{\rho}-\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}-\rho^{\gamma}\mathbb{I})-\rho\nabla
V\textrm{ in }\mathcal{D}^{\prime}((0,T)\times\mathbb{T}^{3})$ (1.6)
$\displaystyle\sqrt{\rho}\mathcal{S}$
$\displaystyle=2\sqrt{\rho}\nabla^{2}\sqrt{\rho}-2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}\,\mbox{
a.e. in }(0,T)\times\mathbb{T}^{3}$ (1.7)

and $C>0$ such that for $a.e.$ $t\in(0,T)$

$\int_{\mathbb{T}^{3}}\left(|\nabla\sqrt{\rho}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}+\frac{1}{2}|\nabla
V|^{2}\right)(t)dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\Lambda|^{2}dsdx\leq C$
(1.8) $\displaystyle\int_{\mathbb{T}^{3}}(\rho(\log\rho-1)+1)(t)dx$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\mathcal{S}|^{2}dsdx+\frac{4}{\gamma}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\nabla\rho^{\frac{\gamma}{2}}|^{2}dsdx$
(1.9)
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho(\rho-g)\;dsdx\leq\int_{\mathbb{T}^{3}}(\rho^{0}(\log\rho^{0}-1)+1)dx.$

Moreover, if in addition, the initial data also satisfy

$\displaystyle\sqrt{\rho_{\epsilon}^{0}}u_{\epsilon}^{0}\to 0\mbox{ strongly
in }L^{2}(\mathbb{T}^{3})$
$\displaystyle\nabla\sqrt{\rho_{\epsilon}^{0}}\to\nabla\sqrt{\rho^{0}}\mbox{
strongly in }L^{2}(\mathbb{T}^{3})$
$\displaystyle\rho_{\epsilon}^{0}\to\rho^{0}\mbox{ strongly in
}L^{\gamma}(\mathbb{T}^{3}),$

then $\rho$ is an energy dissipating weak solution, meaning that in addition
to be a finite energy weak solution for a.e. $t\in(0,T)$ it holds

$\displaystyle\int_{\mathbb{T}^{3}}\left(|\nabla\sqrt{\rho}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}+\frac{1}{2}|\nabla
V|^{2}\right)(t)dx$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\Lambda|^{2}dsdx\leq$
$\displaystyle\int_{\mathbb{T}^{3}}\,|\nabla\sqrt{\rho^{0}}|^{2}dx+\int_{\mathbb{T}^{3}}\frac{{(\rho^{0})^{\gamma}}}{\gamma-1}dx+\int_{\mathbb{T}^{3}}\frac{1}{2}|\nabla
V(0)|^{2}dx.$

Let us notice that the estimates (1.8) and (1.9) yield the boundedness of the
Fisher entropy and the free energy, respectively. On the other hand, the
quantities $\Lambda$ and $\mathcal{S}$ characterized in (1.6) and (1.7)
provide the associated entropy dissipations, in a weaker sense than the
estimates derived in [18] and [27]. Indeed, formally

$\displaystyle\Lambda=2\sqrt{\rho}\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\frac{\nabla\rho^{\gamma}}{\sqrt{\rho}},\qquad\mathcal{S}=\sqrt{\rho}\nabla^{2}\log\rho,$
(1.10)

but, due to the low regularity setting and the possible presence of vacuum
regions it seems not possible to obtain the relations (1.10) in the limit, so
the only available information we have is given by formulas (1.6) and (1.7).
We refer to Remark 2.4 and Proposition 2.10 below for more details on the
tensor $\Lambda$ and $\mathcal{S}$.

An exhaustive list of all references concerning diffusive relaxation limits
and asymptotic behavior for systems of conservation laws with friction, and in
particular for hydrodynamic models for semiconductors, is beyond the interest
of our presentation. For the theory of diffusive relaxation, we refer here to
[17], concerning in particular the case of multidimensional general semilinear
systems, and the reference therein. Moreover, concerning in particular the
case of high friction limits with relative entropy techniques in the context
of Korteweg theories [19], we refer to [34, 14]; see also [33] for the case of
Euler equations with friction. Finally, for the particular case of
Euler–Poisson models for semiconductors, we recall that the rigorous analysis
of the diffusive relaxation limits in the context of weak, entropic solutions
started with the seminal paper [35], where the one dimensional case is treated
using compensated compactness; see also [32, 31] for the multi–$d$ case.

Besides the modeling point of view, there are some other mathematical aspects
which motivate our result. First of all, the study of this singular limit is
related to the asymptotic behavior of solutions to (1.1) for large times. Let
$(\rho^{\star},u^{\star})=(r,0)$ be the stationary solution to (1.1), where,
for $g$ constant, $r=\fint\rho=g$ is the mean value of the particle density.
Then it can be shown that solutions to (1.1) exponentially converge towards
$(\rho^{\star},u^{\star})$ as $t\to\infty$, see [20] for the one-dimensional
problem (with suitable boundary conditions, see also [29] for some extensions)
and [11] for the proof of this result in the framework of finite energy weak
solutions in the three dimensional torus.

On the other hand, it is also interesting to determine the asymptotic dynamics
which governs the exponential convergence to equilibrium. This is indeed
achieved by performing the scaling in (1.3), hence the (QDD) equation (1.2)
also gives the asymptotic dynamics we are interested in.

On a related subject, let us also comment on the inviscid counterpart of
system (1.1), namely the quantum hydrodynamic (QHD) system [1, 2]. Due to the
dissipative term $-\xi\rho u$, also in this case it is possible to show both
the exponential convergence towards the stationary solution [21, 22] and the
relaxation limit [25], again towards the (QDD) equation. However the only
available results here deal with small, regular perturbations around
stationary solutions. This can be seen as due to the lack of regularizing
effect of the viscosity, by means of the BD entropy estimates.

Notice that in Theorem 1.1 the only assumption needed is the initial energy
associated to the system (1.1) to be uniformly bounded at the initial time. In
particular no assumptions on the limiting solution to (1.2) are given.
Consequently, as a byproduct our main Theorem also provides an alternative
proof for the existence of finite energy weak solutions to (1.2), see [27] and
[18]. Furthermore, in the proof of our main result it is possible to see how
the energy and BD entropy estimates, respectively, associated to (1.1) yield,
in the limit $\varepsilon\to 0$, the Fisher and free energy, respectively,
associated to (1.2). Those facts were already noticed, in a similar context,
in the recent preprint [10], see also [9]. More precisely, the Authors in [10]
consider the one dimensional shallow water equations with a nonlinear damping
term. By using a similar scaling as in (1.3), the authors study the
convergence towards a lubrication type model. In particular in [10] the
Authors emphasize how the BD entropy for the hydrodynamical system converge
towards the so called Bernis-Friedman [8] entropy, associated to the limiting
diffusive equation.

Let us remark that also in the context of semiconductor device modeling it
would make sense to consider a nonlinear damping term, as in [10]. Indeed this
would correspond to the case when the relaxation time $\tau$ is no longer a
constant, but a function of the particle density. This is consistent with the
derivation of the hydrodynamic system from kinetic theories, as in general the
relaxation coefficient may depend on the particle density. Finally, our result
can also be seen as related to the derivation of (1.1) and (1.2) from kinetic
equations. These macroscopic models for quantum transport are usually derived
from collisional Wigner-type equations, with a suitable choice of the
collision operator, see [16, 15] and [24] for a more comprehensive discussion
about those issues. In particular, the QNS system with a linear damping was
derived in [28] by applying the moment method to a Wigner-type equation whose
collisional operator is chosen to be the sum of a BGK and a Caldeira-Leggett-
type operator, see also [26] where an alternative derivation is given by
avoiding the Chapman-Enskog expansion. Actually in [28] and [26] the authors
derive the full QNS system, where also the dynamics of the energy density is
given, in our paper we only consider the isentropic dynamics given by (1.1).
We also mention [13] where the QNS system without damping is derived. On the
other hand the QDD equation can also be derived from the same Wigner-type
equation by using a diffusive scaling. In this sense our result, obtained by
using the scaling (1.3), can be seen as linking the two different scalings
used to derive (1.1) and (1.2) directly from kinetic models.

### Organization of the paper

The paper is organized as follows. In Section 2 we give the definition of weak
solution for the Quantum-Navier-Stokes system and the Quantum-Drift-Diffusion
equation, we give a formal proof of the $\epsilon$-independent estimates and
we state the main theorem and in Section 3 we prove the main result of the
paper.

### Notations

We denote with $L^{p}(\mathbb{T}^{n})$ the standard Lebesgue spaces. The
Sobolev space of $L^{p}$ functions with $k$ distributional derivatives in
$L^{p}$ is denoted $W^{k,p}$, in the case $p=2$ we write
$H^{k}(\mathbb{T}^{n})$. The spaces $W^{-k,p}$ and $H^{-k}$ denote the dual
spaces of $W^{k,p^{\prime}}$ and $H^{k}$ where $p^{\prime}$ is the usual
Hölder conjugate of p. Given a Banach space $B$, the classical Bochner space
of real valued function with values in $B$ is denote by $L^{p}(0,T;B)$ and
sometimes also the abbreviation $L^{p}_{t}(B_{x})$ will be used. Given a
function $f\in L^{p}(\mathbb{T}^{3})$ we denote the average of $f$
alternatively $\fint f$ or $\bar{f}$, and throughout the paper we can assume
without loss of generality that $|\mathbb{T}^{3}|=1$. We denote with
$Du=(\nabla u+\nabla u^{t})/2$ the symmetric part of the Jacobian matrix
$\nabla u$ and with $Au=(\nabla u-\nabla u^{t})/2$ the antisymmetric part.
Finally, the subscript $\epsilon$ used to denote sequences of functions has to
be always understood running over a countable set.

##  2\. Definition of Weak Solutions and Main Result

In this section we give the definition of weak solutions for the system (2.1)
and (1.2). The existence of such solutions is out of the main aims of this
paper; we underline here this result can be proved via a compactness argument
as done in [30, 4] with minor modifications; see also [3, 5, 6]. However, it
is worth observing that with this technique, thanks to the particular choice
of the approximating sequence, the constructed weak solutions verify various
appropriate bounds, instrumental for our convergence result. For this reason,
we shall assume these bounds are valid in our framework; see Definition 2.1
and Remarks 2.4 and 2.5 below.

###  2.1. Weak solutions of the Quantum-Navier-Stokes-Poisson equations

Let us consider the following system in $(0,T)\times\mathbb{T}^{3}$ for a
given $g_{\epsilon}:\mathbb{T}^{3}\rightarrow\mathbb{R}$,

$\displaystyle\partial_{t}\rho_{\epsilon}+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}u_{\epsilon})=0$
(2.1)
$\displaystyle\partial_{t}(\rho_{\epsilon}u_{\epsilon})+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}u_{\epsilon}\otimes
u_{\epsilon})-\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}Du_{\epsilon})+\frac{1}{\epsilon}\nabla\rho_{\epsilon}^{\gamma}+\frac{1}{\epsilon}\rho_{\epsilon}\nabla
V_{\epsilon}=\frac{1}{\epsilon}2\rho_{\epsilon}\nabla\left(\frac{\Delta\sqrt{\rho_{\epsilon}}}{\sqrt{\rho_{\epsilon}}}\right)-\frac{1}{\epsilon^{2}}\rho_{\epsilon}u_{\epsilon}$
$\displaystyle-\Delta V_{\epsilon}=\rho_{\epsilon}-g_{\epsilon}$

with initial data

$\begin{split}&\rho_{\epsilon}(0,x)=\rho^{0}_{\epsilon}(x),\\\
&\rho_{\epsilon}(0,x)u_{\epsilon}(0,x)=\rho^{0}_{\epsilon}(x)u^{0}_{\epsilon}(x),\end{split}$
(2.2)

on $\\{t=0\\}\times\mathbb{T}^{3}$ and zero average condition for
$V_{\epsilon}$, namely

$\fint_{\mathbb{T}^{3}}V_{\epsilon}(x,t)dx=0.$ (2.3)

We emphasize that in order to solve (2.1)3, (2.3) we need the following
assumption on the doping profile $g_{\epsilon}$:
$\fint_{\mathbb{T}^{3}}g_{\epsilon}(x)dx=M_{\epsilon}$ and $M_{\epsilon}>0$
where $M_{\epsilon}:=\fint_{\mathbb{T}^{3}}\rho_{\epsilon}^{0}(x)$; see also
Remark 2.6 below.

The definition of weak solution is the following.

######  Definition 2.1.

Given $\rho^{0}_{\epsilon}$ positive and such that
$\sqrt{\rho}^{0}_{\epsilon}\in H^{1}(\mathbb{T}^{3})$ and
$\rho^{0}_{\epsilon}\in L^{\gamma}(\mathbb{T}^{3})$, $g_{\epsilon}\in
L^{2}(\mathbb{T}^{3})$ and $u^{0}_{\epsilon}$ such that
$\sqrt{\rho}^{0}_{\epsilon}u^{0}_{\epsilon}\in L^{2}(\mathbb{T}^{3})$, then
$(\rho_{\epsilon},u_{\epsilon},V_{\epsilon})$ with $\rho_{\epsilon}\geq 0$ and
$V_{\epsilon}$ with zero average is a weak solution of the Cauchy problem
(2.1)-(2.2) if the following conditions are satisfied.

  1. (1)

Integrability conditions:

$\displaystyle\sqrt{\rho_{\epsilon}}\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{3}))\cap L^{2}(0,T;H^{2}(\mathbb{T}^{3})),$
$\displaystyle\sqrt{\rho_{\epsilon}}u_{\epsilon}\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3})),$
$\displaystyle\rho_{\epsilon}^{\gamma}\in
L^{\infty}(0,T;L^{1}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{T}_{\epsilon}\in L^{2}(0,T;L^{2}(\mathbb{T}^{3})),$
$\displaystyle\rho_{\epsilon}\in C(0,T;L^{2}(\mathbb{T}^{3})),$ $\displaystyle
V_{\epsilon}\in C(0,T;H^{2}(\mathbb{T}^{3})).$

  2. (2)

Continuity equation:
For any $\phi\in C^{\infty}([0,T)\times\mathbb{T}^{3};\mathbb{R})$

$\int\rho_{\epsilon}^{0}\phi(0)dx+\iint\rho_{\epsilon}\phi_{t}+\frac{1}{\epsilon}\sqrt{\rho_{\epsilon}}\sqrt{\rho_{\epsilon}}u_{\epsilon}\nabla\phi\;dsdx=0.$
(2.4)

  3. (3)

Momentum equation:
For any $\psi\in C^{\infty}([0,T)\times\mathbb{T}^{3};\mathbb{R}^{3})$

$\displaystyle\int\rho_{\epsilon}^{0}u_{\epsilon}^{0}\psi(0)dx+\iint\sqrt{\rho_{\epsilon}}(\sqrt{\rho_{\epsilon}}u_{\epsilon})\psi_{t}\;dsdx+\frac{1}{\epsilon}\iint\sqrt{\rho_{\epsilon}}u_{\epsilon}\sqrt{\rho_{\epsilon}}u_{\epsilon}\cdot\nabla\psi\;dsdx$
(2.5)
$\displaystyle-\frac{1}{\epsilon}\iint\sqrt{\rho_{\epsilon}}\mathcal{T}_{\epsilon}^{s}:\nabla\psi\;dsdx+\frac{1}{\epsilon}\iint\rho_{\epsilon}^{\gamma}\operatorname{div}\psi\;dsdx-\frac{1}{\epsilon}\iint\rho_{\epsilon}\nabla
V_{\epsilon}\psi\;dsdx$ $\displaystyle-\frac{1}{\epsilon}\iint
2\sqrt{\rho_{\epsilon}}\,\nabla^{2}\sqrt{\rho_{\epsilon}}:\nabla\psi\;dsdx+\frac{1}{\epsilon}\iint
2\nabla\sqrt{\rho_{\epsilon}}\otimes\nabla\sqrt{\rho_{\epsilon}}\,:\nabla\psi\;dsdx+\frac{1}{\epsilon^{2}}\iint\rho_{\epsilon}u_{\epsilon}\psi\;dsdx=0.$

  4. (4)

Poisson equation:
For a.e. $(t,x)\in(0,T)\times\mathbb{T}^{3}$ it holds that

$-\Delta V_{\epsilon}=\rho_{\epsilon}-g_{\epsilon}.$ (2.6)

  5. (5)

Energy dissipation:
For any $\varphi\in C^{\infty}([0,T)\times\mathbb{T}^{3};\mathbb{R})$

$\iint\sqrt{\rho_{\epsilon}}\mathcal{T}_{\epsilon,i,j}\varphi\;dsdx=-\iint\rho_{\epsilon}u_{\epsilon,i}\nabla_{j}\varphi\;dsdx-\iint
2\sqrt{\rho_{\epsilon}}u_{\epsilon,i}\otimes\nabla_{j}\sqrt{\rho_{\epsilon}}\,\varphi\;dsdx.$
(2.7)

  6. (6)

Energy Inequality:
For a.e. $t\in(0,T)$

$\begin{split}&\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho_{\epsilon}|u_{\epsilon}|^{2}+\frac{\rho_{\epsilon}^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho_{\epsilon}}|^{2}+\frac{1}{2}|\nabla
V_{\epsilon}|^{2}\right)(t)dx+\frac{1}{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\mathcal{T}^{s}_{\epsilon}|^{2}dsxd+\frac{1}{\epsilon^{2}}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|u_{\epsilon}|^{2}dsdx\\\
&\leq\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho_{\epsilon}^{0}|u_{\epsilon}^{0}|^{2}+\frac{({\rho_{\epsilon}^{0})}^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho_{\epsilon}^{0}}|^{2}+\frac{1}{2}|\nabla
V_{\epsilon}(0)|^{2}\,\right)dx.\end{split}$ (2.8)

  7. (7)

BD Entropy: Let $w_{\epsilon}=u_{\epsilon}+\nabla\log\rho_{\epsilon}$, then
there exists a tensor $\mathcal{S}_{\epsilon}\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ such that

$\sqrt{\rho_{\epsilon}}\mathcal{S}_{\epsilon}=2\sqrt{\rho_{\epsilon}}\nabla^{2}\sqrt{\rho_{\epsilon}}-2\nabla\sqrt{\rho_{\epsilon}}\otimes\nabla\sqrt{\rho_{\epsilon}}\,\mbox{
a.e. in }(0,T)\times\mathbb{T}^{3}$ (2.9)

we have for a.e. $t\in(0,T)$

$\displaystyle\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\epsilon\rho_{\epsilon}|w_{{\epsilon}}|^{2}+\epsilon\frac{\rho_{\epsilon}^{\gamma}}{\gamma-1}+\epsilon
2|\nabla\sqrt{\rho_{\epsilon}}|^{2}+\frac{1}{2}\epsilon|\nabla
V_{\epsilon}|^{2}+(\rho_{\epsilon}(\log\rho_{\epsilon}-1)+1)\right)(t)dx$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\mathcal{T}_{\epsilon}^{a}|^{2}\;dsdx+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\mathcal{S}_{\epsilon}|^{2}\;dsdx+\frac{4}{\gamma}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\nabla\rho_{\epsilon}^{\frac{\gamma}{2}}|^{2}\;dsdx+\frac{1}{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|u_{\epsilon}|^{2}\;dsdx$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}(\rho_{\epsilon}-g_{\epsilon})\;dsdx\leq$
$\displaystyle\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\epsilon\rho_{\epsilon}^{0}|w_{{\epsilon}}^{0}|^{2}+\epsilon\frac{({\rho_{\epsilon}^{0}})^{\gamma}}{\gamma-1}+\epsilon
2|\nabla\sqrt{\rho_{\epsilon}^{0}}|^{2}++\frac{1}{2}\epsilon|\nabla
V_{\epsilon}(0)|^{2}+(\rho_{\epsilon}^{0}(\log\rho_{\epsilon}^{0}-1)+1)\right)dx.$
(2.10)

  8. (8)

There exists an absolute constant $C$ such that

$\displaystyle\int_{0}^{t}\int_{\mathbb{T}^{3}}|\nabla^{2}\sqrt{\rho_{\epsilon}}|^{2}+|\nabla\rho_{\epsilon}^{\frac{1}{4}}|^{4}$
$\displaystyle dsdx\leq
C\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\epsilon\rho_{\epsilon}^{0}|w_{{\epsilon}}^{0}|^{2}+\epsilon\frac{(\rho_{\epsilon}^{0})^{\gamma}}{\gamma-1}+\epsilon
2|\nabla\sqrt{\rho_{\epsilon}^{0}}|^{2}\right)dx$ (2.11)
$\displaystyle+C\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\epsilon|\nabla
V_{\epsilon}(0)|^{2}+(\rho_{\epsilon}^{0}(\log\rho_{\epsilon}^{0}-1)+1)\right)dx$

The following remarks aim at explaining some peculiar points of the definition
of weak solutions. In particular, we explain the presence of the tensors
$\mathcal{T}_{\epsilon}$ and $\mathcal{S}_{\epsilon}$ in the following
remarks, and the reason why we must further assume the bound (2.11).

######  Remark 2.2 (_Weak formulation of the quantum term_).

We emphasize that in the weak formulation introduced above, the third-order
term in the momentum equation can be written in different ways:

$2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)=\operatorname{div}(\rho\nabla^{2}\log\rho)=\operatorname{div}(2\sqrt{\rho}\nabla^{2}\sqrt{\rho}-2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}),$
(2.12)

and we are using the last expression to give a distributional meaning to the
third-order tensor term.

######  Remark 2.3 (_The velocity field and the vacuum set_).

We stress that in the definition of weak solutions the vacuum region can be of
positive measure. As a consequence, the velocity it is not uniquely defined in
the vacuum set, namely if we change its value on $\\{\rho=0\\}$ we would still
have the same weak solution of (2.1)-(2.2). Moreover, even if we choose $u=0$
on $\\{\rho=0\\}$ we can not deduce any a priori bound in any Lebesgue space.

######  Remark 2.4 (_About the tensors $\mathcal{T}_{\epsilon}$ and
$\mathcal{S}_{\epsilon}$_).

The presence of the tensor $\mathcal{T}_{\epsilon}$ is due to the possible
presence of vacuum regions. Indeed, if the density is bounded away from zero,
(2.7) implies that
$\mathcal{T}_{\epsilon}=\sqrt{\rho_{\epsilon}}\nabla\,u_{\epsilon}$. On the
other hand, even in the case when the vacuum has zero Lebesgue measure,
$\nabla u_{\epsilon}$ also can not be defined in a distributional sense.
Therefore the tensor $\mathcal{T}_{\epsilon}$ arises as a weak $L^{2}$-limit
of the sequence $\\{\sqrt{\rho^{n}}\nabla u_{\epsilon}^{n})\\}_{n}$, where
$\\{\rho_{\epsilon}^{n},u_{\epsilon}^{n}\\}_{n}$ is a suitable sequence of
approximations, see [30, 4]. Analogously, $\mathcal{S}_{\epsilon}$ is again
motivated by the presence of vacuum regions. Indeed, also in this case, if the
density is bounded away from zero, by using the identity (2.12), we have that
$\mathcal{S}_{\epsilon}=\sqrt{\rho_{\epsilon}}\nabla^{2}\log\rho_{\epsilon}$.
As for $\mathcal{T}_{\epsilon}$, the tensor $\mathcal{S}_{\epsilon}$ arises as
an $L^{2}$ weak limit of the sequence
$\\{\sqrt{\rho^{n}}\nabla^{2}\log\rho_{\epsilon}^{n}\\}_{n}$, where
$\\{\rho_{\epsilon}^{n},u_{\epsilon}^{n}\\}_{n}$ is a suitable sequence of
approximations. Notice that the fact that
$\\{\sqrt{\rho^{n}}\nabla^{2}\log\rho_{\epsilon}^{n}\\}_{n}$ is bounded in
$L^{2}((0,T)\times\mathbb{T}^{3})$ because of the BD Entropy, see Proposition
2.7.

######  Remark 2.5 (_About the inequality ( 2.11) _).

Regarding (2.11) we recall the following inequality proved in [27] : There
exists $C>0$ depending only on the dimension, such that for any function
$\rho\geq 0$ with $\sqrt{\rho}\in H^{2}(\mathbb{T}^{3})$

$\iint|\nabla\rho^{\frac{1}{4}}|^{4}\;dsdx+\iint|\nabla^{2}\sqrt{\rho}|^{2}\;dsdx\leq
C\iint\rho|\nabla^{2}\log\rho|^{2}\;dsdx.$ (2.13)

Therefore, (2.11) follows by applying (2.13) to some approximation
$\rho_{\epsilon}^{n}$, for which the quantity
$\sqrt{\rho^{n}}\nabla^{2}\log\rho_{\epsilon}^{n}$ is well-defined, using the
bound on $\sqrt{\rho^{n}}\nabla^{2}\log\rho_{\epsilon}^{n}$ inferred from the
BD Entropy and using the weak lower-semicontinuity of the norms. For
completeness we give the proof of (2.11) in the appendix.

######  Remark 2.6 (_About the integrability conditions and the Poisson
equation_).

We notice that by using the integrability hypothesis and (2.4) the average of
the density is conserved, namely,
$\fint_{\mathbb{T}^{3}}\rho_{\epsilon}(t,x)\,dx=M_{\epsilon}$ for any
$t\in(0,T)$. Therefore, $\rho_{\epsilon}-g$ has zero average for any
$t\in(0,T)$ and then the compatibility for the Poisson equation
$\eqref{qns}_{3}$ is always satisfied.
Next, from the mimimun regularity required for $\rho_{\epsilon}$ to be a weak,
finite energy solution, we readily obtain $\rho_{\epsilon}\in
C([0,T];L^{2}(\mathbb{T}^{3}))$; this (redundant) condition is thus listed
explicitly in Definition 2.1. Hence
$\rho_{\epsilon}|_{t=0}=\rho_{\epsilon}^{0}$ in $L^{2}(\mathbb{T}^{3})$ and we
have that $V_{\epsilon}(0)$ is well-defined and coincides with the solution of
the corresponding elliptic equation at $t=0$, that is:

$-\Delta\,V_{\epsilon}(0)=\rho_{\epsilon}^{0}-g_{\epsilon}.$

Finally, since $\rho_{\epsilon}^{0}$ and $g_{\epsilon}$ are both bounded in
$L^{2}(\mathbb{T}^{3})$ we have that $V_{\epsilon}|_{t=0}=V_{\epsilon}(0)$ is
bounded in $H^{2}(\mathbb{T}^{3})$.

Weak solutions of (2.1)-(2.2) can be obtained by using the same approximation
argument of [30] with minor changes. In particular, being the proof of the
existence based on a compactness argument, (2.8) and (2.10) should be proved
for suitable approximate solutions and then would be obtained trough a
limiting argument. On the other hand, as usual in PDE they can be motivated by
formal estimate. This is exactly the content of the next proposition.

######  Proposition 2.7.

Let $\epsilon>0$ and $(\rho_{\epsilon},u_{\epsilon},V_{\epsilon})$ be a
sequence of smooth solutions of (2.1) with initial data (2.2). Define
$w_{{\epsilon}}=u_{\epsilon}+\nabla\log\rho_{\epsilon}$. Then, for any
$t\in[0,T)$ the pair $(\rho_{\epsilon},u_{\epsilon},V_{\epsilon})$ satisfies
the following estimates:

  1. (1)

(Energy Estimate)

$\displaystyle\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho_{\epsilon}|u_{\epsilon}|^{2}+\frac{\rho_{\epsilon}^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho_{\epsilon}}|^{2}+\frac{1}{2}|\nabla
V_{\epsilon}|^{2}\right)(t)dx+\frac{1}{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|Du_{\epsilon}|^{2}\;dsdx+\frac{1}{\epsilon^{2}}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|u_{\epsilon}|^{2}\;dsdx$
(2.14)
$\displaystyle=\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho_{\epsilon}^{0}|u_{\epsilon}^{0}|^{2}+\frac{({\rho_{\epsilon}^{0})}^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho_{\epsilon}^{0}}|^{2}+\frac{1}{2}|\nabla
V_{\epsilon}(0)|^{2}\right)dx.$

  2. (2)

(BD Entropy)

$\displaystyle\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\epsilon\rho_{\epsilon}|w_{{\epsilon}}|^{2}+\epsilon\frac{\rho_{\epsilon}^{\gamma}}{\gamma-1}+\epsilon
2|\nabla\sqrt{\rho_{\epsilon}}|^{2}+\epsilon\frac{1}{2}|\nabla
V_{\epsilon}|^{2}+(\rho_{\epsilon}(\log\rho_{\epsilon}-1)+1)\right)(t)dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|Au_{\epsilon}|^{2}\;dsdx$
(2.15)
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|\nabla^{2}\log\rho_{\epsilon}|^{2}\;dsdx+\frac{4}{\gamma}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\nabla\rho_{\epsilon}^{\frac{\gamma}{2}}|^{2}\;dsdx+\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}(\rho_{\epsilon}-g_{\epsilon})\;dsdx+\frac{1}{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|u_{\epsilon}|^{2}\;dsdx$
$\displaystyle=\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\epsilon\rho_{\epsilon}^{0}|w_{{\epsilon}}^{0}|^{2}+\epsilon\frac{({\rho_{\epsilon}^{0}})^{\gamma}}{\gamma-1}+\epsilon
2|\nabla\sqrt{\rho_{\epsilon}^{0}}|^{2}+\frac{1}{2}\epsilon|\nabla
V_{\epsilon}(0)|^{2}+(\rho_{\epsilon}^{0}(\log\rho_{\epsilon}^{0}-1)+1)\right)dx.$

###### Proof.

Let us start by recalling the following alternative ways to write the
dispersive term:

$2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)=\operatorname{div}(\rho\nabla^{2}\log\rho)=\operatorname{div}(2\sqrt{\rho}\nabla^{2}\sqrt{\rho}-2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}).$

Then, for smooth solutions $(\rho_{\epsilon},u_{\epsilon})$ of $\eqref{qns}$,
as usual we multiply the continuity equation $\eqref{qns}_{1}$ by
$\gamma\rho_{\epsilon}^{\gamma-1}/(\gamma-1)$ and the momentum equation
$\eqref{qns}_{2}$ by $u_{\epsilon}$ and integrate in space to get:

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho|u_{\epsilon}|^{2}\right)dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho|Du_{\epsilon}|^{2}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\nabla\rho^{\gamma}\cdot
u_{\epsilon}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon}:\nabla
u_{\epsilon}dx$ (2.16) $\displaystyle\
+\frac{1}{\epsilon^{2}}\int_{\mathbb{T}^{3}}\rho|u_{\epsilon}|^{2}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla
V_{\epsilon}u_{\epsilon}dx=0,$
$\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{1}{2}|\nabla
V_{\epsilon}|^{2}\;dx=-\int_{\mathbb{T}^{3}}(\Delta
V_{\epsilon})_{t}V_{\epsilon}\;dx=\int_{\mathbb{T}^{3}}\partial_{t}\rho_{\epsilon}V_{\epsilon}\;dx=-\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\operatorname{div}(\rho_{\epsilon}u_{\epsilon})V_{\epsilon}\;dx=\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla
V_{\epsilon}u_{\epsilon}\;dx,$ (2.17)
$\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{\rho_{\epsilon}^{\gamma}}{\gamma-1}\;dx-\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\nabla\rho_{\epsilon}^{\gamma}u_{\epsilon}dx=0.$
(2.18)

Moreover, using the equation satisfied by the so called effective velocity
$\nabla\log\rho_{\epsilon}$, (see [12] for its formal derivation)

$(\rho_{\epsilon}\nabla\log\rho_{\epsilon})_{t}+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}\nabla\log\rho_{\epsilon}\otimes
u_{\epsilon})+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}^{t}\nabla
u_{\epsilon})=0,$

after multiplication by $\nabla\log\rho_{\epsilon}$ itself and space
integration we obtain

$\frac{d}{dt}\int_{\mathbb{T}^{3}}2|\nabla\sqrt{\rho}_{\epsilon}|^{2}\;dx-\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon}:\nabla
u_{\epsilon}\;dx=0.$ (2.19)

Hence summing $\eqref{es},\eqref{press}$ and $\eqref{effqns}$ and integrating
time we end up to (2.14). For the BD Entropy relation, we take once again
advantage of the effective velocity, and in particular introducing the
quantity $w_{\epsilon}=u_{\epsilon}+\nabla\log\rho_{\epsilon}$. Then, using
the relations

$\displaystyle\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}\nabla\log\rho_{\epsilon})$
$\displaystyle=\frac{1}{\epsilon}\Delta\rho_{\epsilon},$
$\displaystyle(\rho_{\epsilon}\nabla\log\rho_{\epsilon})_{t}$
$\displaystyle=-\frac{1}{\epsilon}\nabla\operatorname{div}(\rho_{\epsilon}u_{\epsilon}),$
$\displaystyle\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}u_{\epsilon}\otimes\nabla\log\rho_{\epsilon}+\rho_{\epsilon}\nabla\log\rho_{\epsilon}\otimes
u_{\epsilon})$
$\displaystyle=\frac{1}{\epsilon}\Delta(\rho_{\epsilon}u_{\epsilon})-\frac{2}{\epsilon}\operatorname{div}(\rho_{\epsilon}Du_{\epsilon})+\frac{1}{\epsilon}\nabla\operatorname{div}(\rho_{\epsilon}u_{\epsilon}),$
$\displaystyle\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}\nabla\log\rho_{\epsilon}\otimes\nabla\log\rho_{\epsilon})$
$\displaystyle=\frac{1}{\epsilon}\Delta(\rho_{\epsilon}\nabla\log\rho_{\epsilon})-\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon}).$

we obtain the following alternative version of $\eqref{qns}$:

$\displaystyle\partial_{t}\rho_{\epsilon}+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}w_{\epsilon})=\frac{1}{\epsilon}\Delta\rho_{\epsilon}$
(2.20)
$\displaystyle\partial_{t}(\rho_{\epsilon}w_{\epsilon})+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}w_{\epsilon}\otimes
w_{\epsilon})+\frac{1}{\epsilon}\nabla\rho_{\epsilon}^{\gamma}+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}Dw_{\epsilon})-\frac{1}{\epsilon}\Delta(\rho_{\epsilon}w_{\epsilon})$
$\displaystyle\
-\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon})+\frac{1}{\epsilon}\rho_{\epsilon}\nabla
V_{\epsilon}+\frac{1}{\epsilon^{2}}\rho_{\epsilon}u_{\epsilon}=0.$

Now we pass to compute the energy associated the the system $\eqref{newqns}$.
To this end, as already done for $\eqref{qns}$, we use the continuity equation
$\eqref{newqns}_{1}$ and multiply the equation $\eqref{newqns}_{2}$ by
$w_{\epsilon}$ to conclude

$\begin{split}&\partial_{t}\left(\frac{1}{2}\rho|w_{\epsilon}|^{2}\right)+\frac{1}{\epsilon}\operatorname{div}\left(\rho_{\epsilon}w_{\epsilon}\frac{1}{2}|w_{\epsilon}|^{2}\right)+\frac{1}{\epsilon}\left(\frac{1}{2}\Delta\rho_{\epsilon}|w_{\epsilon}|^{2}-\Delta(\rho_{\epsilon}w_{\epsilon})\cdot
w_{\epsilon}\right)+\\\ &\frac{1}{\epsilon}\nabla\rho_{\epsilon}^{\gamma}\cdot
w_{\epsilon}+\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}Dw_{\epsilon})\cdot
w_{\epsilon}-\frac{1}{\epsilon}\operatorname{div}(\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon})\cdot
w_{\epsilon}+\frac{1}{\epsilon^{2}}\rho_{\epsilon}u_{\epsilon}w_{\epsilon}+\frac{1}{\epsilon}\rho_{\epsilon}\nabla
V_{\epsilon}w_{\epsilon}=0.\end{split}$

Since

$\frac{1}{\epsilon}\left(\frac{1}{2}\Delta\rho_{\epsilon}|w_{\epsilon}|^{2}-\Delta(\rho_{\epsilon}w_{\epsilon})\cdot
w_{\epsilon}\right)=\frac{1}{\epsilon}\left(\operatorname{div}\left(\nabla\rho_{\epsilon}\frac{1}{2}|w_{\epsilon}|^{2}\right)-\operatorname{div}(\rho_{\epsilon}\cdot\nabla
w_{\epsilon})\cdot w_{\epsilon}\right),$

when we integrate the equality above over $\mathbb{T}^{3}$ we get:

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{1}{2}\rho_{\epsilon}|w_{\epsilon}|^{2}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|\nabla
w_{\epsilon}|^{2}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\nabla\rho_{\epsilon}^{\gamma}\cdot
w_{\epsilon}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla
V_{\epsilon}w_{\epsilon}\;dx$
$\displaystyle-\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}|D(w_{\epsilon})|^{2}dx+\frac{1}{\epsilon^{2}}\int_{\mathbb{T}^{3}}\rho_{\epsilon}u_{\epsilon}w_{\epsilon}dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon}:\nabla
w_{\epsilon}\;dx=0.$ (2.21)

We observe that $|\nabla
w_{\epsilon}|^{2}-|D(w_{\epsilon})|^{2}=|A(w_{\epsilon})|^{2}=|A(u_{\epsilon})|^{2}$,
and, from the definition of $w_{\epsilon}$, we infer

$\displaystyle\nabla\rho_{\epsilon}^{\gamma}\cdot w_{\epsilon}$
$\displaystyle=\nabla\rho_{\epsilon}^{\gamma}\cdot
u_{\epsilon}+\frac{4}{\gamma}|\nabla\rho_{\epsilon}^{\gamma/2}|^{2};$
$\displaystyle\rho_{\epsilon}u_{\epsilon}w_{\epsilon}$
$\displaystyle=\rho_{\epsilon}u_{\epsilon}^{2}+\rho_{\epsilon}u_{\epsilon}\nabla\log\rho_{\epsilon};$
$\displaystyle\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon}:\nabla
w_{\epsilon}$
$\displaystyle=\rho_{\epsilon}\nabla^{2}\log\rho_{\epsilon}:\nabla
u_{\epsilon}+\rho_{\epsilon}|\nabla^{2}\log\rho_{\epsilon}|^{2}$
$\displaystyle\rho_{\epsilon}\nabla V_{\epsilon}w_{\epsilon}$
$\displaystyle=\rho_{\epsilon}\nabla
V_{\epsilon}u_{\epsilon}+\rho_{\epsilon}\nabla
V_{\epsilon}\nabla\log\rho_{\epsilon}$

Moreover, we recall (2.18), (2.19) and (2.17) to compute:

$\frac{d}{dt}\int_{\mathbb{T}^{3}}\big{(}\rho_{\epsilon}(\log\rho_{\epsilon}-1)+1\big{)}dx=\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}u_{\epsilon}\nabla\log\rho_{\epsilon}\;dx,$
$\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla
V_{\epsilon}w_{\epsilon}\;dx=\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{1}{2}|\nabla
V_{\epsilon}|^{2}\;dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla
V_{\epsilon}\nabla\log\rho_{\epsilon}\;dx,$

the last term can be rewrite:

$\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\nabla
V_{\epsilon}\nabla\log\rho_{\epsilon}\;dx=\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\nabla
V_{\epsilon}\nabla\rho_{\epsilon}\;dx=-\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\Delta
V_{\epsilon}\rho_{\epsilon}\;dx=\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}(\rho_{\epsilon}-g_{\epsilon})\;dx$

Therefore we multiply $\eqref{bd2}$ by $\epsilon$ and, using the previous
identities, we obtain the final estimate

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}\left[\epsilon\left(\frac{1}{2}\rho|w_{\epsilon}|^{2}+\frac{\rho_{\epsilon}^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho_{\epsilon}}|^{2}+\frac{1}{2}|\nabla
V_{\epsilon}|^{2}\right)+\rho_{\epsilon}(\log\rho_{\epsilon}-1)+1\right]dx$
$\displaystyle+\int_{\mathbb{T}^{3}}\rho|A(u_{\epsilon})|^{2}\;dx+\frac{4}{\gamma}\int_{\mathbb{T}^{3}}|\nabla\rho_{\epsilon}^{\frac{\gamma}{2}}|^{2}\;dx+\int_{\mathbb{T}^{3}}\rho_{\epsilon}|\nabla^{2}\log\rho_{\epsilon}|^{2}\;dx+\frac{1}{\epsilon}\int_{\mathbb{T}^{3}}\rho_{\epsilon}u_{\epsilon}^{2}\;dx+\int_{\mathbb{T}^{3}}\rho_{\epsilon}(\rho_{\epsilon}-g_{\epsilon})\;dx=0$

which gives $\eqref{newbdestimate}$ upon time integration. ∎

###  2.2. Weak solution of the Quantum-Drift-Diffusion equation

Next, we consider the Quantum Drift–Diffusion–Poisson equation in
$(0,T)\times\mathbb{T}^{3}$ with $g:\mathbb{T}^{3}\rightarrow\mathbb{R}$

$\displaystyle{\partial}_{t}\rho+\operatorname{div}\left(2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\nabla\rho^{\gamma}-\rho\nabla
V\right)=0$ (2.22) $\displaystyle-\Delta V=\rho-g,$

with initial datum

$\rho|_{t=0}=\rho^{0}\geq 0,$ (2.23)

and $V$ such that

$\fint_{\mathbb{T}^{3}}Vdx=0.$ (2.24)

As before, to solve (2.22)2, (2.24) we need the following assumption for $g$:
defining $M:=\fint_{\mathbb{T}^{3}}\rho^{0}$, we assume that
$\fint_{\mathbb{T}^{3}}g=M$.

The following definition specifies the framework of weak solutions to (2.22)
we shall obtain at the limit.

######  Definition 2.8.

Given $g\in L^{2}(\mathbb{T}^{3})$, we say that $(\rho,V)$ with $\rho\geq 0$
is a finite energy weak solution of (2.22) if

  * (1)

Integrability condition:

$\displaystyle\sqrt{\rho}\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{3}))\cap
L^{2}(0,T;H^{2}(\mathbb{T}^{3}));\quad$ $\displaystyle\rho\in
L^{\infty}(0,T;L^{\gamma}(\mathbb{T}^{3}))$ $\displaystyle V\in
C((0,T);H^{2}(\mathbb{T}^{3}))$

  * (2)

Continuity equation:

$\int_{\mathbb{T}^{3}}\rho^{0}\phi(0)\;dx+\iint\big{(}\rho\phi_{t}-\rho^{\gamma}\operatorname{div}\nabla\phi-2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}:\nabla^{2}\phi+2\sqrt{\rho}\nabla^{2}\sqrt{\rho}:\nabla^{2}\phi+\rho\nabla
V\cdot\phi\big{)}\;dsdx=0,$

for any $\phi\in C_{c}^{\infty}([0,T);C^{\infty}(\mathbb{T}^{3}))$,

  * (3)

Poisson equation:
For a.e. $(t,x)\in(0,T)\times\mathbb{T}^{3}$ it holds that

$-\Delta V=\rho-g$ (2.25)

  * (4)

Entropy inequalities:
there exist $\Lambda,\,\mathcal{S}\in L^{2}((0,T)\times\mathbb{T}^{3})$ such
that

$\displaystyle\sqrt{\rho}\Lambda=\operatorname{div}(2\sqrt{\rho}\nabla^{2}\sqrt{\rho}-2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}-\rho^{\gamma}\mathbb{I})-\rho\nabla
V\textrm{ in }\mathcal{D}^{\prime}((0,T)\times\mathbb{T}^{3})$ (2.26)
$\displaystyle\sqrt{\rho}\mathcal{S}=2\sqrt{\rho}\nabla^{2}\sqrt{\rho}-2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}\,\mbox{
a.e. in }(0,T)\times\mathbb{T}^{3}$ (2.27)

and constant $C>0$ such that for a.e. $t\in(0,T)$

$\int_{\mathbb{T}^{3}}\left(|\nabla\sqrt{\rho}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}+\frac{1}{2}|\nabla
V|^{2}\right)(t)\;dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\Lambda|^{2}\;dsdx\leq
C,$ (2.28)
$\displaystyle\int_{\mathbb{T}^{3}}(\rho(\log\rho-1)+1)(t)\;dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\mathcal{S}|^{2}\;dsdx+\frac{4}{\gamma}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\nabla\rho^{\frac{\gamma}{2}}|^{2}\;dsdx+\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho(\rho-g)\;dsdx$
(2.29)
$\displaystyle\leq\int_{\mathbb{T}^{3}}(\rho^{0}(\log\rho^{0}-1)+1)\;dx.$

In addition, $(\rho,V)$ is called _energy dissipating weak solution_ if in
particular it holds that for a.e. $t\in(0,T)$:

$\displaystyle\int_{\mathbb{T}^{3}}\left(|\nabla\sqrt{\rho}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}+\frac{1}{2}|\nabla
V|^{2}\right)(t)\,dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\Lambda|^{2}\,dsdx$
$\displaystyle\leq\int_{\mathbb{T}^{3}}\,|\nabla\sqrt{\rho^{0}}|^{2}\,dx+\int_{\mathbb{T}^{3}}\frac{{\rho}^{0\gamma}}{\gamma-1}\,dx$
(2.30) $\displaystyle+\int_{\mathbb{T}^{3}}\frac{1}{2}|\nabla V(0,x)|^{2}.$

######  Remark 2.9.

The terminology finite energy and energy dissipating weak solutions is
motivated by the fact that both (2.28) and (2.30) arises as a weak limit of
the energy inequality (2.8). As in Proposition 2.7 we derive the formal
estimates in the framework of smooth solutions.

######  Proposition 2.10.

Let $(\rho,V)$ be a smooth solution of (2.22) with data satisfying
(2.23),(2.24). Then, for any $t\in[0,T)$ the pair $(\rho,V)$ satisfies the
following estimates:

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}\left(2|\nabla\sqrt{\rho}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}+\frac{1}{2}|\nabla
V|^{2}\right)dx+\int_{\mathbb{T}^{3}}\rho\left|2\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\gamma\rho^{\gamma-2}\nabla\rho-\nabla
V\right|^{2}\;dx=0,$ (2.31)
$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}\rho(\log\rho-1)+1+\;dx\int_{\mathbb{T}^{3}}\rho|\nabla^{2}\log\rho|^{2}\;dx+\frac{4}{\gamma}\int_{\mathbb{T}^{3}}|\nabla\rho^{\frac{\gamma}{2}}|^{2}\;dx+\int_{\mathbb{T}^{3}}\rho(\rho-g)\;dx=0.$
(2.32)

###### Proof.

The energy (2.32) is achieved by multiplying (2.22) by $\log\rho$ and
integrating by parts:

$\displaystyle\int_{\mathbb{T}^{3}}\rho_{t}\log\rho\;dx=\frac{d}{dt}\int_{\mathbb{T}^{3}}\big{(}(\rho(\log\rho-1))+1\big{)}\;dx,$
$\displaystyle\int_{\mathbb{T}^{3}}\operatorname{div}\left(2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)-\nabla\rho^{\gamma}\right)\log\rho\;dx=\int_{\mathbb{T}^{3}}\operatorname{div}\left(\operatorname{div}\left(\rho\nabla^{2}\log\rho\right)-\nabla\rho^{\gamma}\right)\log\rho\;dx$
$\displaystyle=\int_{\mathbb{T}^{3}}\rho|\nabla^{2}\log\rho|^{2}\;dx+\frac{4}{\gamma}\int_{\mathbb{T}^{3}}|\nabla\rho^{\frac{\gamma}{2}}|^{2}\;dx,$
$\displaystyle-\int_{\mathbb{T}^{3}}\operatorname{div}(\rho\nabla
V)\log\rho\;dx=\int_{\mathbb{T}^{3}}\rho\nabla
V\nabla\log\rho\;dx=-\int_{\mathbb{T}^{3}}\Delta
V\rho\;dx=\int_{\mathbb{T}^{3}}\rho(\rho-g)\;dx.$

Moreover, if we multiply (2.22) by
$-2\Delta\sqrt{\rho}/\sqrt{\rho}+\gamma\rho^{\gamma-1}/(\gamma-1)$, after
integrating by parts we get

$\displaystyle\int_{\mathbb{T}^{3}}\rho_{t}\left(\frac{-2\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\;dx=-\int_{\mathbb{T}^{3}}\operatorname{div}\left(2\frac{\nabla\sqrt{\rho}}{\sqrt{\rho}}\rho_{t}\right)\;dx+\int_{\mathbb{T}^{3}}\nabla\sqrt{\rho}\nabla\left(2\frac{\rho_{t}}{\sqrt{\rho}}\right)\;dx=\frac{d}{dt}\int_{\mathbb{T}^{3}}2|\nabla\sqrt{\rho}|^{2}\;dx,$
$\displaystyle\int_{\mathbb{T}^{3}}\rho_{t}\left(\frac{\gamma}{\gamma-1}\rho^{\gamma-1}\right)\;dx=\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{\rho^{\gamma}}{\gamma-1}\;dx,$
$\displaystyle\int_{\mathbb{T}^{3}}\operatorname{div}\left(2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\right)\left(-2\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}+\frac{\gamma}{\gamma-1}\rho^{\gamma-1}\right)\;dx=$
$\displaystyle=\int_{\mathbb{T}^{3}}\left(4\rho\left|\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\right|^{2}-2\gamma\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\rho^{\gamma-2}\nabla\rho\right)\;dx,$
$\displaystyle\int_{\mathbb{T}^{3}}\operatorname{div}(-\nabla\rho^{\gamma})\left(-2\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}+\frac{\gamma}{\gamma-1}\rho^{\gamma-1}\right)\;dx=\int_{\mathbb{T}^{3}}-\nabla\rho^{\gamma}\nabla\left(2\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)+\rho|\gamma\rho^{\gamma-2}\nabla\rho|^{2}\;dx$
$\displaystyle=\int_{\mathbb{T}^{3}}-2\gamma\rho\rho^{\gamma-2}\nabla\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)+\rho|\gamma\rho^{\gamma-2}\nabla\rho|^{2}\;dx,$
$\displaystyle\int_{\mathbb{T}^{3}}-\operatorname{div}(\rho\nabla
V)\frac{\gamma}{\gamma-1}\rho^{\gamma-1}\;dx=\int_{\mathbb{T}^{3}}\rho\nabla
V\gamma\rho^{\gamma-2}\nabla\rho\;dx=\int_{\mathbb{T}^{3}}\nabla
V\nabla\rho^{\gamma}\;dx=-\int_{\mathbb{T}^{3}}\Delta V\rho^{\gamma}\;dx$
$\displaystyle\int_{\mathbb{T}^{3}}\operatorname{div}(\rho\nabla
V)2\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\;dx=-\int_{\mathbb{T}^{3}}\rho\nabla
V2\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\;dx=-\int_{\mathbb{T}^{3}}\nabla
V\operatorname{div}(\rho\nabla^{2}\log\rho)\;dx$

Finally we multiply (2.22) by $V$, integrating by parts we get:

$\int_{\mathbb{T}^{3}}\rho_{t}V\;dx=-\int_{\mathbb{T}^{3}}\Delta
V_{t}\;V\;dx=\int_{\mathbb{T}^{3}}\nabla V_{t}\nabla
V\;dx=\frac{1}{2}\frac{d}{dt}\int_{\mathbb{T}^{3}}|\nabla V|^{2}\;dx,$
$\int_{\mathbb{T}^{3}}\operatorname{div}\left(2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\right)V\;dx=-\int_{\mathbb{T}^{3}}2\rho\nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right)\nabla
V\;dx=-\int_{\mathbb{T}^{3}}\nabla
V\operatorname{div}(\rho\nabla^{2}\log\rho)\;dx,$
$\int_{\mathbb{T}^{3}}-\operatorname{div}(\nabla\rho^{\gamma})V=\int_{\mathbb{T}^{3}}\nabla\rho^{\gamma}\nabla
V=-\int_{\mathbb{T}^{3}}\rho^{\gamma}\Delta V$

and (2.31) follows by summing up all terms. ∎

######  Remark 2.11.

It is worth to observe that if we perform the Hilbert expansion of
$\eqref{qns}$ the limit solution $(\bar{\rho},\bar{u})$ formally satisfies at
the first non trivial order $O(1/\epsilon)$ (the order $O(1/\epsilon^{2})$
tells us the momentum expansion starts form the power one in $\epsilon$) the
following identities:

$\displaystyle\bar{\rho}\bar{u}=\epsilon\bar{\rho}\left(2\nabla\left(\frac{\Delta\sqrt{\bar{\rho}}}{\sqrt{\bar{\rho}}}\right)-\gamma\bar{\rho}^{\gamma-2}\nabla\bar{\rho}-\nabla\bar{V}\right),$
$\displaystyle\partial_{t}\bar{\rho}+\operatorname{div}\left(2\bar{\rho}\nabla\left(\frac{\Delta\sqrt{\bar{\rho}}}{\sqrt{\bar{\rho}}}\right)-\nabla\bar{\rho}^{\gamma}-\bar{\rho}\nabla\bar{V}\right)=0,$

which is exactly the Quantum Drift–Diffusion equation $\eqref{eq:qdd}$. We
underline also that, with the above definition for $\bar{u}$, namely

$(\bar{\rho},\bar{u})=\left(\bar{\rho},\epsilon\left(2\nabla\left(\frac{\Delta\sqrt{\bar{\rho}}}{\sqrt{\bar{\rho}}}\right)-\gamma\bar{\rho}^{\gamma-2}\nabla\bar{\rho}-\nabla\bar{V}\right)\right),$

in the formulation for the mechanical energy $\eqref{fes}$ for $\eqref{qns}$,
at the limit the latter reduces to $\eqref{eq:form1}$ after a time
integration, that is the corresponding conservation of the mechanical energy
for the Quantum Drift–Diffusion equation. The same happens for the BD entropy,
namely $\eqref{newbdestimate1}$ for $\epsilon\to 0$ reduces to (the time
integrated version of) (2.32). This fact is coherent with our analysis and it
will be validated by Theorem 3.1 below.

######  Remark 2.12.

We remark that if $\rho$ is a weak solution in the sense of the previous
definition the initial datum $\rho^{0}$ is attained for example in the strong
topology of $L^{2}(\mathbb{T}^{3})$.

##  3\. Main result

In this section we prove Theorem 1.1, which we rewrite for reader’s
convenience.

######  Theorem 3.1.

Let $(\rho_{\epsilon},u_{\epsilon},V_{\epsilon})$ be a weak solution of
$\eqref{qns}$ in the sense Definition 2.1 with data (2.2) satisfying

$\displaystyle\\{\rho_{\epsilon}^{0}\\}_{\epsilon}\mbox{ is bounded in
}L^{1}\cap L^{\gamma}(\mathbb{T}^{3})\ \hbox{such that}\
\rho_{\epsilon}^{0}\to\rho^{0}\ \hbox{in}\ L^{q}(\mathbb{T}^{3}),\,q<3$ (3.1)
$\displaystyle\\{\nabla\sqrt{\rho_{\epsilon}^{0}}\\}_{\epsilon}\mbox{ is
bounded in }L^{2}(\mathbb{T}^{3}),$
$\displaystyle\\{\sqrt{\rho_{\epsilon}^{0}}u_{\epsilon}^{0}\\}_{\epsilon}\mbox{
is bounded in }L^{2}(\mathbb{T}^{3}),$
$\displaystyle\\{g_{\epsilon}\\}_{\epsilon}\in L^{2}(\mathbb{T}^{3})\mbox{ and
}g_{\epsilon}\rightharpoonup g\mbox{ weakly in }L^{2}(\mathbb{T}^{3}).$

Then, up to subsequences, there exist $\rho\geq 0$ and $V$ such that

$\displaystyle\sqrt{\rho_{\epsilon}}\rightarrow\sqrt{\rho}\textrm{ strongly in
}L^{2}((0,T);H^{1}(\mathbb{T}^{3}))$ $\displaystyle\nabla
V_{\epsilon}\rightarrow\nabla V\text{ strongly in
}C([0,T);L^{2}(\mathbb{T}^{3}))$

and $(\rho,V)$ is a finite energy weak solution of
$\eqref{eq:qdd}-\eqref{eq:qddid}$ in the sense of Definition 2.8. If in
addition to (3.1), it also hold that

$\displaystyle\sqrt{\rho_{\epsilon}^{0}}u_{\epsilon}^{0}\to 0\mbox{ strongly
in }L^{2}(\mathbb{T}^{3})$ (3.2)
$\displaystyle\nabla\sqrt{\rho_{\epsilon}^{0}}\to\nabla\sqrt{\rho^{0}}\mbox{
strongly in }L^{2}(\mathbb{T}^{3})$
$\displaystyle\rho_{\epsilon}^{0}\to\rho^{0}\mbox{ strongly in
}L^{\gamma}(\mathbb{T}^{3})$

then $(\rho,V)$ is an energy dissipating weak solution.

The proof of Theorem 3.1 requires several preliminaries, collected in the
following section.

###  3.1. Preliminary results

We start by proving the uniform bounds obtained by the Energy estimate and BD
Entropy.

######  Lemma 3.2.

There exists a constant $C>0$ such that the following uniform bounds on the
data

$\displaystyle\sup_{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}\;dx\leq
C;\quad\sup_{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}u_{\epsilon}^{2}\;dx\leq
C;\quad\int_{0}^{T}\int_{\mathbb{T}^{3}}|\nabla\rho_{\epsilon}^{\frac{\gamma}{2}}|^{2}\;dsdx\leq
C;\quad\sup_{t}\int|\nabla\sqrt{\rho_{\epsilon}}|^{2}\;dsdx\leq C;$
$\displaystyle\frac{1}{\epsilon}\int_{0}^{T}\int_{\mathbb{T}^{3}}|\mathcal{T}_{\epsilon}^{s}|^{2}\;dsdx\leq
C;\quad\sup_{t}\int\rho_{\epsilon}^{\gamma}\;dx\leq
C;\quad\frac{1}{\epsilon^{2}}\int_{0}^{T}\int\rho_{\epsilon}u_{\epsilon}^{2}\;dsdx\leq
C;$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{3}}|\nabla^{2}\sqrt{\rho_{\epsilon}}|^{2}\;dsdx\leq
C;\quad\int_{0}^{T}\int_{\mathbb{T}^{3}}|\mathcal{S}_{\epsilon}|^{2}\;dsdx\leq
C;\quad\sup_{t}\int_{\mathbb{T}^{3}}|\nabla^{2}V_{\epsilon}|^{2}\;dx\leq C$
(3.3)

###### Proof.

We first notice that under the hypothesis (3.1) we have that

$\\{\rho_{\epsilon}^{0}\log\rho_{\epsilon}^{0}\\}_{\epsilon}\text{ is bounded
in }L^{1}(\mathbb{T}^{3}).$ (3.4)

This is obtained very easily under the hypothesis (3.1). Indeed, one can
easily show that for $q>p\geq 1$ there exists $C>0$ such that for $s\geq 0$:

$|s\log s|^{q}\leq C(1+s^{p})$

and therefore, by taking $q=\gamma$ and $1\leq p<\gamma$ we have that

$\\{\rho_{\epsilon}^{0}\log\rho_{\epsilon}^{0}\\}_{\epsilon}\text{ is bounded
in }L^{q}(\mathbb{T}^{3}).$ (3.5)

which implies (3.4).

Next, as shown in Remark 2.6, $\\{\nabla V_{\epsilon}|_{t=0}\\}_{\epsilon}$ is
bounded in $L^{2}(\mathbb{T}^{3})$ and we have proved that the right-hand
sides of (2.8) and (2.10) are bounded uniformly in $\epsilon$. Then, to obtain
(3.2) is enough to use the bound

$\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}(\rho_{\epsilon}-g_{\epsilon})\;dsdx\geq\frac{1}{2}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}|^{2}\;dsdx-\frac{1}{2}\int_{0}^{t}\int_{\mathbb{T}^{3}}|g_{\epsilon}|^{2}\;dsdx$

in the energy inequalities (2.8) and (2.10). Finally, the bound on the
potential follows from the Poisson equation. ∎

The following lemma we prove the convergence needed to pass to the limit as
$\epsilon\to 0$.

######  Lemma 3.3.

There exist $\rho\geq 0$ such that $\sqrt{\rho}\in
L^{\infty}(0,T);H^{1}(\mathbb{T}^{3})\cap L^{2}(0,T;H^{2}(\mathbb{T}^{3}))$
and the following hold:

$\displaystyle\rho_{\epsilon}\rightarrow\rho\text{ in
$C([0,T];L^{q}(\mathbb{T}^{3})$ },\,q<3,$ (3.6)
$\displaystyle\sqrt{\rho}_{\epsilon}\rightharpoonup\sqrt{\rho}\text{ weakly in
$L^{2}(0,T;H^{2}(\mathbb{T}^{3}))$},$ (3.7)
$\displaystyle\sqrt{\rho_{\epsilon}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\sqrt{\rho}\text{
weakly* in $L^{\infty}(0,T;H^{1}(\mathbb{T}^{3}))$},$ (3.8)
$\displaystyle\sqrt{\rho_{\epsilon}}\rightarrow\sqrt{\rho}\text{ strongly in
}L^{2}(0,T;H^{1}(\mathbb{T}^{n})).$ (3.9)
$\displaystyle\rho_{\epsilon}(\log\rho_{\epsilon}+1)+1\to\rho(\log\rho+1)+1\text{
strongly in }L^{1}(0,T;L^{1}(\mathbb{T}^{3})),$ (3.10)
$\displaystyle\rho_{\epsilon}^{\gamma}\rightarrow\rho^{\gamma}\text{ in
$L^{1}(0,T;L^{1}(\mathbb{T}^{n}))$}$ (3.11)

###### Proof.

From $\eqref{uniformbounds}$ and Sobolev embedding have that:

$\\{\sqrt{\rho_{\epsilon}}\\}_{\epsilon}\text{ is bounded in
}L^{2}(0,T;L^{q}(\mathbb{T}^{3}))\quad q\in[1,6]$ (3.13)

Therefore since
$\nabla\rho_{\epsilon}=2\nabla\sqrt{\rho_{\epsilon}}\sqrt{\rho_{\epsilon}}$ by
using (3.13) and (3.2) we can infer that

$\\{\rho_{\epsilon}\\}_{\epsilon}\mbox{ is bounded in
}L^{\infty}(0,T;W^{1,\frac{3}{2}}(\mathbb{T}^{3})).$

Next, by using the weak formulation of the continuity equation and the bounds
(3.2) we get that

$\\{\partial_{t}\rho_{\epsilon}\\}_{\epsilon}\mbox{ is bounded in
}L^{2}(0,T;W^{-1,\frac{3}{2}}(\mathbb{T}^{3})).$

Indeed, it is enough to note that we have

$\int_{0}^{T}\|\partial_{t}\rho_{\epsilon}\|^{2}_{W^{-1,\frac{3}{2}}}\;ds\leq\frac{1}{\epsilon^{2}}\int_{0}^{T}\|\rho_{\epsilon}u_{\epsilon}\|^{2}_{L^{\frac{3}{2}}_{x}}\;ds\leq\int_{0}^{T}\|\sqrt{\rho_{\epsilon}}\|_{L_{x}^{6}}^{2}\left\|\sqrt{\rho_{\epsilon}}\frac{u_{\epsilon}}{\epsilon}\right\|^{2}_{L^{2}_{x}}\;ds.$

Since for $q<3$ we have $W^{1,3/2}(\mathbb{T}^{3})\subset
L^{q}(\mathbb{T}^{3})$ with compact embedding and
$L^{q}(\mathbb{T}^{3})\subset W^{-1,3/2}(\mathbb{T}^{3})$ we can apply Aubin-
Lion lemma to deduce that there exists a subsequence not relabelled and
$\rho\geq 0$ such that (3.6) holds. Moreover, by passing to a further
subsequence if necessary, (3.6) implies

$\rho_{\epsilon}\rightarrow\rho\;\;\text{ a.e in }(0,T)\times\mathbb{T}^{3}.$
(3.14)

Then, the convergence (3.7) and (3.11) follow from (3.14), the uniform bounds
(3.2) and standard weak compactness considerations. Next, to prove (3.9) we
start by proving that

$\sqrt{\rho_{\epsilon}}\to\sqrt{\rho}\text{ strongly in
}L^{2}(0,T;L^{2}(\mathbb{T}^{3})).$ (3.15)

First, notice that (3.14) implies that

$\sqrt{\rho_{\epsilon}}\rightarrow\sqrt{\rho}\;\;\text{ a.e in (0,T)
$\times\mathbb{T}^{3}$}.$

Let $M>0$, then

$\begin{split}\int_{0}^{T}\int_{\mathbb{T}^{3}}|\sqrt{\rho_{\epsilon}}-\sqrt{\rho}|^{2}\;dsdx\leq&\int_{0}^{T}\int_{\\{\rho_{\epsilon}>M\\}}|\sqrt{\rho_{\epsilon}}|^{2}\;dsdx+\\\
&\int_{0}^{T}\int_{\mathbb{T}^{3}}|\sqrt{\rho_{\epsilon}}\chi_{\\{\rho_{\epsilon}\leq
M\\}}-\sqrt{\rho}\chi_{\\{\rho\leq M\\}}|^{2}\;dsdx\;+\\\
&\int_{0}^{T}\int_{\\{\rho>M\\}}|\sqrt{\rho}|^{2}\;dsdx.\end{split}$

Then, by using (3.2), (3.13), the fact that $\sqrt{\rho}\in
L^{\infty}(0,T;L^{6}(\mathbb{T}^{3}))$, we have that

$\begin{split}\int_{0}^{T}\int_{\mathbb{T}^{3}}|\sqrt{\rho_{\epsilon}}-\sqrt{\rho}|^{2}\;dsdx\leq&\frac{1}{M^{2}}\int_{0}^{T}\int|\sqrt{\rho_{\epsilon}}|^{6}\;dsdx\\\
&+\int_{0}^{T}\int_{\mathbb{T}^{3}}|\sqrt{\rho_{\epsilon}}\chi_{\\{\rho_{\epsilon}\leq
M\\}}-\sqrt{\rho}\chi_{\\{\rho\leq M\\}}|^{2}\;dsdx\;\\\
&+\frac{1}{M^{2}}\int_{0}^{T}\int|\sqrt{\rho}|^{6}\;dsdx\\\
&\leq\int_{0}^{T}\int_{\mathbb{T}^{3}}|\sqrt{\rho_{\epsilon}}\chi_{\\{\sqrt{\rho_{\epsilon}}\leq
M\\}}-\sqrt{\rho}\chi_{\\{\sqrt{\rho}\leq
M\\}}|^{2}\;dsdx+\frac{2C}{M^{2}}\end{split}$

Then, we conclude by first sending $\epsilon\to 0$ in the second term, where
we use Dominated Convergence Theorem, and then by choosing suitably
$M\to\infty$.

The strong convergence (3.9) of $\sqrt{\rho_{\epsilon}}$ in
$L^{2}(0,T;H^{1}(\mathbb{T}^{3}))$ is a consequence of the following simple
interpolation inequality:

$\|\sqrt{\rho_{\epsilon}}(t)-\sqrt{\rho}(t)\|_{H^{1}}\leq
C\|\sqrt{\rho_{\epsilon}}(t)-\sqrt{\rho}(t)\|_{L^{2}}^{\frac{1}{2}}\|\sqrt{\rho_{\epsilon}}(t)-\sqrt{\rho}(t)\|_{H^{2}}^{\frac{1}{2}}.$
(3.16)

Next, we prove (3.10). We first notice that by using (3.2) and by the very
same argument used to deduce (3.5), we easily have that for some $p>1$

$\displaystyle\rho\log\rho\in L^{p}((0,T)\times\mathbb{T}^{3}),$
$\displaystyle\\{\rho_{\epsilon}\log\rho_{\epsilon}\\}_{\epsilon}\text{ is
bounded in }L^{p}((0,T)\times\mathbb{T}^{3}).$

Moreover, since the function $s\to s\log\,s$ is continuous on $[0,\infty)$ we
have that

$\rho_{\epsilon}\log\rho_{\epsilon}\to\rho\log\rho\text{ a.e. in
}(0,T)\times\mathbb{T}^{3}.$

Let $M>0$, then

$\begin{split}\int_{0}^{T}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}\log\rho_{\epsilon}-\rho\log\rho|\;dsdx\leq&\int_{0}^{T}\int_{\\{\rho_{\epsilon}>M\\}}|\rho_{\epsilon}\log\rho_{\epsilon}|\;dsdx\\\
&+\int_{0}^{T}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}\log\rho_{\epsilon}\chi_{\\{\rho_{\epsilon}\leq
M\\}}-\rho\log\rho\chi_{\\{\rho\leq M\\}}|\;dsdx\;\\\
&+\int_{0}^{T}\int_{\\{\rho>M\\}}|\rho\log\rho|\;dsdx.\\\
&\leq\int_{0}^{T}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}\log\rho_{\epsilon}\chi_{\\{\rho_{\epsilon}\leq
M\\}}-\rho\log\rho\chi_{\\{\rho\leq M\\}}|\;dsdx\;\\\ &+\frac{2C}{(M\log
M)^{p-1}}\end{split}$

and we conclude as in before. Finally, we prove the convergence of the
pressure term. We first note that from (3.2) we have that

$\\{\rho_{\epsilon}^{\frac{\gamma}{2}}\\}_{\epsilon}\mbox{ is bounded in
}L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{T}^{3})).$

Then, by Sobolev embedding

$\\{\rho_{\epsilon}^{\frac{\gamma}{2}}\\}_{\epsilon}\text{ is bounded in
}L^{2}(0,T;L^{6}(\mathbb{T}^{3}))$

and therefore for a.e $t$:

$\|\rho_{\epsilon}^{\frac{\gamma}{2}}(t)\|_{\frac{10}{3}}\leq\|\rho_{\epsilon}^{\frac{\gamma}{2}}(t)\|^{\frac{2}{5}}_{2}\|\rho_{\epsilon}^{\frac{\gamma}{2}}(t)\|_{6}^{\frac{3}{5}}.$

Therefore, by integrating in time and using (3.2) we have that

$\\{\rho_{\epsilon}^{\frac{\gamma}{2}}\\}_{\epsilon}\mbox{ is bounded in
}L^{\frac{10}{3}}((0,T)\times\mathbb{T}^{3}),$

which is equivalent to say that

$\\{\rho_{\epsilon}^{\gamma}\\}_{\epsilon}\mbox{ is bounded in
}L^{\frac{5}{3}}((0,T)\times\mathbb{T}^{3}).$

Moreover, by (3.14) we have that also
$\rho_{\epsilon}^{\gamma}\rightarrow\rho^{\gamma}$ a.e. in
$(0,T)\times\mathbb{T}^{3}$ and by Fatou Lemma we have that

$\rho^{\gamma}\in L^{\frac{5}{3}}((0,T)\times\mathbb{T}^{3}).$

Then, if $M>0$ we have

$\begin{split}\int_{0}^{T}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}^{\gamma}-\rho^{\gamma}|\;dsdx\leq&\int_{0}^{T}\int_{\\{\rho_{\epsilon}>M\\}}\rho_{\epsilon}^{\gamma}\;dsdx\\\
+&\int_{0}^{T}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}^{\gamma}\chi_{\\{\rho_{\epsilon}\leq
M\\}}-\rho^{\gamma}\chi_{\\{\rho\leq M\\}}|\;dsdx\;\\\
+&\int_{0}^{T}\int_{\\{\rho>M\\}}\rho^{\gamma}\;dsdx\\\
\leq&\int_{0}^{T}\int_{\mathbb{T}^{3}}|\rho_{\epsilon}^{\gamma}\chi_{\\{\rho_{\epsilon}\leq
M\\}}-\rho^{\gamma}\chi_{\\{\rho\leq M\\}}|\;dsdx\\\
+&\frac{2C}{M^{\frac{2}{3}\gamma}}\end{split}$

and we conclude as before.

∎

###  3.2. Proof of Theorem 3.1

First, we notice that by (3.2), (3.6) and the compact embedding of
$L^{2}(\mathbb{T}^{3})$ in $H^{-1}(\mathbb{T}^{3})$ for
$\\{g_{\epsilon}\\}_{\epsilon}$, we have that there exists $V\in
C([0,T);H^{2}(\mathbb{T}^{3}))$ such that

$\nabla V_{\epsilon}\rightarrow\nabla V\mbox{ strongly in
}C([0,T);L^{2}(\mathbb{T}^{3})),$ (3.17)

and the Poisson equation is satisfied pointwise.

Regarding the momentum equation, let $\psi\in
C^{\infty}([0,T);C^{\infty}(\mathbb{T}^{3}))$ and consider the weak
formulation of the momentum equations in Definition 2.1, multiplied by
$\epsilon$,

$\displaystyle\epsilon\int\rho_{\epsilon}^{0}u_{\epsilon}^{0}\psi(0)\,dx+\epsilon^{2}\iint\sqrt{\rho_{\epsilon}}\sqrt{\rho_{\epsilon}}\frac{u_{\epsilon}}{\epsilon}\psi_{t}\;dsdx+\epsilon^{2}\iint\sqrt{\rho_{\epsilon}}\frac{u_{\epsilon}}{\epsilon}\otimes\sqrt{\rho_{\epsilon}}\frac{u_{\epsilon}}{\epsilon}:\nabla\psi\;dsdx$
(3.18)
$\displaystyle-\sqrt{\epsilon}\iint\sqrt{\rho_{\epsilon}}\frac{\mathcal{T}_{\epsilon}^{s}}{\sqrt{\epsilon}}:\nabla\psi\;dsdx+\iint\rho_{\epsilon}^{\gamma}\operatorname{div}\psi\;dsdx-\iint
2\sqrt{\rho_{\epsilon}}\,\nabla^{2}\sqrt{\rho_{\epsilon}}:\nabla\psi\;dsdx$
$\displaystyle-\iint\rho_{\epsilon}\nabla V_{\epsilon}\psi\;dsdx+\iint
2\nabla\sqrt{\rho_{\epsilon}}\otimes\nabla\sqrt{\rho_{\epsilon}}\,:\nabla\psi\;dsdx+\iint\rho_{\epsilon}\frac{u_{\epsilon}}{\epsilon}\psi\;dsdx=0.$

We study the convergence in the limit of $\epsilon\to 0$ of all the terms
separately. By using (3.1) and Hölder inequality we conclude

$\epsilon\left|\int\rho_{\epsilon}^{0}u_{\epsilon}^{0}\psi(0)\,dx\right|\leq\epsilon\,\|\psi\|_{L^{\infty}_{t,x}}\|\sqrt{\rho_{\epsilon}^{0}}\|_{L^{2}}\|\sqrt{\rho^{0}_{\epsilon}}u^{0}_{\epsilon}\|_{L^{2}}\leq\epsilon\,C\rightarrow
0\text{ as $\epsilon\rightarrow 0$}.$

Analogously, from (3.2) and Hölder inequality, we get for $\epsilon\rightarrow
0$:

$\displaystyle\epsilon^{2}\left|\iint\sqrt{\rho_{\epsilon}}\sqrt{\rho_{\epsilon}}\frac{u_{\epsilon}}{\epsilon}\psi_{t}\;dsdx\right|\leq\epsilon^{2}\|\psi_{t}\|_{L^{\infty}_{t,x}}\|\sqrt{\rho_{\epsilon}}\|_{L^{2}_{t,x}}\left\|\sqrt{\rho_{\epsilon}}\,\frac{u_{\epsilon}}{\epsilon}\right\|_{L^{2}_{t,x}}\leq\epsilon^{2}C\rightarrow
0,$
$\displaystyle\epsilon^{2}\left|\iint\sqrt{\rho_{\epsilon}}\,\frac{u_{\epsilon}}{\epsilon}\otimes\sqrt{\rho_{\epsilon}}\,\frac{u_{\epsilon}}{\epsilon}:\nabla\psi\;dsdx\right|\leq\epsilon^{2}\|\nabla\psi\|_{L^{\infty}_{t,x}}\left\|\sqrt{\rho_{\epsilon}}\,\frac{u_{\epsilon}}{\epsilon}\right\|^{2}_{L^{2}_{t,x}}\leq\epsilon^{2}C\rightarrow
0,$
$\displaystyle\sqrt{\epsilon}\left|\iint\sqrt{\rho_{\epsilon}}\frac{\mathcal{T}_{\epsilon}^{s}}{\sqrt{\epsilon}}:\nabla\psi\;dsdx\right|\leq\sqrt{\epsilon}\|\nabla\psi\|_{L^{\infty}_{t,x}}\|\sqrt{\rho_{\epsilon}}\|_{L^{2}_{t,x}}\left\|\frac{\mathcal{T}_{\epsilon}^{s}}{\sqrt{\epsilon}}\right\|_{L^{2}_{t,x}}\leq\sqrt{\epsilon}C\rightarrow
0.$

Next, by using (3.7) and (3.9) it follows that for $\epsilon\rightarrow 0$

$\displaystyle\iint
2\sqrt{\rho_{\epsilon}}\nabla^{2}\sqrt{\rho_{\epsilon}}:\nabla\psi\;dsdx\rightarrow\iint
2\sqrt{\rho}\nabla^{2}\sqrt{\rho}:\nabla\psi\;dsdx,$ $\displaystyle\iint
2\nabla\sqrt{\rho_{\epsilon}}\otimes\nabla\sqrt{\rho_{\epsilon}}:\nabla\psi\;dsdx\rightarrow\iint
2\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}:\nabla\psi\;dsdx.$

Moreover, the convergence of $\rho_{\epsilon}^{\gamma}$ in (3.11) implies the
continuity of the pressure term

$\iint\rho_{\epsilon}^{\gamma}\operatorname{div}\psi\;dsdx\rightarrow\iint\rho^{\gamma}\operatorname{div}\psi\;dsdx$

as $\epsilon\rightarrow 0$. Next, we consider the potential term, by using the
(3.6) and (3.17) one gets:

$\iint\rho_{\epsilon}\nabla V_{\epsilon}\psi\;dsdx\rightarrow\iint\rho\nabla
V\psi\;dsdx\text{ as }\epsilon\rightarrow 0.$

For the damping term, we first note that, that from (3.2), we can infer that
there exists $\Lambda$ such that

$\sqrt{\rho_{\epsilon}}\,\frac{u_{\epsilon}}{\epsilon}\rightharpoonup\Lambda\text{
weakly in $L^{2}((0,T)\times\mathbb{T}^{3})$},$ (3.19)

by using also (3.9) we get that

$\iint\sqrt{\rho_{\epsilon}}\sqrt{\rho_{\epsilon}}\,\frac{u_{\epsilon}}{\epsilon}\psi\;dsdx\rightarrow\iint\sqrt{\rho}\Lambda\psi\;dsdx\,\text{
as }\epsilon\rightarrow 0$

Therefore from (3.18) we conclude

$\begin{split}\iint\sqrt{\rho}\Lambda\psi\;dsdx=&-2\iint\nabla\sqrt{\rho}\otimes\sqrt{\rho}:\nabla\psi\;dsdx+2\iint\sqrt{\rho}\nabla^{2}\sqrt{\rho}:\nabla\psi\;dsdx\\\
&-\iint\rho^{\gamma}\operatorname{div}\psi\;dsdx+\iint\rho\nabla
V\psi\;dsdx,\end{split}$

that is

$\sqrt{\rho}\Lambda=\operatorname{div}(-2\nabla\sqrt{\rho}\otimes\sqrt{\rho}+2\sqrt{\rho}\nabla^{2}\sqrt{\rho}-\rho^{\gamma}\mathbb{I})+\rho\nabla
V\quad\text{in }\mathcal{D}^{\prime}((0,T)\times\mathbb{T}^{3}).$ (3.20)

Finally, for the continuity equation we similarly have for $\epsilon\to 0$:

$\int\rho_{\epsilon}^{0}\phi(0)\,dx+\iint\rho_{\epsilon}\phi_{t}+\sqrt{\rho_{\epsilon}}\sqrt{\rho_{\epsilon}}u_{\epsilon}\nabla\phi\;dsdx\to\int\rho^{0}\phi(0)\;dx+\iint\rho\phi_{t}+\sqrt{\rho}\Lambda\nabla\phi\;dsdx,$

and therefore taking into account (3.20) we get that $\rho$ satisfies

$\iint\rho\phi_{t}+\sqrt{\rho}\Lambda\nabla\phi\;dsdx=0,$ (3.21)

or any $\phi\in C^{\infty}_{c}([0,T);C^{\infty}(\mathbb{T}^{3}))$. Next, we
prove (2.27). Again, from (3.2) we have that there exists $\mathcal{S}$ such
that

$\mathcal{S}_{\epsilon}\rightharpoonup\mathcal{S}\text{ weakly in
$L^{2}((0,T)\times\mathbb{T}^{3})$},$ (3.22)

Therefore, for any $\phi\in C^{\infty}(\mathbb{T}^{3})$, we have that

$\displaystyle\iint\sqrt{\rho_{\epsilon}}\mathcal{S}_{\epsilon}\phi\;dsdx\to\iint\sqrt{\rho}\mathcal{S}\phi\;dsdx$
$\displaystyle\iint\sqrt{\rho_{\epsilon}}\nabla^{2}\sqrt{\rho_{\epsilon}}\phi-\nabla\sqrt{\rho_{\epsilon}}\otimes\nabla\sqrt{\rho_{\epsilon}}\;dsdx\to\iint\sqrt{\rho}\nabla^{2}\sqrt{\rho}\phi-\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}\;dsdx$

where we have used (3.9), (3.7) and (3.22). Finally, by using (2.9) we get
(2.27). Next, we prove the entropy inequalities. By lower semicontinuity we
have that for a.e. $t\in(0,T)$

$\displaystyle\int_{\mathbb{T}^{3}}\left(\frac{\rho(t,x)^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho}(t,x)|^{2}+\frac{1}{2}|\nabla
V(t,x)|^{2}\right)\,dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}|\Lambda(t,x)|^{2}\;dsdx$
(3.23) $\displaystyle\leq\liminf_{\epsilon\to
0}\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho_{\epsilon}^{0}|u_{\epsilon}^{0}|^{2}+\frac{({\rho_{\epsilon}^{0})}^{\gamma}}{\gamma-1}+2|\nabla\sqrt{\rho_{\epsilon}^{0}}|^{2}+\frac{1}{2}|\nabla
V_{\epsilon}^{0}|^{2}\right)\;dx,$

and then (2.28) and (2.30) follows by using (3.1) and (3.2), respectively, and
(3.17).

Finally, regarding (2.29), we recall that we only assume (3.1). We first note
that (3.1) implies that, up to a subsequence,

$\rho_{\epsilon}^{0}\to\rho^{0}\,\text{ a.e. in }(0,T)\times\mathbb{T}^{3}$

then by using (3.4) and the very same argument used in Lemma 3.3 to prove
(3.10) we get that

$\displaystyle\rho_{\epsilon}^{0}(\log\rho_{\epsilon}^{0}+1)+1\to\rho^{0}(\log\rho^{0}+1)+1\text{
strongly in }L^{1}(\mathbb{T}^{3}).$

Moreover, we have that

$\lim_{\epsilon\rightarrow
0}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}(\rho_{\epsilon}-g_{\epsilon})\;dsdx=\lim_{\epsilon\rightarrow
0}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}^{2}\;dsdx-\lim_{\epsilon\rightarrow
0}\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho_{\epsilon}g_{\epsilon}\;dsdx=\int_{0}^{t}\int_{\mathbb{T}^{3}}\rho(\rho-g)\;dsdx,$

but this follows directly from (3.6) and the weak convergence of
$g_{\epsilon}$. Therefore considering (2.10) and arguing exactly as done to
deduce (3.23) we get (2.29).

## Appendix A

For completeness, we give the proof of (2.13).

######  Lemma A.

There exists $C>0$, depending only on the dimension, such that for any
function $\rho\in H^{2}(\mathbb{T}^{3})$, with $\rho>0$ a.e. on
$(0,T)\times\mathbb{T}^{3}$

$\iint|\nabla\rho^{\frac{1}{4}}|^{4}+\iint|\nabla^{2}\sqrt{\rho}|^{2}\leq
C\iint\rho|\nabla^{2}\log\rho|^{2}.$

###### Proof.

By a density argument it is enough to prove the lemma for $\rho$ being a
smooth function strictly positive everywhere. We first notice that

$\displaystyle\iint\rho\left|\nabla\left(\frac{\nabla\sqrt{\rho}}{\sqrt{\rho}}\right)\right|^{2}=\iint\rho\left|-\frac{1}{2}\nabla\log\rho\otimes\nabla\log\rho+\frac{1}{2\rho}\nabla^{2}\rho\right|^{2}$
$\displaystyle=\frac{1}{4}\iint\rho|\nabla^{2}\log\rho|^{2}.$ (3.24)

On the other hand we also have

$\displaystyle\iint\rho\left|\nabla\left(\frac{\nabla\sqrt{\rho}}{\sqrt{\rho}}\right)\right|^{2}=\iint\frac{1}{\rho}|\nabla\sqrt{\rho}|^{4}+|\nabla^{2}\sqrt{\rho}|^{2}-2\frac{1}{\sqrt{\rho}}\nabla^{2}\sqrt{\rho}:\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}.$
(3.25)

We have:

$\displaystyle\iint\frac{1}{\sqrt{\rho}}\partial_{x_{i}}\partial_{x_{j}}\sqrt{\rho}\partial_{x_{i}}\sqrt{\rho}\partial_{x_{j}}\sqrt{\rho}=\iint\partial_{x_{i}}\left(\partial_{x_{j}}\sqrt{\rho}\frac{\partial_{x_{i}}\sqrt{\rho}}{\sqrt{\rho}}\partial_{x_{j}}\sqrt{\rho}\right)$
$\displaystyle-\iint\partial_{x_{j}}\sqrt{\rho}\partial_{x_{i}}\left(\frac{\partial_{x_{i}}\sqrt{\rho}}{\sqrt{\rho}}\right)\partial_{x_{j}}\sqrt{\rho}-\iint\frac{1}{\sqrt{\rho}}\partial_{x_{i}}\partial_{x_{j}}\sqrt{\rho}\partial_{x_{i}}\sqrt{\rho}\partial_{x_{j}}\sqrt{\rho}.$

The first term is zero and thus we get

$\displaystyle
2\iint\frac{1}{\sqrt{\rho}}\nabla^{2}\sqrt{\rho}:\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}=-\iint|\nabla\sqrt{\rho}|^{2}\operatorname{div}\left(\frac{\nabla\sqrt{\rho}}{\sqrt{\rho}}\right).$

Then, we use Young inequality

$\displaystyle
2\left|\iint\frac{1}{\sqrt{\rho}}\nabla^{2}\sqrt{\rho}:\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}\right|\leq\frac{1}{2}\iint\frac{|\nabla\sqrt{\rho}|^{4}}{\rho}+2\iint\rho\left|\operatorname{div}\left(\frac{\nabla\sqrt{\rho}}{\sqrt{\rho}}\right)\right|^{2}$
$\displaystyle\leq\frac{1}{2}\iint\frac{|\nabla\sqrt{\rho}|^{4}}{\rho}+2\iint\rho\left|\nabla\left(\frac{\nabla\sqrt{\rho}}{\sqrt{\rho}}\right)\right|^{2},$

and finally we get, using (3.24) and (3.25):

$\displaystyle\iint\frac{|\nabla\sqrt{\rho}|^{4}}{\rho}+|\nabla^{2}\sqrt{\rho}|^{2}\leq
C\iint\rho|\nabla^{2}\log\rho|^{2}$

that gives (2.13), being

$\iint|\nabla\rho^{\frac{1}{4}}|^{4}=\iint\frac{|\nabla\sqrt{\rho}|^{4}}{\rho}.$

∎

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# Elko and Mass Dimension One Fermions: Editorial

Dharam Vir Ahluwalia<EMAIL_ADDRESS>Mountain Physics Camp,
Center for the Studies of the Glass Bead Game
Bir, Himachal Pradesh, 176077 India

###### Abstract

This editorial presents the quantum field theoretic ambiance in which Elko and
mass dimension one fermions come to exist. This may serve not only to
introduce Elko, and the associated quantum field, but it may also open a door
to a new class of spin one half fermions and bosons. These new particles, as
it will be argued in this volume, are first-principle candidates for dark
matter.

††journal: European Physical Journal – Special Topics

Our reader certainly needs no reminder that astrophysical and cosmological
observations reveal that the Standard Model of high energy physics (SM-hep)
accounts for only $5\%$ of the universe. The remaining $95\%$ is dark to us,
except gravitationally. It is bifurcated into dark matter, $25\%$, and dark
energy, $70\%$. The _kinematic structure_ of the SM-hep fermions allows
introduction of certain _local gauge symmetries_. These account for the
‘luminosity’ of our universe, of stability of nuclei and atoms, and
radioactivity. All chemical and nuclear reactions arise from the fields
associated with the local gauge symmetries – with gravity being an important
source of astrophysical and cosmological structures, and primary initiator of
astrophysical nuclear reactions. The essential elements of the kinematic
structure are the spin-$1/2$ quantum fields of Dirac, and the Dirac operator
$\left(i\gamma_{\mu}\partial^{\mu}-m\mathbb{I}\right)$ – and, the definition
of spinorial dual and adjoint. In momentum space, $m^{-1}\gamma_{\mu}p^{\mu}$
serves as the parity operator. Modulo certain phases, its eigenfunctions serve
as the expansion coefficients of the quantum field of Dirac.

Taking a hint from the standard model of high energy physics (SM-hep) we
together may ask: is the kinematic structure of the dark matter based on the
quantum-field theoretic extension of Dirac’s 1928 formalism in such a manner
that it does not support the gauge symmetries of the SM-hep. Is there only one
dark-matter sector, or there are several? The central thesis of this issue of
EPJ-ST is to answer the asked question in the affirmative, and to open up a
logical possibility that there may not be one but several dark-matter sectors.
A result, on the way, would suggest how to tame zero point energies of all the
cosmic matter fields. In undertaking our task we prefer to be conservative:
whatever dark matter is it must be one representation or the other of the
Lorentz algebra, and the well known discrete symmetries. This is our working
conjecture.

The question we ask would seem to be blocked by a no-go theorem inherent in
the 1964 work of Weinberg [1, 2]. The block is evaded by the realisation that
the duals of the spinors in the $(1/2,0)\oplus(0,1/2)$ representation space
are not unique, nor are the adjoints [3, 4, 5, 6]. And, arguments akin to
those of Carl Bender further help evade the no-go theorem [7, 8] .

The story of the SM-hep matter begins with the 1928 paper of Dirac [9]. At its
core, it presents a non-trivial square root of the dispersion relation
$p_{\mu}p^{\mu}=m^{2}$. The square root of the left-hand side that Dirac
obtained was $\gamma_{\mu}p^{\mu}$. The square root of the right-hand side was
implicitly assumed to be $\pm\,m\,\mathbb{I}$, where $\mathbb{I}$ is an
identity matrix in the four dimensional $(1/2,0)\oplus(0,1/2)$ representation
space. Combined, these yield the momentum-space wave equation
$\left(\gamma_{\mu}p^{\mu}\pm
m\,\mathbb{I}\right)\psi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=0$.

A first hint that the kinematic structure suggested by Dirac may not be unique
arises from two observations: (a) casting Dirac’s 1928 work in the manner just
done, immediately tells us that non-trivial square roots of $4\times 4$
identity matrix $\mathbb{I}$ may lead to fundamentally new kinematic
structures; (b) in 2014, Sperança observed that $m^{-1}\gamma_{\mu}p^{\mu}$
serves as the parity operator, with the consequence that
$\psi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$ emerge as eigenspinors
of the parity operator [10].111The parity operator $m^{-1}\gamma_{\mu}p^{\mu}$
is derived without reference to Dirac equation, see [5, Section 5.3] and [10].
A first example of a new kinematic structure arising from observation (a) has
just been published [11]. Observation (b) led to the theoretical discovery of
Elko – eigenspinors of the charge conjugation operator – and mass dimension
one fermions.

To clear a path for our reader, we note that there are two extremes in the
manner in which Dirac’s quantum field is introduced to the un-initiated. In
some of these works ‘derivation’ of Dirac equation contains a serious mistake
[12, page 167] and [13, page 44, in the first edition], and the pairing
between the the annihilation operators and the antiparticle spinors is
incompatible with that required by parity covariance, rotational symmetry, and
cluster decomposition principle. The other extreme is Weinberg’s monograph on
the subject. It is flawless but it requires a serious investment of time.

In the notation of [5], the mistake in the derivation of Dirac equation lies
in the fact that for the particle spinors one indeed has
$\phi_{R}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}=\mbox{\boldmath$\displaystyle\boldsymbol{0}$},\pm\sigma)=+\phi_{L}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}=\mbox{\boldmath$\displaystyle\boldsymbol{0}$},\pm\sigma)$
but for the antiparticle spinors one must have
$\phi_{R}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}=\mbox{\boldmath$\displaystyle\boldsymbol{0}$},\pm\sigma)=-\phi_{L}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}=\mbox{\boldmath$\displaystyle\boldsymbol{0}$},\pm\sigma)$
[14, 15]. The correct pairing, upto a common normalisation factor, of spinors
and phases for the particle and antiparticle spinors must be as follows [2,
page 224]:

$\displaystyle
u(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},+1/2)=+\,\mbox{\boldmath$\displaystyle\boldsymbol{\kappa}$}\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
0\\\ 1\\\ 0\end{array}\right),\quad\mbox{paired
with}~{}a(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},+1/2)$ (5)
$\displaystyle
u(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},-1/2)=+\,\mbox{\boldmath$\displaystyle\boldsymbol{\kappa}$}\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
1\\\ 0\\\ 1\end{array}\right),\quad\mbox{paired
with}~{}a(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},-1/2)$ (10)

and

$\displaystyle
v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},+1/2)=+\,\mbox{\boldmath$\displaystyle\boldsymbol{\kappa}$}\frac{1}{\sqrt{2}}\left(\begin{array}[]{l}0\\\
1\\\ 0\\\ -1\end{array}\right),\quad\mbox{paired
with}~{}b^{\dagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},+1/2)$ (15)
$\displaystyle
v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},-1/2)=-\,\mbox{\boldmath$\displaystyle\boldsymbol{\kappa}$}\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
0\\\ -1\\\ 0\end{array}\right),\quad\mbox{paired
with}~{}b^{\dagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},-1/2)$ (20)

What goes wrong is that in (15) and (20),
$v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},+1/2)$ and
$v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},-1/2)$ are interchanged (see,
for example [16, page 113] and a fair fraction of textbooks on quantum field
theory); without altering the right-hand sides.222I am grateful to Gerald
Folland for a private communication, and his permission to point out the
error. In addition, both the multiplicative phase factors that appear before
$\displaystyle\boldsymbol{\kappa}$ are made $+1$. The boost operator
$\displaystyle\boldsymbol{\kappa}$ in the above equations is

$\mbox{\boldmath$\displaystyle\boldsymbol{\kappa}$}=\sqrt{\frac{E+m}{2m}}\left[\begin{array}[]{cc}\mathbb{I}+\frac{\boldsymbol{\sigma}\cdot\mathbf{p}}{E+m}&\mbox{\boldmath$\displaystyle\boldsymbol{0}$}\\\
\mbox{\boldmath$\displaystyle\boldsymbol{0}$}&\mathbb{I}-\frac{\boldsymbol{\sigma}\cdot\mathbf{p}}{E+m}\end{array}\right]$
(21)

Thus the particle and antiparticle spinors satisfy Dirac equation. Each of
them has a freedom of a multiplicative phase factor. These phase factors can
vary from one spinor to another but when we use these spinors as expansion
coefficients of a quantum field, these become relative – with additional
constraint entering on Majorana-isation of the field. The cluster
decomposition principle, Lorentz and Parity covariance fix these phases to be
as above. These observations, in part, led to the removal [17, 5] of the non-
locality present in the early works on mass dimension one fermions (with Elko
as expansion coefficients) [18, 3].

This discussion sets the ambiance for the kinematic structure of known matter
fields – that is, the field necessary for all the SM-hep matter fields. We
assume that dark-matter sector(s), like its cousin in the SM-hep, is (are)
also described by spin one half particles. We say particles because spin-$1/2$
supports not only an entirely new class of mass dimension one fermions but
also bosons! Since it was too late to include a detailed review article on the
subject, the reader is referred to two publications in press [19, 20]

To bring this editorial to a close, we define a generalised quantum field of
the form

$\psi(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int\frac{d^{3}p}{(2\pi)^{3/2}}\sum_{\sigma}\left[u(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)e^{-ip\cdot
x}a(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)+v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)e^{ip\cdot
x}b^{\ddagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)\right]$
(22)

where

$\displaystyle
u(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)~{}\mbox{and}~{}v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)=$
a orthonormal set of expansion coefficients in the finite dimensional
representtaion space of the Lorentz algebra

and $\ddagger$ is an adjoint (which for the purposes of this editorial shall
be taken as the usual hermitian adjoint $\dagger$). The
$a(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$ and
$a^{\dagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$, and
similarly $b(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$ and
$b^{\dagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$, satisfy
the usual bosonic or fermionic commutators or anticommutators. The dual of the
expansion coefficients, jointly called
$\xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$, is defined as

$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)=\left[\Xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)\right]^{\ddagger}\eta$
(23)

where $\Xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$ is an invertible
map with $\Xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})^{2}=\mathbb{I}$,
and $\eta$ is determined by the constraints

$\displaystyle\left[\zeta_{i},\eta\right]=0,\quad
i=x,y,z;\qquad\left\\{\kappa_{i},\eta\right\\}=0\quad i=x,y,z$ (24)

The $\displaystyle\boldsymbol{\zeta}$ and $\displaystyle\boldsymbol{\kappa}$
are the generators of rotations and boosts in the representation space in
which the $\xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$ reside.
For spin one half, setting
$\Xi(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=\mathbb{I},\forall\,\mbox{\boldmath$\displaystyle\boldsymbol{p}$}$,
one recovers the usual Dirac dual of spinors, and the standard adjoint.

Original mass dimension one fermions arose from taking the
$u(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$ and
$v(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$ as eigenspinors of
the charge conjugation (Elko). It was discovered that under the Dirac dual
these massive spinors had a norm that identically vanished. This observation
motivated the introduction of a new dual first introduced in [21], and later
taken to its final form in [17, 5].333That said, the subject is still under
active investigation. We refer the reader to [6] and references therein for a
full flavour of current research activity. In a series of papers it was argued
that the newly discovered fermions carried a natural darkness with respect to
the SM-hep fields. In addition they had a natural dimension four quartic self
interaction. First arguments to this effect can be traced back to [18, 3].

Editor’s monograph provides an elementary and pedagogic introduction to Elko
and mass dimension one fermions [5]. Equipped with that knowledge Cheng-Yang
Lee’s opening review to this volume provides a slightly different perspective
and extends the formalism to higher spins [22]. One loop calculations are
presented in two papers. First of these is the paper by Gustavo Pazzini de
Brito and colleagues [23], the second is the contribution by Alekha Nayak and
Tripurari Srivastava [24]. Elko have drawn significant attention of studies on
branes. The review by Xiang-Nan Zhou and Yu-Xiao Liu is devoted to this
emerging field of research [25]. Following this Saulo Pereira et al. review
the cosmology based on the new spinors [26]. It is intended to serve as a
primer on Elko cosmology. It also gives references to earlier pioneering
studies on the subject. Following this are two papers, one by Luca Fabbri who
looks at Elko in polar form [27]. The second, by Marco Dias et al. concludes
this volume by presenting a tutorial approach to phenomenological analysis in
the context of mass dimension one fermions [28].

This editorial only sets the stage for the invited authors to sow seeds of
their contributions. The stage is now theirs, and I hope we have succeeded in
making the stage well-lighted for the readers, our audience.

Acknowledgements. It is now time to thank Balasubramanian Ananthanarayan who
first envisioned a need for this volume and invited me to consider composing
an EPJ-ST volume on Elko and Mass dimension one fermions. I thank him, and
also Sandrine Karpe of the editorial and production offices of EPJ-ST for
attending to various formalities and needs. Thanks are also extended to all
the referees, and to all the authors for making this project take this form.
And thank you Sweta Sarmah for all your good contributions to this project.
Again, thank you all: cheers!

Funding. The research presented here is entirely supported by the personal
funds of the author.

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