Our method is based on the construction of weak solutions that are renormalized in the velocity variable

NAVIER-STOKES EQUATION AND ITS SEMI-CLASSICAL LIMIT

INGRID LACROIX-VIOLET AND ALEXIS F. VASSEUR

Abstract. This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy.

The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. Our method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence, and stability of these solutions do not need the Mellet-Vasseur inequality.

1. Introduction

Quantum models can be used to describe superfluids [12], quantum semiconductors [6], weakly interacting Bose gases [8] and quantum trajectories of Bohmian mechanics [16].

They have attracted considerable attention in the last decades due, for example, to the development of nanotechnology applications.

In this paper, we consider the barotropic compressible quantum Navier-Stokes equa- tions, which has been derived in [5], under some assumptions, using a Chapman-Enskog expansion in Wigner equation. In particular, we are interested in the existence of global weak solutions together with the associated semi-classical limit. The quantum Navier- Stokes equation that we are considering read as:

ρ_t+ div(ρu) = 0,

(ρu)_t+ div(ρu⊗u) +∇ρ^γ−2div(√

νρSν+√

κρSκ) =√

ρf +√

κdiv(√

ρM), (1.1) where

√νρSν =ρDu, div(√

κρSκ) =κρ∇

∆√

√ ρ ρ

, (1.2)

and with initial data

ρ(0, x) =ρ₀(x), (ρu)(0, x) = (ρ₀u₀)(x) in Ω, (1.3) whereρis the density,γ >1,u⊗uis the matrix with componentsuiuj,Du= ¹₂ ∇u+∇u^T is the symetric part of the velocity gradient, and Ω =T^dis thed−dimensional torus, here d= 2 or 3. The vector valued function f, and the matrix valued function M are source terms.

Date: April 26, 2019.

2010Mathematics Subject Classification. 35Q35, 76N10.

Key words and phrases. Global weak solutions, compressible Quantum Navier-Stokes Equations, vac- uum, degenerate viscosity.

Acknowledgment. A. F. Vasseur was partially supported by the NSF Grant DMS 1209420.

1

The relation (1.2) between the stress tensors and the solution (√ ρ,√

ρu) will be proved in the following form. For the quantic part, it will be showed that

2√

κρSκ = 2κ√ ρ

∇²√

ρ−4(∇ρ^1/4⊗ ∇ρ^1/4)

. (1.4)

For the viscous term, the matrix valued function Sν is the symmetric part of a matrix valued function Tν, where

√νρTν =ν∇(ρu)−2ν√

ρu· ∇√

ρ. (1.5)

Whenever,ρis regular and away from zero, the quantic part of (1.2) is equivalent to (1.4), and the matrix function Tν is formally √

νρ∇u. However, the a priori estimates do not allow to define 1/√

ρ and ∇u.

The energy of the system is given by E(√

ρ,√ ρu) =

Z

Ω

ρ|u|²

2 + ρ^γ

γ−1 + 2κ|∇√ ρ|²

dx, with dissipation of entropy (in the case without source term)

D_E(Sν) = 2 Z

Ω

|Sν|²dx, which is formally:

2ν Z

Ω

ρ|Du|²dx.

In [2, 4], Bresch and Desjardins introduced a new entropy of the system, now known as the BD entropy:

E_BD(ρ, u) = Z

Ω

ρ|u+ν∇lnρ|²

2 + ρ^γ

γ−1+ 2κ|∇√ ρ|²

dx, with associated dissipation (again without source term)

D_BD(ρ, u) =ν Z

Ω

4

γ|∇ρ^γ/2|²+κρ|∇²lnρ|²+ 2ρ|Au|²

dx,

whereAis the antisymmetric part of the matrix∇u. The function lnρis not controled by the a priori estimates. But we can use two other quantities,E₀ and D_E⁰, which are defined as follows:

E₀(√ ρ,√

ρu) = Z

Ω

ρ|u|²

2 + (2κ+ 4ν²)|∇√

ρ|²+ρ^γ

dx, D_E⁰(√

ρ,√ ρu) =

Z

Ω

ν|∇ρ^γ/2|²+νκ

|∇ρ^1/4|⁴+|∇²√ ρ|²

+|Tν|² dx.

From J¨ungel [9] the functions E₀ and D⁰_E, are equivalent, respectively, to E(√ ρ,√

ρu) + E_BD(√

ρ,√

ρu) and D_E(√ ρ,√

ρu,Sν) +D_BD(ρ, u), whenever each term can be defined, and Sν is the symmetric part ofTν with Tν =√

νρ∇u. Namely there exists a universal

constantC∗ such that for any such (ρ, u):

1 C∗

(E(√ ρ,√

ρu) +E_BD(ρ, u))≤ E₀(√ ρ,√

ρu)≤C∗(E(√ ρ,√

ρu) +E_BD(ρ, u)), 1

C∗

(D_E(Sν) +D_BD(ρ, u))≤ D⁰_E(√ ρ,√

ρu,Tν)≤C∗(De(Sν) +D_BD(ρ, u)).

The aim of this paper is to construct weak solutions for the system (1.1) using the a priori estimates provided by the energy and BD entropy inequalities. The main idea is to introduce a slightly stronger notion of weak solution that we call renormalized solutions.

They are defined in the following way. For any function ϕ∈ W^2,∞(R^d) compactly sup- ported, there exists two measuresR_ϕ, R_ϕ ∈ M(R⁺×Ω) such that the following is verified in the sense of distribution:

∂_t(ρϕ(u)) + div(ρuϕ(u)) +ϕ⁰(u)· ∇ρ^γ

−2div(√

ρϕ⁰(u)(√

νSν+√ κSκ))

=√

ρϕ⁰(u)·f+√

κdiv(√

ρϕ⁰(u)M) +Rϕ,

(1.6) withSκverifies (1.4), andSν is the symmetric part ofTν such that for everyi, j, kbetween 1 and d:

√νρϕ⁰_i(u)[Tν]jk =ν∂j(ρϕ⁰_i(u)uk)−2ν√

ρukϕ⁰_i(u)∂j

√ρ+Rϕ, (1.7) and

kR_ϕk_M(R+×Ω)+kR_ϕk_M(R+×Ω) ≤Ckϕ⁰⁰k_L^∞.

The precise definition of weak solutions and renormalized weak solutions is laid out in definitions 2.1 and 2.2. Note that, taking a sequence of function ϕn such that ϕn(y) converges to y_i, but kϕ⁰⁰k_L^∞ converges to 0, we can retrieve formally the equation (1.1).

We will show that it is actually true that any renormalized weak solution is, indeed, a weak solution.

For every

f ∈

L¹(R⁺;L²(Ω))d

, M∈

L²(R⁺;L²(Ω))d×d

, and every (√

ρ0,√

ρ0u0) such thatE₀(√ ρ0,√

ρ0u0) is bounded, we define M₀(√

ρ₀,√

ρ₀u₀, f,M) =E₀(√ ρ₀,√

ρ₀u) +kfk_L1(R⁺;L²(Ω))+kMk_L2(R⁺;L²(Ω)). (1.8) The main theorem of the paper is the following.

Theorem 1.1. (1) There exists a universal constantC∗>0such that the following is true. Let √

ρ₀,√

ρ₀u₀, f,M be such that M₀(√ ρ₀,√

ρ₀u₀, f,M) is bounded. Then, for anyκ≥0, there exists a renormalized solution (√

ρ,√

ρu) of (1.1) with E₀(√

ρ(t),√

ρ(t)u(t))≤M₀(√ ρ₀,√

ρ₀u₀, f,M), for t >0, Z ∞

0

D⁰_E(√ ρ(t),√

ρ(t)u(t))dt≤M0(√ ρ₀,√

ρ0u0, f,M).

Moreover, for everyϕ∈W^2,∞(R^d),

kR_ϕk_M(R+×Ω)+kR_ϕk_M(R+×Ω)≤C∗kϕ⁰⁰k_L^∞M0(√ ρ₀,√

ρ0u0, f,M).

Moreover, ρ ∈ C⁰(R⁺;L^p(Ω)) for 1≤p < sup(3, γ), and ρu∈ C⁰(R⁺;L^3/2(Ω)− weak)∩C⁰(R⁺;L^γ+12γ (Ω)−weak).

(2) Any renormalized solution of (1.1)is a weak solution of (1.1)with the same initial value.

(3) Consider any sequences κn ≥ 0, converging to κ ≥ 0, νn > 0 converging to ν >

0, (√ ρ_0,n,√

ρ_0,nu_0,n, f_n,Mn) such that M₀(√ ρ_0,n,√

ρ_0,nu_0,n, f_n,Mn) is uniformly bounded, and an associated weak renormalized solution (√

ρ_n,√

ρ_nu_n) to (1.1).

Then, there exists a subsequence (still denoted withn), and(√ ρ,√

ρu)renormalized solution to (1.1)with initial value (√

ρ₀,√

ρ₀u₀) and Planck constantκ, such that ρn converges to ρ in C⁰(R⁺;L^p(Ω)) for 1 ≤ p < sup(3, γ), and ρnun converges to ρu in C⁰(R⁺;L^3/2(Ω)−weak)∩C⁰(R⁺;L

2γ

γ+1(Ω)−weak). The function Tν ,n

converges weakly inL²(R⁺×Ω)toTν. Moreover, for every functionϕ∈W^2,∞(R^d) compactly supported, √

ρnϕ(un) converges strongly in L^p_loc(R⁺×Ω) to√

ρϕ(u) for 1≤p <6.

Note that all the results hold for any values ofκ, including the Navier-Stokes caseκ= 0.

For κ >0 we can have from the a priori estimates, better controls on the solutions, and convergence in stronger norms. The stability part of the result includes the follwoing case of the semi-classical limits 0< κ_n→0.

Corollary 1.1. The semi-classical limit. Consider(√ ρ0,√

ρ0u0, f,M)such that the quan- tity M₀(√

ρ₀,√

ρ₀u₀, f,M) is bounded, and consider an associated weak renormalized so- lution (√

ρ_κ,√

ρ_κu_κ) to the quantum Navier-Stokes equations (1.1) with κ > 0.Then, there exists a subsequence (still denoted with κ), and (√

ρ,√

ρu) renormalized solution to the Navier-Stokes equations ((1.1) with κ = 0) with same initial value such that ρ_κ converges to ρ in C⁰(R⁺;L^p(Ω)) for 1 ≤ p < sup(3, γ), and ρ_κu_κ converges to ρu in C⁰(R⁺;L^3/2(Ω)−weak)∩C⁰(R⁺;L

2γ

γ+1(Ω)−weak). The function Tν κ converges weakly in L²(R⁺×Ω) to Tν. Moreover, for every function ϕ∈W^2,∞(R^d) compactly supported,

√ρκϕ(uκ) converges strongly in L^p_loc(R⁺×Ω) to √

ρϕ(u) for 1≤p <6.

For this problem, the a priori estimates include control on the gradient of some density quantities. This provides compactness on both the density ρ and the momentum ρu.

The difficulty is due to the fact that we have only a control of ρu² in L^∞(R⁺, L¹(Ω)).

This cannot prevent concentration phenomena in the construction of solutions in the term ρu⊗u. When κ = 0, the same problem arises for the term √

ρu∇√

ρ, since the only a priori estimates available on both √

ρuand ∇√

ρare in L^∞(R⁺, L²(Ω)).

This problem can be avoided by the introduction of additional terms as drag forces or cold pressure, as proposed by Bresch and Desjardins [3]. In the case of drag forces, the system is

ρ_t+ div(ρu) = 0,

(ρu)t+ div(ρu⊗u) +∇ρ^γ−2div(√

νρSν+√ κρSκ)

=−r₀u−r₁ρ|u|²u+√

ρf +√

κdiv(√ ρM),

(1.9) still endowed with (1.5) and (1.4). The solutions of this kind of augmented systems can be explicitly constructed via a Faedo-Galerkin method. This was first performed by Zatorska

in [17] for the classical case (κ = 0) with chemical reactions and where the drag forces were replaced by cold pressure terms as∇(ρ^−N), forN big enough. In the quantum case solutions have been constructed in [7] in the case with cold pressure, and in [15] in the case of drag forces as (1.9).

When considering the system without additional terms (as drag forces or cold pressure), it has been shown in [13] that, in the classical caseκ= 0, solutions of (1.1) verify formally

that Z

Ω

ρ|u|²ln(1 +|u|²)dx (1.10)

is uniformly bounded in time, provided that the bound is valid at t = 0. It was also shown that a sequence of solutions, such that this quantity is uniformly bounded att= 0 (together with the energy and BD entropy), will converge, up to a subsequence, to a solution of the Navier-Stokes equation.

The standard way to construct weak solutions of systems verifying the weak stability is to construct solutions to an approximated problem (as the Faedo Galerkin method) for which the a priori estimates are still uniformly valid. Usually, those solutions are smooth so that every formal computation is actually true. However, in this context, the approximated problem needs to be compatible with the usual energy, the BD entropy, and the additional mathematical inequality (1.10). Only recently, such an approximated problem has been found [10] in dimension two, and in dimension three with unphysical stress tensors. Moreover, the regularity of the associated solutions is limited (notC^∞). For solutions constructed via a Faedo-Galerkin method, the energy and BD entropy estimates can be verified at the approximated level, since u is an admissible test function. This is not the case for the mathematical inequality (1.10). In [14], the construction of solutions to (1.1) with κ= 0 was obtained following a different strategy. It was shown that limits of solutions to (1.9) whenκconverges to 0 verify (1.10), even if it is not verified forκ >0.

The idea was to show that the quantity Z

Ω

ρϕ(|u|)dx

are uniformly bounded for smooth and bounded functionsϕ. This allowed to recover (1.10) forκ= 0 thanks to a sequence of approximations ϕn of the function y→y²ln(1 +y²).

Spririto and Antonelli showed in [1] that, formally, an estimate on (1.10) can still be obtained on solutions to (1.1) for a range of κ close (or bigger) thanν. But this estimate cannot be used for the semi-classical limit.

The notion of renormalized solutions is inspired from [14]. However, this notion is not used to recover an estimate on (1.10), which is known to be not verified for some range of κ. The idea is that we can obtain the stability of renormalized solutions, since the notion avoids the problem of concentration. Consider, for instance, the term

ρnunϕ(un).

Sinceρnandρnunare compact inL^p, for ap >1, we can show that, up to a subsequence, ρn converges almost everywhere to a function ρ, and un converges almost everywhere on {(t, x) | ρ(t, x) >0} to a function u. Hence ρ_nu_nϕ(u_n) converges almost everywhere to

ρuϕ(u). The function ϕ prevent concentration, and so ρ_nu_nϕ(u_n) converges strongly to ρuϕ(u).

The challenge is then to show that the renormalized solutions, are indeed, weak solutions in the general sense (see Definition 2.1). It is obtained by considering a sequence of bounded functionsϕn, uniformly dominated byy→ |y|, and converging almost everywhere to y → y_i, for a fixed direction i. This provides the momentum equation for ρu_i at the limit n → ∞. The key point, is that, while performing this limit, the functions ρ, u are fixed. Considering, for example, the term

ρuϕ_n(u).

The function ϕ_n(u) converges almost everywhere to u_i. And, thanks to the Lebesgue’s dominated convergence Theorem,ρuϕn(u) converges inL¹ toρuui. Note that the bound- edness of ρuu_i inL¹ is enough for this procedure. Choosing the sequence ofϕ_n such that kϕ⁰⁰_nk_L^∞ converges to 0, we show that the extra termsRϕn andRϕn converge to 0 whenn converges to ∞.

The main difference with [14] is that we do not need to reconstruct the energy inequality nor the control on (1.10) via the sequence of functions ϕn. Hence, we do not need an explicit form of the terms involving second derivatives ofϕin the definition of renormalized solutions. Those terms (for which we do not have stability) are dumped in the extra terms Rϕ and Rϕ.

2. Preliminary results and main ideas

We are first working on the System (1.9) with drag forces. The definitions will be valid for all the range of parameter, r0 ≥ 0, r1 ≥ 0, κ ≥ 0, ν > 0. The energy and the BD entropy on solutions to (1.9) provide controls on

E_r(√ ρ,√

ρu) = Z

Ω

ρ|u|²

2 + (2κ+ 4ν²)|∇√

ρ|²+ρ^γ+r₀(ρ−lnρ)

dx, D^r_E(√

ρ,√ ρu) = Z

Ω

ν|∇ρ^γ/2|²+νκ

|∇ρ^1/4|⁴+|∇²√ ρ|²

+|Tν|²+r0|u|²+r1ρ|u|⁴ dx.

From these quantities, we can obtain the following a priori estimates. For the sake of completeness we show how to obtain them in the appendix.

√ρ∈L^∞(R⁺;L²(Ω)), ∇√

ρ∈L^∞(R⁺;L²(Ω)), ∇ρ^γ/2∈L²(R⁺;L²(Ω))

√ρu∈L^∞(R⁺;L²(Ω)), Tν ∈L²(R⁺;L²(Ω)), √ κ∇²√

ρ∈L²(R⁺;L²(Ω)), κ^1/4∇ρ^1/4∈L⁴(R⁺;L⁴(Ω)), r^1/4₁ ρ^1/4u∈L⁴(R⁺;L⁴(Ω)),

r^1/2₀ u∈L²(R⁺;L²(Ω)), r₀lnρ∈L^∞(R⁺;L¹(Ω)).

(2.1)

Note that those a priori estimates are not sufficient to define∇uas a function. The state- ment that√

ρ∇uis bounded inL² means that there exists a functionTν ∈L²(R⁺;L²(Ω)) such that:

√ν√

ρTν = div(ρu)−√

ρu· ∇√ ρ,

which is, formally,ρ∇u. The definition of weak solutions and renormalized weak solutions for the system (1.9) are as follows.

Definition 2.1. We say that (√ ρ,√

ρu) is a weak solution to (1.9), if it verifies the a priori estimates (2.1), and for any functionψ∈C_c^∞(R⁺×Ω):

Z ∞ 0

Z

Ω

(ρψ_t+ρu· ∇ψ)dx dt= 0, Z ∞

0

Z

Ω

(ρuψ_t+ (ρu⊗u−2√

νρSν−2√

κρSκ−√

κρM)· ∇ψ+F ψ)dx dt= 0, with Sν the symmetric part ofTν verifying (1.5),Sκ verifying (1.4), and

F =−2ρ^γ/2∇ρ^γ/2−r0u−r1ρ|u|²u+√

ρf, (2.2)

and for any ψ∈C_c^∞(R):

limt→0

Z

Ω

ρ(t, x)ψ(x)dx= Z

Ω

ρ₀(x)ψ(x)dx, limt→0

Z

Ω

ρ(t, x)u(t, x)ψ(x)dx= Z

Ω

ρ0(x)u0(x)ψ(x)dx.

Definition 2.2. We say that (√ ρ,√

ρu) is a renormalized weak solution to (1.9), if it veri- fies the a priori estimates (2.1), and for any functionϕ∈W^2,∞(R^d) compactly supported, there exists two measuresRϕ, Rϕ∈ M(R⁺×Ω), with

kR_ϕk_M(R+×Ω)+kR_ϕk_M(R+×Ω)≤Ckϕ⁰⁰k_L∞(R), where the constant C depends only on the solution (√

ρ,√

ρu), and for any function ψ∈C_c^∞(R⁺×Ω),

Z ∞ 0

Z

Ω

(ρψt+ρu· ∇ψ)dx dt= 0, Z ∞

0

Z

Ω

ρϕ(u)ψ_t+ ρϕ(u)u−(2√

νρSν + 2√

κρSκ+√

κρM)ϕ⁰(u)

· ∇ψ +F·ϕ⁰(u)ψ

dx dt=hR_ϕ, ψi,

withSν the symmetric part ofTν verifying (1.7), Sκ verifying (1.4), f given by (2.2), and for any ψ∈C_c^∞(R):

limt→0

Z

Ω

ρ(t, x)ψ(x)dx= Z

Ω

ρ0(x)ψ(x)dx, limt→0

Z

Ω

ρ(t, x)u(t, x)ψ(x)dx= Z

Ω

ρ₀(x)u₀(x)ψ(x)dx.

Let us define M_r(√

ρ₀,√

ρ₀u₀, f,M) =E_r(√ ρ₀,√

ρ₀u₀) +kfk_L1(R⁺;L²(Ω))+kMk_L2(R⁺;L²(Ω)). The main theorem proved in this paper is the following.

Theorem 2.1. (1) There exists a universal constantC∗>0such that the following is true. Let √

ρ₀,√

ρ₀u₀, f,M be such that M_r(√ ρ₀,√

ρ₀u₀, f,M) is bounded. Then,

for any κ ≥0, r₀ ≥0, r₁ ≥0, there exists a renormalized solution (√ ρ,√

ρu) of (1.9) with

E_r(√ ρ(t),√

ρ(t)u(t))≤Mr(√ ρ₀,√

ρ0u0, f,M), for t >0, Z ∞

0

D^r_E(√ ρ(t),√

ρ(t)u(t))dt≤M_r(√ ρ₀,√

ρ₀u₀, f,M).

Moreover, for every function ϕ∈W^2,∞(R^d) compactly supported, kR_ϕk_M(

R⁺×Ω)+kR_ϕk_M(

R⁺×Ω)≤C∗kϕ⁰⁰k_L^∞M₀(√ ρ₀,√

ρ₀u₀, f,M).

Moreover, ρ ∈ C⁰(R⁺;L^p(Ω)) for 1≤p < sup(3, γ), and ρu∈ C⁰(R⁺;L^3/2(Ω)− weak)∩C⁰(R⁺;L

2γ

γ+1(Ω)−weak).

(2) Any renormalized solution of (1.9)is a weak solution of (1.9)with the same initial value.

(3) Ifr0 >0,r1 >0, andκ >0, then any weak solution to (1.9)is also a renormalized solution to (1.9) with the same initial value.

(4) Consider any sequences κn ≥ 0, r0,n ≥ 0, r1,n ≥ 0, νn > 0, converging respec- tively to κ ≥ 0, r₀ ≥ 0, r₁ ≥0, and ν >0, (√

ρ_0,n),√

ρ_0,nu_0,n, f_n,Mn) such that M_rn(√

ρ_0,n,√

ρ_0,n, f_n,Mn) is uniformly bounded, and an associated weak renor- malized solution (√

ρn,√

ρnun) to (1.9). Then, there exists a subsequence (still denoted with n), and (√

ρ,√

ρu) renormalized solution to (1.9) with initial value (√

ρ₀,√

ρ₀u₀) Planck constant κ, and drag forces coefficients r₀, r₁ such that ρ_n converges to ρ in C⁰(R⁺;L^p_loc(Ω)) for 1 ≤ p < sup(3, γ), and ρ_nu_n converges to ρuinC⁰(R⁺;L^3/2(Ω)−weak)∩C⁰(R⁺;L^γ+12γ (Ω)−weak). The functionTν ,n con- verges weakly in L²(R⁺×Ω) to Tν. Moreover, for every function ϕ∈W^2,∞(R^d) compactly supported, √

ρ_nϕ(u_n) converges strongly in L^p_loc(R⁺×Ω) to√

ρϕ(u) for 1≤p <10γ/3.

Remark 2.1. We can actually show in (1) of the previous theorem that there is one solution verifying

R_ϕ,i,k,j =

d

X

l=1

√ρu_jϕ⁰⁰_i,l(u)Tν k,l.

But this is not needed to show that a renormalized solution is a weak solution. We cannot do the same for the term Rϕ.

Note that Theorem 2.1 together with [15] implies Theorem 1.1. Indeed, [15] provides the construction of weak solutions to (1.9) with positive r0, r1, κ. Part (3) in Theorem 2.1 insures that this solution is actually a renormalized solution. Considering sequences r0,n ≥0 and r1,n≥0 both converging to 0, part (4) of Theorem 2.1 provides at the limit a renormalized solution to (1.1).

3. From weak solutions to renormalized solutions in the presence of drag forces

This section is dedicated to the proof of part (3) of Theorem 2.1. In the whole section we will assume that κ >0,r₀ >0, andr₁ >0, and we will consider a fixed weak solution

(√ ρ,√

ρu) as in Definition (2.1). Let us define g_ε for any functiong as g_ε(t, x) =ηε∗g(t, x), t > ε,

where

ηε(t, x) = 1

ε^d+1η1(t/ε, x/ε),

with η₁ a smooth nonnegative even function compactly supported in the space time ball of radius 1, and with integral equal to 1.

Formally, we can show that a weak solution is also a renormalized solution by multiply- ing the equation byϕ⁰(u). However, solutions of (1.9) have a limited amount of regularity.

This has to be performed carefully. Let us explain the difficulties. First let us focus on the term

div(√ νρSν).

One way to obtain the renormalized equation from the weak one, is to consider the family of test functionsψϕ⁰(u_ε). We need to pass to the limit in the expression

Z ∞ 0

Z

Ω

ψ√

νρSν∇ϕ⁰(u_ε)dx dt.

But we cannot pass into the limit for the term √

νρ∇ϕ⁰(u_ε) =√

νρϕ⁰⁰(u_ε)∇u_ε. Note that this term is different from Tν ε. The problem is that∇u is not bounded in any functional space.

An other difficulty is to obtain, in the sense of distribution, the equality ϕ⁰(u)∂_t(ρu) =∂_t(ρϕ(u)) + (ϕ⁰(u)u−ϕ(u))∂_tρ.

Indeed, we have absolutely no estimate available on ∂_tu. Following Di Perna and Li- ons, [11], this can be obtained using commutators estimates which requires more a priori estimates than can be formally intuited.

To solve these problems, we need to introduce a cut-off function inρ,φ_m(ρ) whereφ_m is defined for everym >0 as

φ_m(y) =



























0, for 0≤y≤ 1

2m, 2my−1, for 1

2m ≤y≤ 1 m,

1, for 1

m ≤y≤m, 2−y/m, for m≤y ≤2m,

0, for y≥2m.

(3.1)

We will now work on

v_m =φ_m(ρ)u (3.2)

instead ofu. Note that

∇v_m= φm(ρ)

√νρ Tν+ 4√

ρφ⁰_m(ρ)ρ^1/4u∇ρ^1/4.

For m fixed, ^φ√m_νρ^(ρ) and √

ρφ⁰_m(ρ) are bounded, and so ∇v_m is bounded in L², thanks to the a priori estimates obtained from κ >0 andr1>0 in (2.1).

Obtaining the equation onv_m is pretty standard, thanks to the extra regularity on the density provided by the quantum term, and the BD entropy. However, to highlight where are the difficulties, we will provide a complete proof.

3.1. Preliminary lemmas. In this subsection, we introduce two standard lemmas to clarify the issues. The second one use the commutator estimates of Di Perna and Lions [11].

Lemma 3.1. Let g ∈ L^p(R⁺×Ω) and h ∈ L^q(R⁺×Ω) with 1/p+ 1/q = 1 and H ∈ W^1,∞(R). We denote by∂ a partial derivative with respect to one of the dimension (time or space). Then we have:

g_ε converges strongly to g in L^p(R⁺×Ω), Z ∞

0

Z

Ω

g_εh dx dt= Z ∞

0

Z

Ω

gh_εdx dt,

ε→0lim Z ∞

0

Z

Ω

g_εh dx dt= Z ∞

0

Z

Ω

gh dx dt,

∂g_ε= [∂g]_ε,

ε→0limkH(g_ε)−H(g)k_Ls

loc(R⁺×Ω)= 0, for any 1≤s <∞, For 1≤r <∞,if f ∈L^r(R⁺×Ω),then

H(g_ε)f converges strongly to H(g)f inL^r(R⁺×Ω), For 1≤r <∞,as long as ∂g∈L^r(R⁺×Ω), we have :

∂H(g)∈L^r(R⁺×Ω) and ∂H(g_ε) converges weakly to ∂H(g) in L^r(R⁺×Ω).

Proof. The sequence g_ε converges strongly to g inL^p(R⁺×Ω), so, up to a subsequence, it converges almost everywhere, and H(g_ε) converges almost everywhere to H(g). Note that:

kH(g_ε)f −H(g)fk^r_Lr(R⁺×Ω)= Z

R

Z

Ω

|H(g_ε)−H(g)|^rf^rdx dt.

But H is bounded, so |H(g_ε)−H(g)|^rf^r is dominated by (2 sup|H|)^r|f|^r which does not depend on ε and lies in L¹(R⁺×Ω). We obtain that H(g_ε)f converges to H(g)f in L^r(R⁺×Ω), using the Lebesgue theorem. By the uniqueness of the limit, the whole sequence is converging. The rest of the lemma is very standard and then we omit the

proof.

We use also in the sequel the following lemma due to Lions (see [11]).

Lemma 3.2. Let ∇_t,x be the gradient in time and space ∇_t,x = (∂_t,∇_x). Let g,∇_t,xg ∈ L^p(R⁺×Ω), h∈L^q(R⁺×Ω) with1≤p, q≤ ∞, and ¹_p +¹_q ≤1. Then, we have

k[∇_t,x(gh)]_ε− ∇_t,x(gh_ε)k_Lr(R⁺×Ω) ≤Ck∇_t,xgk_Lp(R⁺×Ω)khk_Lq(R⁺×Ω)

for some constant C ≥ 0 independent of ε, g and h, and with r given by ¹_r = ¹_p +¹_q. In addition,

[∇_t,x(gh)]_ε− ∇_t,x(gh_ε)→0 in L^r(R⁺×Ω)

as ε→0 if r <∞.

We can easily localize this lemma in time, as in this corollary.

Corollary 3.1. Let T₁ < T₂. Let ∇_t,x be the gradient in time and space ∇_t,x = (∂_t,∇_x).

Let g,∇_t,xg ∈ L^p((0, T₂)×Ω), h ∈ L^q((0, T₂)×Ω) with 1 ≤ p, q ≤ ∞, and ¹_p + ¹_q ≤ 1.

Then, for ε≤T₂−T₁, we have

k[∇_t,x(gh)]_ε−∇_t,x(gh_ε)k_Lr((0,T1)×Ω) ≤C(k∇_t,xgk_Lp((0,T2)×Ω)+kgk_Lp((0,T2)×Ω))khk_Lq((0,T2)×Ω)

for some constant C ≥ 0 independent of ε, g and h, and with r given by ¹_r = ¹_p +¹_q. In addition,

[∇_t,x(gh)]_ε− ∇_t,x(ghε)→0 in L^r((0, T1)×Ω) as ε→0 if r <∞.

Proof. Consider a smooth non negative cut-off functionψ:R⁺ →R⁺ such that ψ(t) = 1 for t < T1 and ψ(t) = 0 for t > T2. The result is obtained by applying Lemma 3.2 toψg

and ψh.

3.2. Equation onv_m. This subsection is dedicated to show that for anyψ∈C_c^∞(R⁺×Ω), and any m >0, we have

Z ∞ 0

Z

Ω

∂tψφm(ρ)−ψ

u· ∇φ_m(ρ) +φ⁰_m(ρ)

√ρ

√νTr(Tν)

dx dt= 0. (3.3)

Z ∞ 0

Z

Ω

∂tψρvm+∇ψ· ρu⊗vm−φm(ρ)√ ρ 2√

νSν + 2√

κSκ+√

κM

+ψ

−

√ρ

√νtr(Tν)φ⁰_m(ρ)ρu−√

κρM∇φ_m(ρ) +φm(ρ)F

dx dt= 0,

(3.4)

whereF is defined in (2.2), vm in (3.2),Sκ is in (1.4), andSν is the symmetric part ofTν

defined in(1.5).

We begin with a list of a priori estimates. Note that they depends on all the parameters r0>0,r1 >0,κ >0, andm.

Lemma 3.3. There exists a constantC >0depending only on the fixed solution(√ ρ,√

ρu), and Cm depending also on m such that

kρk_L5(R⁺×Ω)+kρuk_L5/2(R⁺×Ω)+kρ|u|²+√

ρ(|Sν|+|Sκ|+|M|+|f|)k_L5/3(R⁺×Ω)

+kρ^γ/2∇ρ^γ/2k_L5/4(R⁺×Ω)+kr₀uk_L2(R⁺×Ω)+kr₁ρ|u|²uk_L5/4(R⁺×Ω) ≤C k∇φ_m(ρ)k_L4(R⁺×Ω)+k∂_tφm(ρ)k_L2(R⁺×Ω) ≤Cm.

Proof. From (2.1), √

ρ ∈ L^∞(R⁺, L²(Ω)), ∇√

ρ ∈L^∞(R⁺, L²(Ω)) and ∇²√

ρ ∈L²(R⁺× Ω). Hence, using Gagliardo-Nirenberg inequality, √

ρ ∈ L^∞(R⁺;L⁶(Ω)), and ∇√ ρ ∈ L²(R⁺;L⁶(Ω)). Since∇ρ= 2√

ρ∇√

ρ,∇ρ∈L²(R⁺, L³(Ω)). And ρ² ∈L^∞(R⁺;L^3/2(Ω)), so∇(ρ²) = 2ρ∇ρ∈L²(R⁺;L^3/2(Ω)). This gives

ρ² ∈L²(R⁺;L³(Ω))∪L^∞(R⁺;L^3/2(Ω)).

By interpolation, ρ² lies in all the spaceL^p(L^q) with α

2 = 1

p, α

3 +2(1−α)

3 = 1

q,

for 0≤α≤1. Forα= 4/5, we obtainρ² ∈L^5/2(R⁺×Ω). We have ρu=ρ^3/4ρ^1/4u.

From the r₁ term of D_E^r, ρ^1/4u ∈ L⁴(R⁺ ×Ω), and we have ρ^3/4 ∈ L^20/3(R⁺ ×Ω).

Hence ρu∈L^5/2(R⁺×Ω). The term|Sν|+|Sκ|+|M|+|f|+√

ρ|u|² ∈L²(R⁺×Ω) and

√ρ∈L¹⁰(R⁺×Ω), soρ|u|²+√

ρ(|Sν|+|Sκ|+|M|+|f|)∈L^5/3(R⁺×Ω). From the a priori estimates, we have ∇ρ^γ/2 ∈ L²(R⁺×Ω), and ρ^γ/2 ∈L^∞(R⁺, L²(Ω)). Using Galgliardo- Nirenberg inequality we have ρ^γ/2 ∈L^10/3(R⁺×Ω). So ρ^γ/2∇ρ^γ/2∈L^5/4(R⁺×Ω).

The estimate on r₀u comes directly from the a priori estimates. Since √

ρ|u|² is in L²(R⁺, L²(Ω)) and ρ^1/4u is inL⁴(R⁺×Ω), we have ρ^1/4u√

ρ|u|² ∈ L^4/3(R⁺×Ω). Then using ρ^1/4 ∈L²⁰(R⁺×Ω) we obtain the result forr1ρ|u|²u.

From Lemma 3.1, ∇φ_m(ρ) = 4ρ^3/4φ⁰_m(ρ)∇ρ^1/4. But y → 4y^3/4φ⁰_m(y) is bounded, and by (2.1)∇ρ^1/4 ∈L⁴(R⁺×Ω), so∇φ_m(ρ)∈L⁴(R⁺×Ω). From Lemma 3.1, and (1.5),

∂_tφ_m(ρ) =φ⁰_m(ρ)∂_tρ=−φ⁰_m(ρ)div(ρu) =−φ⁰_m(ρ)(p

ρ/νTr(Tν) + 2√

ρu· ∇√ ρ)

=−√

ρφ⁰_m(ρ)(p

1/νTr(Tν) + 4ρ^1/4u· ∇ρ^1/4).

Hence ∂_tφ_m(ρ)∈L²(R⁺×Ω).

We use the first equation in Definition 2.1 with test function [φ⁰_m(ρ_ε)ψ]_ε.From Lemma 3.1, and (1.5) we get the following.

0 = Z ∞

0

Z

Ω

n

∂tψφ⁰_m(ρ_ε)_ερ+ρu· ∇ψφ⁰_m(ρ_ε)_ε o

dx dt

=− Z ∞

0

Z

Ω

ψφ⁰_m(ρ_ε)∂tρ_ε+ div(ρu_ε)ψφ⁰_m(ρ_ε) dx dt

= Z ∞

0

Z

Ω

(

∂_tψφ_m(ρ_ε)−ψφ⁰_m(ρ_ε)

"√

√ρ

νTr(Tν)

ε

+ 2√

ρu· ∇√ ρ_ε

#) dx dt.

We have ρ ∈ L⁵(R⁺×Ω) and ∂_t[φ⁰_m(ρ)ψ] ∈L^5/4(R⁺×Ω), since ∂_tρ ∈L²(R⁺×Ω) and ψ is C¹ compactly supported. From Lemma 3.3, ρu ∈ L^5/2(R⁺×Ω), and∇[φ⁰_m(ρ)ψ]∈ L^5/3(R⁺×Ω), since ψ is regular and compactly supported, and ∇φ⁰_m(ρ) ∈L⁴(R⁺×Ω).

So we can pass to the limit ε→0 using Lemma 3.1. We obtain:

0 = Z ∞

0

Z

Ω

∂_tψφ_m(ρ)−ψφ⁰_m(ρ) √

√ρ

νTr(Tν) + 2√

ρu· ∇√ ρ

dx dt.

Since ψ∇φ_m(ρ) lies inL⁴(R⁺×Ω) and is compactly supported, andu is in L²(R⁺×Ω), using the Lemma 3.1 we find

0 = Z ∞

0

Z

Ω

∂tψφm(ρ)−ψ

φ⁰_m(ρ)

√ρ

√νTr(Tν) +u· ∇φ_m(ρ)

dx dt,

which is (3.3).

Forεsmall enough, we consider the functionψφ_m(ρ)_εas test function in the Definition 2.1. In the same way than above, from Lemma 3.3 and Lemma 3.1, passing into the limit inε, we get

Z ∞ 0

Z

Ω

∂_tψρv_m+∇ψ· ρu⊗v_m−φ_m(ρ)√ ρ 2√

νSν + 2√

κSκ+√ κM +ψ(∂tφm(ρ) +u· ∇φ_m(ρ))ρu+ψ(−√

κρM∇φ_m(ρ) +φm(ρ)F)} dx dt= 0, Using (3.3) this gives (3.4).

3.3. Equation of renormalized solutions. We use the function ψϕ⁰(v_mε)_ε as a test function in (3.4), and investigate the limit whenεgoes to 0.

Step 1: Note that

∇v_m = ∇

φm(ρ) ρ ρu

= ∇

φ_m(ρ) ρ

ρu+φ_m(ρ) ρ ∇(ρu)

= ∇

φm(ρ) ρ

ρu+ φm(ρ)

√ν√

ρTν+ 2φm(ρ) ρ

√ρu· ∇√ ρ

= 4√

ρφ⁰_m(ρ)ρ^1/4u· ∇ρ^1/4. Note that √

ρφ⁰_m(ρ) is bounded. So, thanks to the third line of (2.1), we have ∇v_m ∈ L²(R⁺×Ω), and so∇ϕ⁰(vm)∈L²(R⁺×Ω). Therefore, using Lemma 3.1

∇(ψϕ⁰(vmε)) converges weakly to ∇(ψϕ⁰(vm)) in L²(R⁺×Ω). (3.5) Step 2: Using Lemma 3.1, we find:

Z ∞ 0

Z

Ω

∂_th

ψϕ⁰(v_mε)_εi

ρv_m+∇h

ψϕ⁰(v_mε)_εi

: (ρu⊗v_m) dx dt

=− Z ∞

0

Z

Ω

ψϕ⁰(v_mε)

∂_t[ρv_m]_ε+ div(ρu⊗v_m)_ε

dx dt. (3.6)

The goal is now to show that it has the same limit when εconverges to 0 as

− Z ∞

0

Z

Ω

ψϕ⁰(v_mε) (∂_t(ρv_mε) + div (ρu⊗v_mε))dx dt.

To this end, let us first consider the first part of the righthand side of (3.6).

Thanks to (2.1) and Lemma 3.3, we can use Corollary 3.1, with g = ρ and h = v_m. Indeed,ψis compactly supported,vm ∈L⁴(R⁺×Ω), and∇_t,xρ∈L²(R⁺;L^3/2(Ω)) (which implies ∇_t,xρ ∈ L²((0, T);L^3/2(Ω)) and then ∇_t,xρ ∈ L^4/3((0, T)×Ω) for any T > 0) thanks to the estimates in the proof of Lemma 3.3. So,

− Z ∞

0

Z

Ω

ψϕ⁰(vmε)∂t[ρvm]_εdx dt has the same limit whenε goes to 0 as

− Z ∞

0

Z

Ω

ψϕ⁰(vmε)∂t(ρvmε)dx dt.

We show now that the quantity

− Z ∞

0

Z

Ω

ψϕ⁰(v_mε)div(ρu⊗v_m)_εdx dt= Z ∞

0

Z

Ω

∇(ψϕ⁰(v_mε)) : (ρu⊗v_m)_εdx dt (3.7) has the same limit as εconverges to 0 as

− Z ∞

0

Z

Ω

ψϕ⁰(v_mε)div(ρu⊗v_mε)dx dt= Z ∞

0

Z

Ω

∇(ψϕ⁰(v_mε)) : (ρu⊗v_mε)dx dt (3.8) Indeed, ρu⊗vm = ρ^1/4u⊗ρ^3/4Φm(ρ)u is bounded in L²(R⁺ ×Ω) since ρ^1/2Φm(ρ) is bounded. So, thanks to Lemma 3.1, (ρu⊗vm)_ε converges strongly in L²(R⁺ ×Ω) to ρu⊗v_m. Using (3.5), (3.7) converges, whenεgoes to 0, to

Z ∞ 0

Z

Ω

∇(ψϕ⁰(vm)) : (ρu⊗vm)dx dt. (3.9) The function ϕ is compactly supported, so there exists a compactly supported smooth functionH :R³→R³ such thatH(y) =y on the support ofϕ, and so

∇(ψϕ⁰(v_mε)) : (ρu⊗v_mε) =∇(ψϕ⁰(v_mε)) : (ρu⊗H(v_mε)),

∇(ψϕ⁰(vm)) : (ρu⊗vm) =∇(ψϕ⁰(vm)) : (ρu⊗H(vm)).

Moreover ρu=ρ^1/4uρ^3/4 is also inL²(R⁺×Ω) sinceρ^1/4uis inL⁴(R⁺×Ω), andρ^3/4 is in L^∞(R⁺;L⁴(Ω)) (indeed √

ρ ∈L^∞(R⁺;L⁶(Ω)) by the proof of Lemma 3.3). Hence, from Lemma 3.1, ρu⊗H(vmε) converges strongly inL²(R⁺×Ω) toρu⊗H(vm). Finally, using (3.5), (3.8) converges also to (3.9).

Then we have now shown that (3.6) has the same limit whenεconverges to 0 as

− Z ∞

0

Z

Ω

ψϕ⁰(v_mε) (∂_t(ρv_mε) + div (ρu⊗v_mε))dx dt.

Thanks to the first equation in Definition 2.1 this is equal to

− Z ∞

0

Z

Ω

ψϕ⁰(v_mε) (ρ∂_tv_mε+ρu· ∇v_mε)dx dt

=− Z ∞

0

Z

Ω

ψ(ρ∂_tϕ(v_mε) +ρu· ∇ϕ(v_mε)) dx dt

= Z ∞

0

Z

Ω

ϕ(v_mε) (ρ∂_tψ+ρu· ∇ψ))dx dt, which converges, when εgoes to 0, to

Z ∞ 0

Z

Ω

ϕ(vm) (ρ∂tψ+ρu· ∇ψ))dx dt. (3.10) Step 3: Thanks to Lemma 3.1 we can pass into the limit in the other terms and find

Z ∞ 0

Z

Ω

−∇ ψϕ⁰(vm)

)φm(ρ)√ ρ 2√

νSν+ 2√

κSκ+√ κM

+ψϕ⁰(vm)

−

√ρ

√νTr(Tν)φ⁰_m(ρ)ρu−√

κρM∇φ_m(ρ) +φm(ρ)F

dx dt. (3.11) Putting (3.10) and (3.11) together gives

Z ∞ 0

Z

Ω

{ϕ(v_m) (ρ∂_tψ+ρu· ∇ψ))

−∇ψϕ⁰(v_m)φ_m(ρ)√ ρ 2√

νSν + 2√

κSκ+√ κM

−ψϕ⁰⁰(v_m)∇v_mφ_m(ρ)√ ρ 2√

νSν+ 2√

κSκ+√ κM +ψϕ⁰(vm)

−

√ρ

√νTr(Tν)φ⁰_m(ρ)ρu−√

κρM∇φ_m(ρ) +φm(ρ)F

dx dt.

We now pass into the limit m goes to infinity. From the a priori estimates (2.1), r0lnρ lies in L^∞(R⁺;L¹(Ω)), soρ >0 almost everywhere, and

φ_m(ρ) converges to 1, for almost every (t, x)∈R⁺×Ω, vm converges to u, for almost every (t, x)∈R⁺×Ω,

|ρφ⁰_m(ρ)| ≤2, and converges to 0 for almost every (t, x)∈R⁺×Ω.

Now, using thatφ_m is compactly supported inR⁺, we get

√ρ∇v_m = φ_m(ρ)

√ρ ρ∇u+ 4ρ^1/4uρφ⁰_m(ρ)∇ρ^1/4

= φm(ρ)

√ρ [∇(ρu)−2√ ρu·∇√

ρ] + 4ρ^1/4uρφ⁰_m(ρ)∇ρ^1/4

= φ_m(ρ)√Tν

ν + 4ρ^1/4uρφ⁰_m(ρ)∇ρ^1/4,

which, thanks to the dominated convergence theorem, converges toTν/√

ν inL²(R⁺×Ω).

Hence, passing into the limitm→ ∞, we find Z ∞

0

Z

Ω

ϕ(u) (ρ∂_tψ+ρu· ∇ψ))− ∇ψϕ⁰(u)√ ρ 2√

νSν + 2√

κSκ+√ κM

−ψϕ⁰⁰(u)√Tν

ν 2√

νSν+ 2√

κSκ+√ κM

+ψϕ⁰(u)F

dx dt.

This gives the second equation in the definition of renormalized solutions with Rϕ=ϕ⁰⁰(u)√Tν

ν 2√

νSν+ 2√

κSκ+√ κM

.

We want now to show (1.7). As above, multiplying (1.5) by [φ_m(ρ)ϕ⁰_i(u)]_ε, and passing into the limit εgoes to 0, we find

φm(ρ)ϕ⁰_i(u)√

νρTν =ν∇(φ_m(ρ)ϕ⁰_i(u)ρu)−4ρφ⁰_m(ρ)∇ρ^1/4ρ^1/4u√ ρϕ⁰_i(u)

−ϕ⁰⁰_i(u)Tν

√ν

√ρuφ_m(ρ)−2νφ_m(ρ)ϕ⁰_i(u)√ ρu·∇√

ρ.

Passing into the limit m goes infinity, we recover (1.7), with Rϕ =−ϕ⁰⁰_i(u)√Tν

ν

√ρu.

Hence, (√ ρ,√

ρu) is a renormalized solution.